0 and a subsequence (*n**m*)*m* ∈ *N* of natural numbers such that *ε* ≤ *D*(*A**n**m*, *A*) for each *m* ∈ N. Since *A* ⊆ *A**n* for each *n* ∈ N, the definition of the Hausdorff distance (Definition **[def:Hausdorff]** above) and the fact that the function *x* ↦ *d*(*x*, *A*) is continuous on the closed and hence compact set *A**n* for each *n* ∈ N (and thus attains a maximum value there), imply that for each *m* ∈ N there exists *x**m* ∈ *A**n**m* such that *ε* ≤ *d*(*x**m*, *A*). Let (*x**m**l*)*l* ∈ N be a convergent subsequence of the sequence (*x**m*)*m* = 1∞, and let *x* be its limit. Since the sequence (*A**n*)*n* ∈ N is decreasing, it follows that *x**m**l*ʹ ∈ *A**n**m**l* whenever *l*, *l*ʹ ∈ N and *l* ≤ *l*ʹ. Therefore *x* ∈ *A**n**m**l* for all *l* ∈ N because *A**n**m**l* is closed. Thus *x* ∈ ⋂*l* = 1∞*A**n**m**l*. But ⋂*l* = 1∞*A**n**m**l* = *A* because the intersection is decreasing. Hence 0 = *d*(*x*, *A*) = lim*l* → ∞*d*(*x**m**l*, *A*) ≥ *ε*, a contradiction.\nNow we turn to proving that *B* = lim*n* → ∞*B**n* (the statement that *B* ≠ ∅ is obvious). Suppose by way of contradiction that this is not true. Then there exists some *ε* > 0 and a subsequence (*n**m*)*m* = 1∞ of natural numbers such that *ε* ≤ *D*(*B**n**m*, *B*) for each *m* ∈ N. Since *B**n* ⊆ *B* for each *n* ∈ N, the definition of the Hausdorff distance and the fact that for each *m* ∈ N the function *x* ↦ *d*(*x*, *B**n**m*) is continuous on the closed and hence compact set *B* for each *n* ∈ N (and thus attains a maximum value there) imply that there exists *x**m* ∈ *B* such that *ε* ≤ *d*(*x**m*, *B**n**m*). Let (*x**m**l*)*l* ∈ N be a convergent subsequence of the sequence (*x**m*)*m* = 1∞, and let *x* be its limit. Then *d*(*x*, *x**m**l*) < *ε*/4 for all *l* ∈ N large enough. Since *B* is closed it follows that *x* ∈ *B*, so by the definition of *B* there exists *n*0 ∈ N such that *d*(*x*, *B**n*0) < *ε*/2. But *B**n*0 ⊆ *B**n* for each *n* ∈ N larger than *n*0, and so *d*(*x*, *B**n*) ≤ *d*(*x*, *B**n*0) < *ε*/2 for all such *n*. In particular, *d*(*x*, *B**n**m**l*) < *ε*/2 for *l* ∈ N sufficiently large. Thus\n \n*ε* ≤ *d*(*x**m**l*, *B**n**m**l*) ≤ *d*(*x**m**l*, *x*) + *d*(*x*, *B**n**m**l*) < 3*ε*/4, \na contradiction.\nThe next claims discuss several subtler properties of ${\\textnormal{dom}}$ and ${\\textnormal{Dom}}$. Lemma **[lem:AntiIntersecdom]** and Proposition **[prop:mDomM]** generalize and respectively, and are partly inspired by them.\n[lem:Dom3] Let (*X*, *d*) be a metric space and let *P* = (*P**k*)*k* ∈ *K* be a tuple of nonempty subsets of *X* having the property that for all *k* ∈ *K*,\n \n*r**k* : = inf{*d*(*P**k*, *P**j*) : *j* ≠ *k*} > 0. \nFor *Q* ⊆ *X* nonempty and *r* > 0, let *B*(*Q*, *r*) : = {*x* ∈ *X* : *d*(*x*, *Q*) < *r*}. We denote by $({\\textnormal{Dom}}P)\\_k$ the *k*-th component of ${\\textnormal{Dom}}P$. Then:\n1. [item:B(P,r)Dom2] $B(P\\_k,r\\_k/4)\\subseteq ({\\textnormal{Dom}}^2 P)\\_k\\subseteq ({\\textnormal{Dom}}^{\\gamma} P)\\_k$ for each integer *γ* ≥ 1 and any *k* ∈ *K*.\n2. [item:r] If (*X*, *d*) is a geodesic metric space, then *d*(*x*, *B*(*Q*, *r*)) + *r* = *d*(*x*, *Q*) for each *Q* ⊆ *X* nonempty, each *r* > 0 and each $x\\notin B(Q,r)$.\n3. [item:Dom3Disjoint] If *d*(*x*, *B*(*P**k*, *r*)) < *d*(*x*, *P**k*) for all *k* ∈ *K*, *r* > 0, and $x\\notin \\overline{P\\_k}$, then for each integer *γ* ≥ 2 the components of ${\\textnormal{Dom}}^{\\gamma}P $ are disjoint. Moreover, if *X* is a geodesic metric space, then for all *j*, *k* ∈ *K*,*j* ≠ *k* and all integer *γ* ≥ 2, the following inequality holds: $(r\\_k/8)+(r\\_j/8)\\leq d(({\\textnormal{Dom}}^{\\gamma}P)\\_k,({\\textnormal{Dom}}^{\\gamma} P)\\_j)$.\n4. [item:rk] Suppose that (*X*, *d*) is a geodesic metric space. Given *j*, *k* ∈ *K*, *j* ≠ *k*, and an integer *γ* ≥ 2, if $x\\in ({\\textnormal{Dom}}^{\\gamma}P)\\_j$, then *r**k*/4 ≤ *d*(*x*, *P**k*).\n1. Let *k* ∈ *K* and suppose that *x* ∈ *B*(*P**k*, *r**k*/4), i.e., *d*(*x*, *P**k*) < *r**k*/4. Definition **[def:zone]** implies that $({\\textnormal{Dom}}^2 P)\\_k={\\textnormal{dom}}\\bigl(P\\_k,\\bigcup\\_{j\\neq k}{\\textnormal{dom}}\\bigl(P\\_j,\\bigcup\\_{i\\neq j}P\\_i\\bigr)\\bigr)$. Hence, in order to prove that $x\\in ({\\textnormal{Dom}}^2 P)\\_k$ it suffices to prove that *r**k*/4 ≤ *d*(*x*, *y*) for all $y\\in \\bigcup\\_{j\\neq k}{\\textnormal{dom}}(P\\_j,\\bigcup\\_{i\\neq j}P\\_i)$. Given some *y* in this union, it belongs to ${\\textnormal{dom}}\\bigl(P\\_j,\\bigcup\\_{i\\neq j}P\\_i\\bigr)$ for some *j* ∈ *K* ∖ {*k*}. This fact combined with the condition *k* ≠ *j* imply that\n \n$$d(y,P\\_j)\\leq d\\biggl(y,\\bigcup\\_{i\\neq j}P\\_i\\biggr)\\leq d(y,P\\_k)\\leq d(y,x)+d(x,P\\_k)<d(y,x)+(r\\_k/4).$$ \nThis inequality, the triangle inequality, and the definition of *r**k*, all imply that\n \n*r**k* ≤ *d*(*P**j*, *P**k*) ≤ *d*(*P**j*, *y*) + *d*(*y*, *x*) + *d*(*x*, *P**k*) ≤ 2*d*(*x*, *y*) + 2*r**k*/4, \ni.e., *d*(*x*, *P**k*) < *r**k*/4 ≤ *d*(*x*, *y*) and the assertion follows. Finally, Lemma **[lem:DomPk]** implies that $I^{(2)}={\\textnormal{Dom}}^2 P\\subseteq {\\textnormal{Dom}}^{\\gamma} P$ for any integer *γ* ≥ 1, and hence $({\\textnormal{Dom}}^2 P)\\_k\\subseteq ({\\textnormal{Dom}}^{\\gamma} P)\\_k$ for each *k* ∈ *K*.\n2. Let *p* ∈ *Q*. Since $x\\notin B(Q,r)$, a simple argument shows that the intersection of the compact segment [*x*, *p*] with the closed set ∂*B*(*Q*, *r*) is attained at some point *y* ∈ [*x*, *p*], e.g., at *y* = *γ*(*t*) for *t* = inf{*u* ∈ [0, *d*(*x*, *p*)] : *γ*(*u*) ∈ *B*(*Q*, *r*)} (where *γ* : [0, *d*(*x*, *p*)] → *X*, is the isometric function which maps [0, *d*(*x*, *p*)] onto [*x*, *p*]). It must be that *s* : = *r* − *d*(*y*, *p*) ≤ 0. Indeed, if this inequality does not hold, then *s* > 0. Hence the open ball *B*(*y*, *s*) is well defined and for each *z* in this ball we have *d*(*z*, *Q*) ≤ *d*(*z*, *p*) ≤ *d*(*z*, *y*) + *d*(*y*, *p*) < *s* + *d*(*y*, *p*) = *r*. Thus *z* ∈ *B*(*Q*, *r*) for all *z* ∈ *B*(*y*, *s*) and hence *y* is in the interior of *B*(*Q*, *r*), a contradiction to the fact that *y* ∈ ∂(*B*(*Q*, *r*)). Therefore, since *y* ∈ [*x*, *p*], we obtain *d*(*x*, *p*) = *d*(*x*, *y*) + *d*(*y*, *p*) ≥ *d*(*x*, *y*) + *r*. Thus, for all *p* ∈ *Q*\n \n$$d(x,B(Q,r))=d(x,\\overline{B(Q,r)})\\leq d(x,y)\\leq d(x,p)-r.$$ \nAs a result, *d*(*x*, *B*(*Q*, *r*)) + *r* ≤ inf{*d*(*x*, *p*) : *p* ∈ *Q*} = *d*(*x*, *Q*). To see that actually *d*(*x*, *B*(*Q*, *r*)) + *r* = *d*(*x*, *Q*), let *ε* > 0 be arbitrary. By the definition of *d*(*x*, *B*(*Q*, *r*)) there exists *z* ∈ *B*(*Q*, *r*) satisfying *d*(*x*, *z*) < *d*(*x*, *B*(*Q*, *r*)) + *ε*. This inequality, the triangle inequality, and the fact that *z* ∈ *B*(*Q*, *r*) imply that *d*(*x*, *Q*) ≤ *d*(*x*, *z*) + *d*(*z*, *Q*) < *d*(*x*, *B*(*Q*, *r*)) + *ε* + *r*. Since *ε* can be an arbitrary small positive number, we conclude that *d*(*x*, *Q*) ≤ *d*(*x*, *B*(*Q*, *r*)) + *r*. Hence *d*(*x*, *B*(*Q*, *r*)) + *r* = *d*(*x*, *Q*).\n3. Suppose that we know that the components of ${\\textnormal{Dom}}^{3}P $ are disjoint. Then for each integer *γ* ≥ 2 the components of ${\\textnormal{Dom}}^{\\gamma}P $ are disjoint, because ${\\textnormal{Dom}}^{\\gamma}P\\subseteq {\\textnormal{Dom}}^{3}P$ by Lemma **[lem:DomPk]**. We now prove that indeed the components of ${\\textnormal{Dom}}^{3}P $ are disjoint. Let *k*1 ≠ *k*2 be two indices in *K*, and assume to the contrary that $x\\in ({\\textnormal{Dom}}^3 P)\\_{k\\_1}\\bigcap ({\\textnormal{Dom}}^3 P)\\_{k\\_2}$. By the definition of ${\\textnormal{Dom}}$ and by Part **([item:B(P,r)Dom2])** we have\n \n$$\\begin{aligned}\nd(x,P\\_{k\\_1})\\leq d\\biggl(x,\\bigcup\\_{j\\neq k\\_1}({\\textnormal{Dom}}^2 P)\\_j\\biggr) \\leq d(x,({\\textnormal{Dom}}^2 P)\\_{k\\_2})\\\\\n\\leq d(x,B(P\\_{k\\_2},r\\_{k\\_2}/4))\\leq d(x,P\\_{k\\_2}).\\end{aligned}$$ \nHence $x\\notin \\overline{P\\_{k\\_2}}$, because if $x\\in \\overline{P\\_{k\\_2}}$, then from the above inequality *d*(*x*, *P**k*1) ≤ *d*(*x*, *P**k*2) = 0, namely $x\\in \\overline{P\\_{k\\_1}}$, a contradiction to the assumption that 0 < *r**k*1 ≤ *d*(*P**k*1, *P**k*2). Thus *d*(*x*, *B*(*P**k*2, *r**k*2/4)) < *d*(*x*, *P**k*2) by assumption, so actually *d*(*x*, *P**k*1) < *d*(*x*, *P**k*2). In the same way *d*(*x*, *P**k*2) < *d*(*x*, *P**k*1), a contradiction.\nFinally, suppose that *X* is a geodesic metric space. Let *j*, *k* ∈ *K* be different. Since $({\\textnormal{Dom}}^{\\gamma}P)\\_i\\subseteq ({\\textnormal{Dom}}^3 P)\\_i$ for each integer *γ* ≥ 2 and each *i* ∈ *K* (by Lemma **[lem:DomPk]**), it suffices to show that (*r**k*/8) + (*r**j*/8) ≤ *d*(*x*, *y*) for all $x\\in ({\\textnormal{Dom}}^3 P)\\_{k}$ and $y\\in({\\textnormal{Dom}}^3 P)\\_j$. By definition, the triangle inequality and parts **([item:B(P,r)Dom2])**, **([item:r])**,\n \n$$\\begin{aligned}\nd(x,P\\_k)\\leq d\\biggl(x,\\bigcup\\_{i\\neq k}({\\textnormal{Dom}}^2 P)\\_i\\biggr)& \\leq d(x,({\\textnormal{Dom}}^2 P)\\_j)\n\\leq d(x,B(P\\_j,r\\_{j}/4))\\\\\n&\\leq d(x,P\\_j)-r\\_j/4\\leq d(x,y)+d(y,P\\_j)-(r\\_j/4).\\end{aligned}$$ \nIn the same way *d*(*y*, *P**j*) ≤ *d*(*y*, *x*) + *d*(*x*, *P**k*) − (*r**k*/4). Now we add these two inequalities and do additional elementary arithmetic operations which imply that indeed (*r**k*/8) + (*r**j*/8) ≤ *d*(*x*, *y*).\n4. If $x\\in ({\\textnormal{Dom}}^{\\gamma}(P))\\_j$, then $x\\notin B(P\\_k,r\\_k/4)\\subseteq ({\\textnormal{Dom}}^{\\gamma}(P))\\_k$ from Parts **([item:B(P,r)Dom2])** and **([item:Dom3Disjoint])**. Hence Part **([item:r])** implies that *r**k*/4 ≤ *d*(*x*, *P**k*) − *d*(*x*, *B*(*P**k*, *r**k*/4)) ≤ *d*(*x*, *P**k*).\n[lem:AntiIntersecdom] Let (*X*, *d*) be a metric space, let *P* ⊆ *X* be nonempty and suppose that {*C**γ*}*γ* = 1∞ is a family of subsets of *X* such that ⋂*γ* = 1∞*C**γ* ≠ ∅. If\n \n$$\\label{eq:domInLess}\n{\\textnormal{dom}}\\Biggl(P,\\bigcap\\_{\\gamma=1}^{\\infty} C\\_{\\gamma}\\Biggr)= \\overline{\\Biggl\\{x\\in X:\nd(x,P)<d\\Biggl(x,\\bigcap\\_{\\gamma=1}^{\\infty} C\\_{\\gamma}\\Biggr)\\Biggr\\}},$$ \nand\n \n$$\\label{eq:ALimAk}\nd\\Biggl(y,\\bigcap\\_{\\gamma=1}^{\\infty} C\\_{\\gamma}\\Biggr)=\\limsup\\_{\\gamma\\to \\infty}d(y,C\\_{\\gamma}),\\quad \\forall y\\in X,$$ \nthen\n \n$$\\label{eq:AntiCont}\n{\\textnormal{dom}}\\Biggl(P,\\bigcap\\_{\\gamma=1}^{\\infty} C\\_{\\gamma}\\Biggr)=\\overline{\\bigcup\\_{\\gamma=1}^{\\infty}{\\textnormal{dom}}(P,C\\_{\\gamma})}.$$ \nBy the antimonotonicity of ${\\textnormal{dom}}(P,\\cdot)$ we have ${\\textnormal{dom}}\\bigl(P,\\bigcap\\_{j=1}^{\\infty}\nC\\_j\\bigr)\\supseteq{\\textnormal{dom}}(P,C\\_{\\gamma})$ for all *γ* ∈ N. Since ${\\textnormal{dom}}\\bigl(P,\\bigcap\\_{j=1}^{\\infty}\nC\\_j\\bigr)$ is a closed set, the inclusion ${\\textnormal{dom}}(P,\\bigcap\\_{\\gamma=1}^{\\infty}\nC\\_{\\gamma})\\supseteq\\overline{\\bigcup\\_{\\gamma=1}^{\\infty}{\\textnormal{dom}}(P,C\\_{\\gamma})}$ follows. For the reverse inclusion, let *ε* > 0 be given and suppose that $x\\in {\\textnormal{dom}}(P,\\bigcap\\_{\\gamma=1}^{\\infty} C\\_{\\gamma})$. We should prove that there are *γ* ∈ N and $y\\in {\\textnormal{dom}}(P,C\\_{\\gamma})$ such that *d*(*y*, *x*) < *ε*.\nBy **([eq:domInLess])** there is *y* ∈ *X* such that *d*(*x*, *y*) < *ε* and $r:=d\\bigl(y,\\bigcap\\_{\\gamma=1}^{\\infty} C\\_{\\gamma}\\bigr)-d(y,P)>0$, and by **([eq:ALimAk])** there is *γ* ∈ N large enough such that $\\bigl|d\\bigl(y,\\bigcap\\_{j=1}^{\\infty} C\\_j\\bigr)-d(y,C\\_{\\gamma})\\bigr|<r/2$. Hence\n \n$$d(y,P)+r/2<d\\biggl(y,\\bigcap\\_{\\gamma=1}^{\\infty} C\\_{\\gamma}\\biggr)<d(y,C\\_{\\gamma})+r/2.$$ \nThus *d*(*y*, *P*) < *d*(*y*, *C**γ*), and so $y\\in {\\textnormal{dom}}(P,C\\_{\\gamma})$.\n[prop:mDomM] Let (*X*, *d*) be a geodesic metric space and let (*P**k*)*k* ∈ *K* be a tuple of nonempty closed sets in *X* satisfying *r* : = inf{*d*(*P**k*, *P**j*) : *j*, *k* ∈ *K*, *j* ≠ *k*} > 0. For each nonnegative integer *γ* and each *k* ∈ *K* let\n \n$$\\label{eq:R\\_gammaA\\_k}\nR^{\\gamma}:={\\textnormal{Dom}}^{2\\gamma+1}(P\\_k)\\_{k\\in K}=O^{(\\gamma)}, \\quad\nA\\_{\\gamma,k}:=\\bigcup\\_{j\\neq k} R\\_j^{\\gamma},\\quad A\\_k:=\\bigcap\\_{\\gamma=1}^{\\infty}A\\_{\\gamma,k}.$$ \n1. [item:UnionIntersect] The following equality holds for all *k* ∈ *K*:\n \n⋃*j* ≠ *k*⋂*γ* = 1∞*R**j**γ* = ⋂*γ* = 1∞⋃*j* ≠ *k**R**j**γ* = *A**k*.\n2. [item:AkNonEmpty] *A**γ*, *k* is a closed set for each *k* ∈ *K* and each *γ* ≥ 1. *A**k* is nonempty, closed, and satisfies *d*(*P**k*, *A**k*) ≥ *r*/4 for each *k* ∈ *K*.\n3. [item:DomCapCup] If for each *k* ∈ *K* both **([eq:domInLess])** and **([eq:ALimAk])** hold with *C**γ* = *A**γ*, *k*, then\n \n$$\\label{eq:DomCapCup}\n{\\textnormal{Dom}}\\Biggl(\\bigcap\\_{\\gamma=0}^{\\infty} {\\textnormal{Dom}}^{2\\gamma+1}(P\\_k)\\_{k\\in K}\\Biggr)=\n\\overline{\\bigcup\\_{\\gamma=0}^{\\infty}{\\textnormal{Dom}}^{2\\gamma}(P\\_k)\\_{k\\in K}}.$$\n4. [item:StrictlyConvexProper] If, in addition, (*X*, *d*) is proper and has the geodesic inclusion property, then **([eq:domInLess])**–**([eq:ALimAk])** (with *C**γ* = *A**γ*, *k*, *k* ∈ *K* arbitrary) and hence **([eq:DomCapCup])** hold.\n1. Fix *k* ∈ *K*. If *x* ∈ ⋃*j* ≠ *k*⋂*γ* = 1∞*R**j**γ*, then *x* ∈ ⋂*γ* = 1∞*R**j**γ* for some *j* ≠ *k*. Since *R**j**γ* ⊆ ⋃*i* ≠ *k**R**i**γ* for all *γ* ∈ N, it follows that ⋂*γ* = 1∞*R**j**γ* ⊆ ⋂*γ* = 1∞⋃*i* ≠ *k**R**i**γ*, so *x* ∈ ⋂*γ* = 1∞⋃*i* ≠ *k**R**i**γ*. On the other hand, let *x* ∈ ⋂*γ* = 1∞⋃*j* ≠ *k**R**j**γ*. Given *γ* ∈ N, Lemma **[lem:Dom3]****([item:Dom3Disjoint])** (note that 2*γ* + 1 ≥ 3 and $R^{\\gamma}={\\textnormal{Dom}}^{2\\gamma+1}(P\\_k)\\_{k\\in K}$) implies that *R**i**γ* ∩ *R**j**γ* = ∅ for indices *j* ≠ *i*. Hence for each *γ* ∈ N there exists exactly one index *j* ≠ *k* such that *x* ∈ *R**j**γ*. It must be that all these indices coincide. Indeed, if this is not true, then *x* ∈ *R**j**γ* ∩ *R**j*ʹ*γ*ʹ for two natural numbers *γ*ʹ > *γ* and two indices *j*ʹ ≠ *j*. But *R**j*ʹ*γ*ʹ ⊆ *R**j*ʹ*γ* by Lemma **[lem:DomPk]**. Thus *x* ∈ *R**j**γ* ∩ *R**j*ʹ*γ*, a contradiction to Lemma **[lem:Dom3]****([item:Dom3Disjoint])**. As a result *x* ∈ ⋂*γ* = 1∞*R**j**γ* for some *j* ≠ *k*, and we conclude that *x* ∈ ⋃*j* ≠ *k*⋂*γ* = 1∞*R**j**γ*. This inclusion, the one proved before, and the definition of *A**k*, all imply **([eq:UnionIntersect])**.\n2. Lemma **[lem:DomPk]** implies that *P**j* ⊆ ⋂*γ* = 1∞*R**j**γ* for each *j* ∈ *K*, and so **([eq:UnionIntersect])** implies that ∅ ≠ ⋃*j* ≠ *k**P**j* ⊆ ⋃*j* ≠ *k*⋂*γ* = 1∞*R**j**γ* = *A**k* for all *k* ∈ *K*. For each *γ* ∈ N and *k* ∈ *K* we have *r*/4 ≤ *d*(*P**k*, *A**γ*, *k*) ≤ *d*(*P**k*, *A**k*), where the left inequality follows from Lemma **[lem:Dom3]****([item:rk])** and the definition **([eq:RgammaAk])** of *A**γ*, *k* (we use Lemma **[lem:Dom3]****([item:rk])** with each *x* ∈ *A**γ*, *k* and recall that *r* ≤ *r**k*) and the right inequality follows from the inclusion *A**k* ⊆ *A**γ*, *k*. From Lemma **[lem:DomPk]** the intersection in **([eq:RgammaAk])** which defines *A**k* is decreasing. By Lemma **[lem:dom]****([item:closure])** and Lemma **[lem:Dom3]****([item:Dom3Disjoint])** each *A**γ*, *k* is a closed set, because it is a union of closed and disjoint sets with a positive distance (at least *r*/4) between any two different members in the union. Therefore *A**k* is the intersection of closed sets and thus closed.\n3. By Part **([item:AkNonEmpty])** we know that *A**k* ≠ ∅ for any *k* ∈ *K*. Given *k* ∈ *K*, our assumption implies that **([eq:domInLess])**–**([eq:ALimAk])** hold with *C**γ* = *A**γ*, *k* for each *γ* ∈ N. By Part **([item:UnionIntersect])** we know that **([eq:UnionIntersect])** holds. These facts, together with Lemma **[lem:DomPk]**, Lemma **[lem:AntiIntersecdom]**, the definition of ${\\textnormal{Dom}}$, and the fact that intersection, union, and closure on tuples are taken component-wise, imply\n \n$$\\begin{gathered}\n{\\textnormal{Dom}}\\Biggl(\\bigcap\\_{\\gamma=0}^{\\infty}{\\textnormal{Dom}}^{2\\gamma+1}(P\\_k)\\_{k\\in K}\\Biggr)\n={\\textnormal{Dom}}\\Biggl(\\bigcap\\_{\\gamma=0}^{\\infty}R^{\\gamma}\\Biggr)\n={\\textnormal{Dom}}\\Biggl(\\bigcap\\_{\\gamma=1}^{\\infty}R^{\\gamma}\\Biggr)\\\\\n=\\Biggl({\\textnormal{dom}}\\Biggl(P\\_k,\\bigcup\\_{j\\neq k}\\bigcap\\_{\\gamma=1}^{\\infty}R\\_j^{\\gamma}\\Biggr)\\Biggr)\\_{k\\in K}\n=\\Biggl({\\textnormal{dom}}\\Biggl(P\\_k,\\bigcap\\_{\\gamma=1}^{\\infty}\\bigcup\\_{j\\neq k}R\\_j^{\\gamma}\\Biggr)\\Biggr)\\_{k\\in K}\\\\\n=\\Biggl({\\textnormal{dom}}\\Biggl(P\\_k,\\bigcap\\_{\\gamma=1}^{\\infty}A\\_{\\gamma,k}\\Biggr)\\Biggr)\\_{k\\in K}\n=\\Biggl(\\overline{\\bigcup\\_{\\gamma=1}^{\\infty}{\\textnormal{dom}}(P\\_k,\\bigcup\\_{j\\neq k}R\\_j^{\\gamma})}\\Biggr)\\_{k\\in K}\\\\\n=\\overline{\\Biggl(\\bigcup\\_{\\gamma=1}^{\\infty}\\Biggl({\\textnormal{Dom}}(R^{\\gamma})\\Biggr)\\_k\\Biggr)\\_{k\\in K}}\n=\\overline{\\bigcup\\_{\\gamma=1}^{\\infty}{\\textnormal{Dom}}(R^{\\gamma})}\n=\\overline{\\bigcup\\_{\\gamma=0}^{\\infty}{\\textnormal{Dom}}^{2\\gamma}(P\\_k)\\_{k\\in K}}.\\end{gathered}$$\n4. Since for each *k* ∈ *K* the intersection in **([eq:RgammaAk])** which defines *A**k* is decreasing (from Lemma **[lem:DomPk]**) of closed subsets (part **([item:AkNonEmpty])**), Lemma **[lem:DistIntersect]** implies that *d*(*x*, *A**k*) = lim*γ* → ∞*d*(*x*, *A**γ*, *k*) for each *x* ∈ *X* and hence **([eq:ALimAk])** holds with *C**γ* : = *A**γ*, *k* for every *γ* ∈ N. Because *d*(*P**k*, *A**k*) ≥ *r*/4 > 0 by Part **([item:AkNonEmpty])** and because (*X*, *d*) is a proper geodesic metric space having the geodesic inclusion property, we conclude from Theorem **[thm:BoundaryInterior]** (equality **([eq:closure])**) that **([eq:domInLess])** holds with *P**k* instead of *P* and *A**k* instead of *A*. From part **([item:DomCapCup])** we conclude that **([eq:DomCapCup])** holds.\nNow it is possible to prove the assertions formulated in Section **[sec:ConvergenceResult]**.\n[**Proof of Theorem **[thm:Metric]****] Lemma **[lem:dom]****([item:DomClosure])** and Lemma **[lem:GeneralDistance]****([item:DomUnionIntersect])** imply that\n \n$$\\label{eq:Dom(m)M}\n{\\textnormal{Dom}}(m)={\\textnormal{Dom}}\\Biggl(\\overline{\\bigcup\\_{n=0}^{\\infty}I^{(n)}}\\Biggr)\n={\\textnormal{Dom}}\\Biggl(\\bigcup\\_{n=0}^{\\infty}I^{(n)}\\Biggr)\n=\\bigcap\\_{n=0}^{\\infty}{\\textnormal{Dom}}(I^{(n)})=\\bigcap\\_{n=0}^{\\infty}O^{(n)}=M.$$ \n[**Proof of Theorem **[thm:AlgUniConvex]****] By assumption (*X*, *d*) is a proper geodesic metric space which has the geodesic inclusion property and the sites satisfy **([eq:d(Pk,Pj)>0])**. Hence we can apply Proposition **[prop:mDomM]****([item:StrictlyConvexProper])** which implies that\n \n$$\\label{eq:Dom(M)m}\n{\\textnormal{Dom}}(M)={\\textnormal{Dom}}\\Biggl(\\bigcap\\_{\\gamma=0}^{\\infty}{\\textnormal{Dom}}^{2\\gamma+1}(P)\\Biggr)\n=\\overline{\\bigcup\\_{\\gamma=0}^{\\infty}{\\textnormal{Dom}}^{2\\gamma}(P)}=\\overline{\\bigcup\\_{n=0}^{\\infty}I^{(n)}}=m.$$ \nThus $m={\\textnormal{Dom}}(M)$. This and Theorem **[thm:Metric]** imply that $M={\\textnormal{Dom}}(m)={\\textnormal{Dom}}^2(M)$ and $m={\\textnormal{Dom}}(M)={\\textnormal{Dom}}^2(m)$. Hence both *m* and *M* are double zone diagrams. From Lemma **[lem:DomPk]****([item:In R On])** and Lemma **[lem:dom]****([item:DomClosure])** it can be seen that *m* and *M* are, respectively, the least and the greatest double zone diagrams. Finally, if ∣*K*∣ = 2, then **([eq:Dom(m)M])**, **([eq:Dom(M)m])**, and the definition of ${\\textnormal{Dom}}$, imply that $(M\\_1,M\\_2)={\\textnormal{Dom}}(m)=(({\\textnormal{dom}}(P\\_1,m\\_2),{\\textnormal{dom}}(P\\_2,m\\_1))$ and $(m\\_1,m\\_2)={\\textnormal{Dom}}(M\\_1,M\\_2)=({\\textnormal{dom}}(P\\_1,M\\_2),{\\textnormal{dom}}(P\\_2,M\\_1))$. Hence $m\\_1={\\textnormal{dom}}(P\\_1,M\\_2)$ and $M\\_2={\\textnormal{dom}}(P\\_2,m\\_1)$. But the definition of ${\\textnormal{Dom}}$ implies that ${\\textnormal{Dom}}(m\\_1,M\\_2)=({\\textnormal{dom}}(P\\_1,M\\_2),{\\textnormal{dom}}(P\\_2,m\\_1))$. Therefore one has ${\\textnormal{Dom}}(m\\_1,M\\_2)=(m\\_1,M\\_2)$, namely, (*m*1, *M*2) is a zone diagram. In the same way (*M*1, *m*2) is a zone diagram.\n[**Proof of Corollary **[cor:CompactLimit]**] Fix *k* ∈ *K* and for each *n* ∈ N ∪ {0} denote *A**n* : = *O**k*(*n*) and *B**n* : = *I**k*(*n*). Lemma **[lem:DomPk]** implies that (*A**n*)*n* = 0∞ is decreasing and (*B**n*)*n* = 0∞ is increasing. As a result, since *X* is compact **([eq:Mmk])** follows from Lemma **[lem:HausdorffUnionIntersection]**.** \n[**Proof of Corollary **[cor:ZoneDoubleZone]****] If the least and the greatest double zone diagram coincide, then, without any restriction on the sites or the space, there exists a unique zone diagram which is equal to both of them. Since in the proof of it was proved that the least double zone diagram coincides with the greatest one when (*X*, *d*) is a finite dimensional Euclidean space, the assertion follows from Theorem **[thm:AlgUniConvex]**.\nConcluding remarks and open problems\n====================================\nWe conclude the paper with the following remarks.\nWe believe that the methods and ideas presented in this paper can be extended to other implicit geometric objects. One of the reasons behind this belief is the fact that an important component in the iterative algorithm described in this paper for computing zone and double zone diagrams (Sections **[sec:QualitativeDescription]**, **[sec:ActualCompute]** above) is the approximation algorithm for computing Voronoi diagrams of general sites introduced in, and the computation of other implicit geometric objects, such as *k*-sectors, makes heavy use of Voronoi diagrams of general sites.\nA very interesting open problem is to establish error estimates for the convergence rate (speed) of (*I*(*n*))*n* = 0∞ and (*O*(*n*))*n* = 0∞. One of the reasons that this problem is interesting, is the observed fast convergence of these sequences (usually 4-5 iterations suffice for obtaining a very good approximation). We feel that the approach and error estimates developed in may help here. We also feel, but not entirely sure, that the rate of convergence is geometric (that is, linear).\nThe availability of the implementation enables one to preform various interesting experiments and to discover new and unexplained phenomena. For instance, although in general a zone diagram is not necessarily unique, even in the case of R2 with two point sites (see e.g., or Example **[ex:mM]** above), and although our theoretical results, when restricted to normed spaces, are limited to normed spaces which are strictly convex, experiments show that Algorithm **[alg:InOn]** actually converges to a unique zone diagram most of the times even when the considered normed space is not strictly convex. This claim is illustrated in Figures **[fig:Zone2D-10Sites-1inSite-L1-UnitSphere005-It2-Endpoints]**–**[fig:ZoneLinfty]**.\nAnother interesting phenomenon which has been observed experimentally is the geometric stability of zone diagrams with respect to certain changes. More precisely, two types of stability have been observed. The first type is that small perturbations in the shapes of the sites (e.g., due to a translation or a distortion) lead to small change in the shapes of the regions of the corresponding zone diagram (formally, the changes are measured with respect to the Hausdorff distance). Figures **[fig:Zone2D-5Sites-55inSite-ShapesType3-L2-UnitSphere005-It5-Rays]**–**[fig:Zone2D-5Sites-55inSite-ShapesType2-L2-UnitSphere005-It5-Rays]** illustrate this phenomenon. Similarly, small changes in the norm of the space (where the sites are fixed) yield small changes in the shapes of the regions of the zone diagram, as illustrated in Figures **[fig:Zone2D-10Sites-1inSite-L1-UnitSphere005-It2-Endpoints]**–**[fig:Zone2D-10Sites-1inSite-L1-UnitSphere0005-It6-Rays]** comparing to Figure **[fig:IntroZone]** and also in Figures **[fig:Zone2D-5Sites-55inSite-ShapesType2-L2-UnitSphere005-It5-Rays]**–**[fig:Zone2D-5Sites-55inSite-ShapesType2-Lalphadelta-UnitSphere005-It5-Rays]**. Here it is worth saying a few words on the formal way to measure these changes, since some caution is needed. Indeed, if we want to measure the changes with respect to the Hausdorff distance, then we face a problem because the Hausdorff distance depends on the norm and what we change is the norm itself. A possible solution to this issue is to consider all of the involved sets, namely the sites and the regions before and after the perturbations, as being sets in the Euclidean plane, and to measure the changes with respect to the Hausdorff distance induced by this Euclidean norm. This choice is natural since in the real world, when we compare the difference between the shapes, we simply consider them, using our eyes, as sets located on the flat page on which the pictures (before and after the perturbations) are embedded, and due to the Euclidean nature of our world, this flat page is modeled by the Euclidean plane.\nAnyway, it is an open problem to explain the above-mentioned phenomena, and we believe that the approach presented in can help here.\n[t]0.45\n\n[fig:Zone2D-10Sites-1inSite-L1-UnitSphere005-It2-Endpoints]\n[t]0.45\n\n[fig:Zone2D-10Sites-1inSite-L1-UnitSphere005-It3-EndpointsConnect]\n[t]0.45\n\n[fig:Zone2D-10Sites-1inSite-L1-UnitSphere0005-It6-Rays]\n[t]0.45\n\n[fig:ZoneLinfty]\n[t]0.45\n\n[fig:Zone2D-5Sites-55inSite-ShapesType3-L2-UnitSphere005-It5-Rays]\n[t]0.45\n\n[fig:Zone2D-5Sites-55inSite-ShapesType4-L2-UnitSphere005-It5-Rays]\n[t]0.45\n\n[fig:Zone2D-5Sites-55inSite-ShapesType2-L2-UnitSphere005-It5-Rays]\n[t]0.45\n\n[fig:Zone2D-5Sites-55inSite-ShapesType2-Lalphadelta-UnitSphere005-It5-Rays]\nIt is interesting to find out whether Theorem **[thm:AlgUniConvex]** can be generalized to other settings, for instance to all proper geodesic metric spaces, or at least to all finite dimensional normed spaces and to some “nice” manifolds.\nIt will be interesting to extend the theory of geodesic metric spaces which has the geodesic inclusion property (Section **[sec:Geodesic]** above) and, in particular, to give more examples of geodesic metric spaces which have this property. For instance to prove or disprove whether locally compact, complete and connected smooth (or piecewise smooth) Riemannian manifolds, as well as Hadamard spaces (in particular, hyperbolic spaces ) have this property.\n[rem:WeakSolution] In Section **[sec:Intro]** we discussed briefly some similarities and differences between implicit computational geometry and differential equations. In what follows we mention another similarity: the relation between double zone diagrams and the concept of weak solutions. A weak solution to a partial differential equation is a function that does not satisfy the original equation but rather it satisfies a weaker condition which any true solution (also called “a strong” or “a classical” solution) to the original equation (if such a solution exists) must satisfy. Hence a weak solution generalizes the concept of a strong solution. It is useful to consider weak solutions since in many cases it is possible to prove their existence. Once this is done, the weak solution becomes a candidate to be a strong solution. Now one needs to prove that the weak solution is indeed a strong solution, and in many cases this can be done. Returning to zone and double zone diagrams, we see that a possible technique to prove the existence of zone diagrams is to consider first a certain double zone diagram (as was already said before, the existence of double zone diagrams is known in a general setting ) and then to show that such a double zone diagram must in fact be a zone diagram. This technique has been used in various works. Hence, in some sense double zone diagrams can be regarded as weak solutions to the zone diagram equation $R={\\textnormal{Dom}}(R)$.\nAcknowledgments\n===============\nThis paper is the output of a long and challenging iterative process. Any honest and constructive feedback that has been given to me during this process is appreciated. I want to use this opportunity to thank the editors of DCG for their patience and to thank the referee for valuable comments. I also want to thank Mario Ponce for inviting me for a visit in the Pontifical Catholic University of Chile (Santiago) during June 2015, for helpful discussions, warm hospitality, and for allowing me to speak there (and in the Pontifical Catholic University of Valparaíso) on issues related to implicit computational geometry in general and to this paper in particular. A very sad moment along the way was when I discovered that Jiří (Jirka) Matoušek, one of the founders of implicit computational geometry, passed away. Unfortunately for me, I have not had the opportunity to meet him, but at least I had the honor to communicate with him electronically and to be a coauthor of him. It was through these communications and also through his scientific outcome that I have discovered his exceptional abilities. I hope and believe that his contributions will have a lasting value and I dedicate this paper to him.\nParts of this work were done during various years, when I have been associated with the following places: The Technion, Haifa, Israel (2009-2010, 2016-2017), the University of Haifa, Haifa, Israel (2010-2011), the National Institute for Pure and Applied Mathematics (IMPA), Rio de Janeiro, Brazil (2011-2013), and the Institute of Mathematics and Computational Sciences (ICMC), University of São Paulo, São Carlos, Brazil (2014-2015), and it is an opportunity for me to thank the BSF and FAPESP. Special thanks are for a special postdoc fellowship (“Pós-doutorado de Excelência”) given to me when I was at IMPA.\n10 [1]#1 urlstyle[1]DOI #1\nAsano, T., Kirkpatrick, D.: Distance trisector curves in regular convex distance metrics. In: Gavrilova, M.L. (ed.) Proceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2006), pp. 8–17. 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In: Kalantari, B., Gavrilova, M.L. (eds.) Proceedings of the 9th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2012), pp. 2–16. IEEE Computer Society, Los Alamitos (2012)\n"},"data_id":{"kind":"string","value":"arxiv_0182368"}}},{"rowIdx":45590,"cells":{"content":{"kind":"string","value":" sign of tidal or other processes that could be connected to their outstanding chemical peculiarity. In contrast, the transition between circular and eccentric orbits seems to occur at periods of ∼ 4 − 800 days for binaries with former AGB companions.\nFurther information on the presently unseen companions can be derived by considering the volume available to them during their evolution; i.e. their Roche-lobe radii, which can be computed from the observed orbital separation (assuming no exchange or loss of mass and angular momentum) and a range of assumed stellar masses. We first adopt a common mass of 0.8 $M\\_\\sun$ for the observed EMP giants and assume that the companion is a (sub)dwarf (i.e. unevolved) star of mass 0.6 $M\\_\\sun$, at least three magnitudes fainter than the star we do see. The present radius of the latter can be estimated from the adopted mass and a typical log *g* ∼ 1.5 dex, i.e. R ∼ 30 $R\\_\\sun$.\nWith the observed orbital elements, notably *a*1sin*i* and the mass function, we then adjust *i* until the computed *M*2 reaches 0.6 $M\\_\\sun$, and calculate the corresponding Roche-lobe radii, which are given in Table [tbl-4]. For HE 0442 − 1234, the minimum secondary mass is 0.67 $M\\_\\sun$ already for *i* = 90$\\degr$; for HE 1044 − 2509 and HE 1523 − 0901, we find *i* = 62$\\degr$ and 1.3$\\degr$, respectively, for a secondary mass of 0.6 $M\\_\\sun$.\nHowever, we might alternatively assume that the companions were initially *more* massive than 0.8 $M\\_\\sun$, i.e. in the range of 1-8 $M\\_\\sun$, where they would have gone through the AGB phase and likely have evolved into now-invisible white dwarfs (WDs) with typical masses of 0.6 $M\\_\\sun$, and thus identical R*R**o**c**h**e* to the ones found above. It is possible that the WD might also have the maximum WD mass of 1.4 $M\\_\\sun$, in which case the entire system would be much larger and the Roche-lobe radius of the companion a larger share of that again. Nominal Roche-lobe radii for this case are also given in Table [tbl-4], and illustrate the dramatic change in the space available to the star during its evolution, depending on its assumed mass.\nFor HE 1044 − 2509, even this R*R**o**c**h**e* is still too small to accommodate a typical AGB star of ∼ 200 $R\\_\\sun$. In any case, the observed absence of any *s*-process signatures indicates that mass transfer from a putative AGB companion did not happen in any of these systems, while a supernova explosion of a putative, even more massive companion would likely have disrupted the binary system.\nIn summary, binary systems seem to occur as a normal part of the formation of *r*-process-enhanced metal-poor stars, and are unrelated to the process by which they acquired their outstanding chemical anomalies.\nOrigin of *r*-I and *r*-II stars\n--------------------------------\nThe separation of *r*-process-enhanced stars into the *r*-I and *r*-II classes was originally a matter of convenience, since the heavy-element abundance patterns of the most enhanced stars could be more easily observed due to their relatively stronger lines at low metallicity. As the sample of such stars has grown, it has become clear that these classes also exhibit rather different behaviour with metallicity; the *r*-II stars are found in a relatively narrow range of metallicity near [Fe/H] ∼ − 3.0, while the *r*-I stars cover a larger range of metallicity, − 3.5 < [Fe/H] < − 1.5. It remains unclear whether this constraint implies that different classes of astrophysical progenitors might be responsible for the *r*-I and *r*-II stars, or whether the initial *r*-process content in the natal clouds of the *r*-I stars has simply been diluted by the elemental mix of standard chemical evolution.\nAll of our programme stars are giants. However, it is important to note that have shown that the star is a cool (T$\\_{\\rm{eff}} \\sim $5000 K) main-sequence dwarf with [Fe/H] = − 3.4, and [Eu/Fe] = + 2.0, making it simultaneously the lowest metallicity and most Eu-enriched *r*-II star yet found. This star is of particular interest, as all previous *r*-process-enhanced stars identified to date have been in more evolved stages of evolution. Since dwarfs with the temperature of SDSS J2357 − 0052 do not have convective atmospheres, it can be reasonably concluded that the *r*-process-enhancement phenomenon is not due to some chemical peculiarity arising from the presence of a convective envelope in such stars. Unfortunately, although this star is sufficiently bright (*V* ∼ 15.6) for the programme described above, it was discovered too late to be included in our target list, but the results of radial-velocity monitoring over about a year by Aoki et al. did not reveal any evidence of significant variation.\nThe nine newly recognized *r*-process-enhanced stars (based on high-resolution spectroscopic follow-up of HK survey stars) by include subgiants and the field equivalents of red horizontal-branch stars, reaffirming that *r*-process enhancement is not an evolutionary effect in the stars. also compare the abundance pattern of the light elements in their *r*-process-enhanced stars with that of *non-* *r*-process-enhanced stars with similar stellar parameters. No evidence was found to indicate that the *r*-process-enhanced stars have different abundance patterns for the light elements than for the comparison sample, leading the authors to conclude that the event(s) producing the high levels of *r*-process material seen in these stars do not produce a distinct light-element abundance pattern. Neither does the large *r*-process enhancement seem to be coupled to the carbon and nitrogen abundances in the stars, although a few of the *r*-II stars are found to also be enhanced in carbon, the most well-known example being CS 22892 − 052.\n explored the Sr, Ba and Eu abundances for a number of *r*-I and *r*-II stars. They found very similar [Ba/Eu] abundance ratios for the two groups ([Ba/Eu] ∼ + 0.60), but the mean [Sr/Eu] ratio differed by 0.36 dex between the two groups, with the *r*-II stars having the lowest ratio, [Sr/Eu] = − 0.93. The authors argued that elements from the first and second *r*-process peak are of common origin in the *r*-II stars, whereas for the *r*-I stars the picture is less clear.\nCompeting scenarios for the origin of the *r*-I and *r*-II stars invoke non-spherical, jet-producing supernova explosions or neutrino winds from merging neutron star binaries. Detailed observations of the abundance patterns predicted from the two competing scenarios may be the best guide to identifying the production site(s), but the frequency of these stars ( ∼ 3% among VMP and EMP stars; ) provides another clue; any jets in such scenarios must be highly collimated in order to selectively enrich only a small fraction of molecular clouds in the early ISM.\nOne circumstance is worthy of note, although its interpretation is currently unclear: Of the two *r*-II stars with secure detections of uranium, exhibits the so-called “actinide boost”, and has now been shown to be a single star, while the detailed *r*-process abundance pattern of the newly identified binary does not. While this may just be a result of small-number statistics on these very rare objects, it should be kept in mind as the sample grows; the details of their chemical-abundance patterns would appear to offer our most reliable clue to the origin of this difference.\nConclusions\n===========\nIn order to detect a possible link between the nature of the binary population among stars that exhibit moderate to strong enhancements in their *r*-process-element abundances and the origin of their peculiar abundance patterns, we have monitored the radial velocities of 17 such stars with high precision ( ∼ 100 m s− 1) over a period of eight years.\nOf the 17 programme stars, 14 exhibit no radial-velocity variations during this period, and are thus presumably single, while three are binaries with normal orbital periods and eccentricities, yielding a normal binary frequency of ∼ 18% among these stars. Hence, there is no evidence that the *r*-process enhancement seen in the *r*-I and *r*-II stars is causally linked to the binary nature of the stars. Furthermore, the detection of *r*-process-element rich stars in dwarf, subgiant, giant, and horizontal-branch evolutionary phases shows that the enhancement phenomenon is also not linked to the evolution of the stars.\nWe conclude that the moderate-to-high *r*-process-element abundances derived for these stars must be an imprint of the clouds from which the stars were formed. Detailed abundance analysis of larger samples of *r*-I and *r*-II stars is required to better constrain the nature of the objects and the chain of events that polluted the natal clouds of these stars. Moreover, the mechanism(s) for such sporadic, inhomogeneous enrichment of the ISM in early galaxies, and their implication for understanding the observed chemical composition of high-redshift DLA systems, should be considered in the next generation of models for the formation and early chemical evolution of galaxies.\nThis paper is based on observations made with the Nordic Optical Telescope, operated by the Nordic Optical Telescope Scientific Association at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.\nWe thank several NOT staff members and students for obtaining most of the observations for us in service mode.\nThe work of T.H. was supported by Sonderforschungsbereich SFB 881 “The Milky Way System” (subproject A4) of the German Research Foundation (DFG). T.C.B. and J.Y. acknowledge partial support for this work from grants PHY 08-22648; Physics Frontier Center/Joint Institute or Nuclear Astrophysics (JINA), and PHY 14-30152; Physics Frontier Center/JINA Center for the Evolution of the Elements (JINA-CEE), awarded by the US National Science Foundation. J.A. and B.N. gratefully acknowledge financial support from the Danish Natural Science Research Council and the Carlsberg Foundation. B.N. gratefully acknowledges partial support from the National Science Foundation under Grant No. NSF PHY11-25915.\nFurthermore, we thank the referee for helpful comments.\nAoki, W., Honda, S., Beers, T. C., & Sneden, C., 2003,, 586, 506\nAoki, W., Beers, T.C., Honda, S., & Carollo, D., 2010,, 723, L201\nAoki, W., Beers, T.C., Lee, Y.S., et al. 2013,, 145, 13\nBarklem, P.S., Christlieb, N., Beers, T.C., et al. 2005,, 439, 129\nBartkevicius, A., Sperauskas, J., Rastorguev, A. S., & Tokovinin, A. A., 1992, Baltic Astronomy, 1, 47\nBeers, T.C., 2013, Nuclei in the Cosmos XII, Proceedings of Science, PoS (NIC-XII), 56\nBeers, T.C., & Christlieb N., 2005,, 43, 531\nBeers, T.C., Flynn, C., Rossi, S., et al., 2007,, 168, 128\nBeers, T.C., Preston, G.W.,& Shectman, S.A., 1985,, 90, 2089\nBeers, T.C., Preston, G.W.,& Shectman, S.A., 1992,, 103, 1987\nBond, H. E., 1980,, 44, 517\nBonifacio, P., Spite, M., Cayrel, R., et al., 2009,, 501, 519\nBuchhave, L.A., 2010, PhD thesis, Univ. of Copenhagen\nBuchhave, L., Latham, D.W., Johansen, A., et al., 2012,, 486, 375\nCarney, B. W., & Latham, D. W., 1986,, 92, 60\nCarney, B.W., Latham, D.W., Stefanik, R.P., et al., 2003,, 125, 293\nCarney, B. W., Gray, D. 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U., et al., 2011,, 742, 54\nHonda, S., Aoki, W., Ando, H., et al., 2004,, 152, 113\nIvezic, Z., Beers, T.C., & Juric, M. 2012,, 50, 251\nKordopatis, G., Gilmore, G., Steinmetz, M., et al., 2013,, 146, 134\nMashonkina, L., Christlieb, N., Barklem, P.S., et al., 2010,, 516, A46\nMathieu, R.D., Latham, D.W., & Griffin, R.F., 1990,, 100, 1899\nMcClure, R.D., 1984,, 280, L31\nMcClure, R.D., Fletcher, J.M., & Nemec, J.M., 1980,, 238, L35\nMcWilliam, A., Preston, G.W., Sneden, C., & Searle, L., 1995,, 109, 2757\nMermilliod, J.-C., Andersen, J., Latham, D.W., & Mayor, M., 2007,, 473, 829\nMermilliod, J.-C., Mayor, M., & Udry, S., 2008,, 473, 829\nNorris, J. E., Ryan, S. G., & Beers, T. C., 1996,, 107, 391\nPreston, G.W., & Sneden, C., 2001,, 122, 1545\nRiebel, D., Meixner, M., Fraser, O., et al., 2010,, 723, 1195\nRoederer, I., Cowan, J.J, Preston, G.W., et al., 2014b,, 445, 2970\nRoederer, I. U., Preston, G. W., Thompson, I. B., et al., 2014a,, 147, 136\nRoederer, I., Lawler, J. E., Sneden, C., Cowan, J. J., Sobeck, J. S., & Pilachowski, C. A., 2008,, 675, 723\nSneden, C., Cowan, J.J., Ivans, I.I., et al., 2000,, 533, L139\nSoubiran, C., Bienaymé, O, Mishenina, T. V., & Kovtyukh, V. V., 2008,, 480, 91\nTauris, T. M., & Takens, R. J., 1998,, 330, 1047\nTsangarides, S. A., Ryan, S. G., & Beers, T. C., 2003, Astronomical Society of the Pasific Conference Seried, CNO in the Universe, ed: Charbonnel, C., Schaerer, D., & Meynet, G., 304, 133\nUdry, S., Mayor, M., & Queloz, D., 1999, PASPC, 185, 367\nWallerstein, G., Greenstein, J. L., Parker, R., Helfer, H. L., & Aller, L. H., 1963,, 137, 280\nWestin, J., Sneden, C., Gustafsson, B., & Cowan, J.J., 2000,, 530, 783\nWilson, R. E., 1953, Carnegie Institute of Washington D.C. Publication 601\nWilson, R. E., & Joy, A. 1950,, 111, 221\nWood, P.R., 2000,, 17, 18\nWoolley, R., & Harding, G.A., 1965, Royal Observatory Bulletin, 93, 363\nYong, D., Norris, J.E., Bessell, M.S., et al., 2013,, 762, 26\nIndividual heliocentric radial velocities measured for the programme stars\n==========================================================================\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.661653 | − 57.892 | 0.033 |\n| 2454338.615723 | − 57.891 | 0.022 |\n| 2454373.612652 | − 57.900 | 0.024 |\n| 2454819.293270 | − 57.983 | 0.030 |\n| 2455126.580295 | − 57.946 | 0.041 |\n| 2455175.394755 | − 57.978 | 0.023 |\n| 2455503.480513 | − 57.855 | 0.070 |\n| 2455776.652777 | − 57.940 | 0.022 |\n| 2455796.657395 | − 57.927 | 0.025 |\n| 2455859.514789 | − 57.862 | 0.034 |\n| 2456139.703103 | − 57.863 | 0.027 |\n| 2456241.446457 | − 57.899 | 0.064 |\n| 2456529.679820 | − 57.934 | 0.023 |\n| 2456917.579578 | − 57.924 | 0.027 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454373.642892 | 104.329 | 0.116 |\n| 2454705.631179 | 104.873 | 0.065 |\n| 2454780.500993 | 105.409 | 0.292 |\n| 2454819.320354 | 105.561 | 0.084 |\n| 2455175.419215 | 105.010 | 0.063 |\n| 2455415.686599 | 104.941 | 0.061 |\n| 2455439.606743 | 105.124 | 0.060 |\n| 2455796.671082 | 105.434 | 0.077 |\n| 2455858.545060 | 105.060 | 0.060 |\n| 2456191.589163 | 105.050 | 0.045 |\n| 2456530.687786 | 104.829 | 0.054 |\n| 2456956.535485 | 104.471 | 0.094 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.681742 | 138.985 | 0.044 |\n| 2454338.656269 | 139.075 | 0.036 |\n| 2454396.548819 | 139.095 | 0.050 |\n| 2454480.372487 | 139.271 | 0.050 |\n| 2454705.645953 | 139.049 | 0.048 |\n| 2454780.529216 | 139.152 | 0.052 |\n| 2454819.336946 | 139.207 | 0.036 |\n| 2455059.684154 | 139.053 | 0.031 |\n| 2455149.486943 | 139.131 | 0.047 |\n| 2455207.337039 | 139.026 | 0.034 |\n| 2455232.321077 | 139.059 | 0.036 |\n| 2455415.734837 | 139.058 | 0.032 |\n| 2455439.622078 | 138.995 | 0.027 |\n| 2455485.577689 | 138.927 | 0.032 |\n| 2455503.470979 | 138.867 | 0.139 |\n| 2455776.741087 | 138.965 | 0.034 |\n| 2455821.588602 | 138.997 | 0.025 |\n| 2455882.542281 | 139.101 | 0.047 |\n| 2456139.725335 | 138.991 | 0.040 |\n| 2456276.485443 | 139.163 | 0.039 |\n| 2456519.736731 | 138.956 | 0.024 |\n| 2456574.672228 | 139.197 | 0.047 |\n| 2456956.570319 | 139.252 | 0.078 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454338.702068 | − 64.948 | 0.983 |\n| 2454373.722874 | − 64.876 | 1.097 |\n| 2454396.715153 | − 66.294 | 1.602 |\n| 2454406.637662 | − 64.942 | 1.710 |\n| 2454459.590478 | − 62.264 | 2.232 |\n| 2454480.431485 | − 65.026 | 1.202 |\n| 2454516.372208 | − 64.832 | 0.592 |\n| 2454780.591463 | − 64.462 | 1.896 |\n| 2454819.450079 | − 64.460 | 1.351 |\n| 2455075.713863 | − 65.352 | 1.544 |\n| 2455176.552628 | − 64.543 | 0.684 |\n| 2455232.419859 | − 65.290 | 1.106 |\n| 2455531.609740 | − 64.654 | 1.303 |\n| 2455620.393797 | − 63.236 | 2.281 |\n| 2455944.449119 | − 66.678 | 2.308 |\n| 2456191.699846 | − 64.856 | 1.104 |\n| 2456545.689792 | − 64.892 | 1.340 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454338.746568 | 231.895 | 0.043 |\n| 2454373.755033 | 231.272 | 0.022 |\n| 2454396.675349 | 231.387 | 0.045 |\n| 2454406.673221 | 231.488 | 0.047 |\n| 2454480.469787 | 231.625 | 0.037 |\n| 2454496.400729 | 231.149 | 0.135 |\n| 2454516.413321 | 231.777 | 0.020 |\n| 2454705.704003 | 230.516 | 0.030 |\n| 2454780.549248 | 230.670 | 0.046 |\n| 2454793.526401 | 230.398 | 0.173 |\n| 2454819.485376 | 231.059 | 0.041 |\n| 2454852.550506 | 231.299 | 0.045 |\n| 2455059.709842 | 254.627 | 0.039 |\n| 2455079.721105 | 255.868 | 0.030 |\n| 2455126.674591 | 252.221 | 0.032 |\n| 2455149.614597 | 250.514 | 0.037 |\n| 2455171.540400 | 248.975 | 0.021 |\n| 2455207.416924 | 246.988 | 0.029 |\n| 2455232.382885 | 245.710 | 0.030 |\n| 2455439.745696 | 240.146 | 0.032 |\n| 2455478.744628 | 239.853 | 0.067 |\n| 2455531.646433 | 239.862 | 0.033 |\n| 2455821.705466 | 236.600 | 0.024 |\n| 2455944.486938 | 236.352 | 0.046 |\n| 2455971.360048 | 236.071 | 0.040 |\n| 2456191.737737 | 234.847 | 0.023 |\n| 2456574.706961 | 233.444 | 0.020 |\n| 2456956.723997 | 231.920 | 0.055 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454396.640800 | 255.070 | 0.185 |\n| 2454406.717056 | 255.306 | 0.110 |\n| 2454480.496860 | 255.712 | 0.098 |\n| 2454820.481773 | 255.614 | 0.142 |\n| 2455149.688274 | 255.624 | 0.084 |\n| 2455207.491478 | 255.546 | 0.104 |\n| 2455503.608363 | 255.614 | 0.551 |\n| 2455531.544536 | 255.478 | 0.171 |\n| 2455882.598085 | 255.358 | 0.108 |\n| 2456005.376504 | 255.236 | 0.360 |\n| 2456207.754219 | 255.340 | 0.082 |\n| 2456603.661353 | 255.443 | 0.088 |\n| 2456956.750015 | 255.185 | 0.196 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454909.489857 | 342.628 | 3.089 |\n| 2454930.426948 | 369.639 | 1.671 |\n| 2455174.767340 | 379.567 | 2.399 |\n| 2455207.625989 | 362.229 | 1.939 |\n| 2455620.563564 | 384.868 | 1.939 |\n| 2455704.385256 | 343.075 | 2.077 |\n| 2455712.405032 | 335.836 | 1.872 |\n| 2455718.397062 | 358.699 | 2.243 |\n| 2455722.397011 | 375.517 | 1.896 |\n| 2455725.396840 | 386.965 | 3.807 |\n| 2455730.395197 | 385.629 | 2.338 |\n| 2455733.391004 | 378.210 | 3.132 |\n| 2455738.401973 | 354.412 | 1.072 |\n| 2456796.438819 | 363.768 | 1.346 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454459.682587 | 76.353 | 1.799 |\n| 2454464.668667 | 76.746 | 1.762 |\n| 2454516.585694 | 75.947 | 1.514 |\n| 2454909.650314 | 75.405 | 0.794 |\n| 2455232.610390 | 76.608 | 0.991 |\n| 2455344.429621 | 76.620 | 1.860 |\n| 2455531.735132 | 76.372 | 1.831 |\n| 2455662.477811 | 76.280 | 1.239 |\n| 2456033.449115 | 75.442 | 1.403 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454459.748479 | 229.676 | 2.402 |\n| 2454481.662602 | 229.183 | 1.172 |\n| 2454964.473253 | 228.850 | 1.277 |\n| 2455232.656932 | 228.583 | 1.005 |\n| 2455620.602643 | 229.334 | 1.314 |\n| 2455662.541359 | 228.736 | 1.308 |\n| 2456458.454152 | 229.734 | 0.725 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454481.728699 | 163.210 | 3.750 |\n| 2454625.460187 | 162.919 | 1.680 |\n| 2455620.657226 | 163.474 | 2.343 |\n| 2456090.413526 | 161.166 | 2.464 |\n| 2456652.702430 | 161.310 | 2.342 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454219.667720 | − 107.225 | 0.276 |\n| 2454314.443374 | − 108.199 | 0.203 |\n| 2454459.780220 | − 107.456 | 0.161 |\n| 2454464.791534 | − 107.738 | 0.217 |\n| 2454480.776288 | − 107.848 | 0.177 |\n| 2454625.420967 | − 107.371 | 0.217 |\n| 2454930.698058 | − 107.699 | 0.173 |\n| 2454951.700884 | − 107.748 | 0.256 |\n| 2454987.473997 | − 107.662 | 0.181 |\n| 2455232.782505 | − 107.709 | 0.145 |\n| 2455344.579755 | − 107.138 | 0.179 |\n| 2455620.687317 | − 107.719 | 0.243 |\n| 2455662.702207 | − 107.560 | 0.173 |\n| 2455704.614991 | − 107.359 | 0.157 |\n| 2455738.545590 | − 108.383 | 0.218 |\n| 2455776.455326 | − 108.773 | 0.204 |\n| 2456005.755704 | − 108.120 | 0.622 |\n| 2456033.623529 | − 108.201 | 0.206 |\n| 2456078.504562 | − 107.171 | 0.187 |\n| 2456712.726996 | − 107.900 | 0.651 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454219.705449 | − 163.546 | 0.016 |\n| 2454254.642226 | − 163.593 | 0.039 |\n| 2454285.507316 | − 163.348 | 0.015 |\n| 2454314.462351 | − 163.385 | 0.012 |\n| 2454625.508366 | − 163.091 | 0.010 |\n| 2454930.677427 | − 163.244 | 0.016 |\n| 2454951.718880 | − 162.971 | 0.017 |\n| 2454964.675531 | − 162.951 | 0.022 |\n| 2454987.509060 | − 163.041 | 0.014 |\n| 2455344.542474 | − 162.849 | 0.013 |\n| 2455415.386096 | − 163.549 | 0.010 |\n| 2455439.363821 | − 163.724 | 0.016 |\n| 2455620.738975 | − 162.875 | 0.017 |\n| 2455662.681656 | − 163.110 | 0.012 |\n| 2455704.633870 | − 163.372 | 0.016 |\n| 2455738.564258 | − 163.723 | 0.020 |\n| 2455776.438140 | − 163.732 | 0.016 |\n| 2456005.728789 | − 163.530 | 0.020 |\n| 2456078.539561 | − 163.442 | 0.015 |\n| 2456307.786409 | − 163.457 | 0.016 |\n| 2456351.717323 | − 163.347 | 0.014 |\n| 2456372.687769 | − 163.477 | 0.014 |\n| 2456399.619896 | − 163.371 | 0.015 |\n| 2456426.543882 | − 163.298 | 0.021 |\n| 2456474.451216 | − 163.108 | 0.011 |\n| 2456488.426825 | − 162.960 | 0.024 |\n| 2456489.500440 | − 163.000 | 0.015 |\n| 2456520.424407 | − 162.818 | 0.020 |\n| 2456529.377789 | − 162.938 | 0.017 |\n| 2456664.770980 | − 163.461 | 0.012 |\n| 2456685.790500 | − 163.626 | 0.011 |\n| 2456712.759468 | − 163.414 | 0.020 |\n| 2456796.646910 | − 162.880 | 0.011 |\n| 2456813.610868 | − 162.988 | 0.015 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.576915 | 13.178 | 0.053 |\n| 2454338.541838 | 13.553 | 0.136 |\n| 2454373.414188 | 13.559 | 0.053 |\n| 2454390.364640 | 13.579 | 0.079 |\n| 2454625.684458 | 13.594 | 0.058 |\n| 2454665.611641 | 13.917 | 0.063 |\n| 2454705.568239 | 13.747 | 0.051 |\n| 2455009.693164 | 13.480 | 0.098 |\n| 2455059.511399 | 13.695 | 0.076 |\n| 2455174.314885 | 13.374 | 0.050 |\n| 2455415.522393 | 13.471 | 0.061 |\n| 2455531.326532 | 13.467 | 0.108 |\n| 2455704.706380 | 13.742 | 0.090 |\n| 2455776.612557 | 13.367 | 0.049 |\n| 2455822.600538 | 13.505 | 0.103 |\n| 2455882.349889 | 13.624 | 0.051 |\n| 2456139.644767 | 13.413 | 0.087 |\n| 2456191.399426 | 13.585 | 0.099 |\n| 2456488.673600 | 13.584 | 0.066 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.626987 | − 113.002 | 0.070 |\n| 2454338.455766 | − 113.041 | 0.061 |\n| 2454373.470335 | − 113.043 | 0.113 |\n| 2454625.646394 | − 112.867 | 0.147 |\n| 2454665.630399 | − 112.981 | 0.118 |\n| 2454705.522460 | − 113.104 | 0.115 |\n| 2455009.711530 | − 113.138 | 0.099 |\n| 2455059.529868 | − 113.020 | 0.056 |\n| 2455070.548978 | − 112.989 | 0.159 |\n| 2455175.444551 | − 113.221 | 0.079 |\n| 2455344.685686 | − 112.945 | 0.119 |\n| 2455415.497735 | − 112.930 | 0.102 |\n| 2455439.426753 | − 112.968 | 0.103 |\n| 2455503.356818 | − 113.402 | 0.352 |\n| 2455531.410900 | − 113.220 | 0.136 |\n| 2455704.677515 | − 113.216 | 0.095 |\n| 2455776.507942 | − 112.924 | 0.093 |\n| 2455796.580352 | − 112.817 | 0.094 |\n| 2455882.328613 | − 113.245 | 0.106 |\n| 2456078.720195 | − 113.064 | 0.081 |\n| 2456140.610920 | − 113.674 | 0.186 |\n| 2456191.379978 | − 113.152 | 0.059 |\n| 2456488.693610 | − 113.205 | 0.084 |\n| 2456956.553882 | − 112.876 | 0.303 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454338.580931 | 147.980 | 1.346 |\n| 2454373.501898 | 148.051 | 1.007 |\n| 2454390.393739 | 148.275 | 1.573 |\n| 2454665.675576 | 147.791 | 1.192 |\n| 2455059.563356 | 147.724 | 1.446 |\n| 2455149.413670 | 148.224 | 1.470 |\n| 2455415.573319 | 148.194 | 1.123 |\n| 2455439.540189 | 147.532 | 1.306 |\n| 2455478.531528 | 147.567 | 0.899 |\n| 2455738.686456 | 147.855 | 0.872 |\n| 2455776.556273 | 147.947 | 1.562 |\n| 2455822.637317 | 148.094 | 1.569 |\n| 2456191.476165 | 148.111 | 0.922 |\n| 2456545.578750 | 147.649 | 1.362 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.636048 | − 121.161 | 0.010 |\n| 2454338.524441 | − 121.179 | 0.009 |\n| 2454373.528280 | − 121.107 | 0.009 |\n| 2454390.345759 | − 121.253 | 0.010 |\n| 2454406.586097 | − 121.114 | 0.011 |\n| 2454480.397616 | − 121.164 | 0.013 |\n| 2454625.656022 | − 121.267 | 0.012 |\n| 2454705.500584 | − 121.138 | 0.017 |\n| 2454780.452621 | − 121.111 | 0.012 |\n| 2454793.471559 | − 121.182 | 0.011 |\n| 2454820.348668 | − 121.321 | 0.014 |\n| 2455009.719853 | − 121.078 | 0.011 |\n| 2455059.695625 | − 121.185 | 0.009 |\n| 2455071.639351 | − 121.199 | 0.017 |\n| 2455171.399245 | − 121.340 | 0.008 |\n| 2455344.693948 | − 121.230 | 0.011 |\n| 2455415.506520 | − 121.259 | 0.012 |\n| 2455439.446714 | − 121.227 | 0.008 |\n| 2455503.346059 | − 121.358 | 0.014 |\n| 2455531.470241 | − 121.412 | 0.011 |\n| 2455704.717098 | − 121.256 | 0.009 |\n| 2455738.710786 | − 121.253 | 0.018 |\n| 2455776.728437 | − 121.333 | 0.012 |\n| 2455796.566301 | − 121.342 | 0.011 |\n| 2455859.532089 | − 121.249 | 0.009 |\n| 2455892.424873 | − 121.011 | 0.029 |\n| 2455915.320942 | − 121.166 | 0.077 |\n| 2456140.740348 | − 121.075 | 0.010 |\n| 2456241.340390 | − 121.056 | 0.013 |\n| 2456488.721460 | − 121.000 | 0.010 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.742671 | − 168.937 | 0.074 |\n| 2454373.581743 | − 169.408 | 0.066 |\n| 2454705.601838 | − 169.668 | 0.074 |\n| 2455059.649134 | − 169.963 | 0.071 |\n| 2455175.377253 | − 168.867 | 0.130 |\n| 2455415.665073 | − 169.830 | 0.078 |\n| 2455776.636347 | − 169.582 | 0.071 |\n| 2455796.640879 | − 169.184 | 0.055 |\n| 2455821.555408 | − 169.165 | 0.065 |\n| 2455859.497621 | − 169.214 | 0.092 |\n| 2455882.472080 | − 169.087 | 0.100 |\n| 2456139.682491 | − 169.008 | 0.242 |\n| 2456530.669987 | − 169.196 | 0.120 |\n| 2456987.357305 | − 169.738 | 0.077 |\nLiterature data for the programme stars\n=======================================\n[ht]\n[tbl-B1]\n| | | | | | |\n| --- | --- | --- | --- | --- | --- |\n| Star ID | ΔT Total | $\\bar{RV}$ (this work) | $\\bar{RV}$ (lit) | N | Ref |\n| | (days) | (km s− 1) | (km s− 1) | | |\n| | 10011 | − 57.914 | − 57.4 | 3 | |\n| | | | − 57.2 | 13 | |\n| | | | − 57.5 | 1 | |\n| | 4742 | + 105.008 | + 105.1 | 1 | |\n| | 5193 | + 139.068 | + 139.1 | 8 | |\n| | | | + 138.9 | 2 | |\n| | | | + 138.2 | 1 | |\n| | | | + 138.9 | 1 | |\n| | | | + 139.4 | 1 | |\n| | | | + 139.0 | 1 | |\n| | | | + 138.4 | 1 | |\n| | | | + 138.9 | 1 | |\n| | 3582 | − 64.800 | − 66.6 | 1 | |\n| | 4032 | + 255.425 | + 255.3 | 1 | |\n| | 3267 | + 76.197 | + 77.0 | 1 | |\n| | 3785 | + 229.157 | + 228.5 | 1 | |\n| | 3885 | + 162.416 | + 163.6 | 1 | |\n| | | | + 163.1 | 5 | |\n| | 3942 | − 107.749 | − 107.4 | 1 | |\n| | 8788 | + 13.549 | + 13.1 | 10 | |\n| | | | + 13.6 | 1 | |\n| | | | + 12.5 | 15 | |\n| | | | + 13.2 | 1 | |\n| | | | + 12.7 | 1 | |\n| | | | + 14.5 | 1 | |\n| | | | + 13.3 | 1 | |\n| | | | + 13.0 | 2 | |\n| | 3968 | − 113.085 | − 112.3 | 1 | |\n| | 3960 | + 147.928 | + 148.1 | 1 | |\n| | 10921 | − 121.201 | − 119.0 | 4 | |\n| | | | − 123.7 | 3 | |\n| | | | − 121.8 | 18 | |\n| | | | − 120.7 | 1 | |\n| | | | − 121.7 | 1 | |\n| | | | − 121.8 | 1 | |\n| | 4741 | − 169.346 | − 169.2 | 1 | |\n[ht] [tbl-B2]\n| | | | | |\n| --- | --- | --- | --- | --- |\n| Star ID | ΔT Total | $\\bar{RV}$ | N | Ref |\n| | (days) | (km s− 1) | | |\n| | 8812 | − 27.6 | 3 | |\n| | | − 26.2 | 2 | |\n| | | − 27.1 | 2 | |\n| | | − 27.2 | 1 | |\n| | 21575 | − 26.1 | 3 | |\n| | | − 26.5 | 6 | |\n| | | − 24.9 | 6 | |\n| | | − 26.0 | 1 | |\n| | | − 26.0 | 1 | |\n| | | − 26.4 | 1 | |\n| | | − 25.6 | 2 | |\n| | | − 26.1 | 3 | |\n---\n1. http://www.not.iac.es/instruments/fies/[↩](#fnref1)\n\nThe role of binaries in the enrichment of the early Galactic halo.\n==================================================================\nI. *r*-process-enhanced metal-poor stars\n========================================\nOur aim is to determine the frequency and orbital properties of binaries among these chemically peculiar stars. This information provides the basis for deciding whether local mass transfer from a binary companion is necessary and sufficient to explain their unusual compositions. This paper discusses our study of a sample of 17 moderately (*r*-I) and highly (*r*-II) *r*-process-element enhanced VMP and EMP stars. High-resolution, low signal-to-noise spectra of the stars were obtained at roughly monthly intervals over eight years with the FIES spectrograph at the Nordic Optical Telescope. From these spectra, radial velocities with an accuracy of ∼ 100 m s− 1 were determined by cross-correlation against an optimized template. Fourteen of the programme stars exhibit no significant radial-velocity variation over this temporal window, while three are binaries with orbits of typical eccentricity for their periods, resulting in a normal binary frequency of ∼ 18 ± 6% for the sample. Our results confirm our preliminary conclusion from 2011, based on partial data, that the chemical peculiarity of the *r*-I and *r*-II stars is not caused by any putative binary companions. Instead, it was imprinted on the natal molecular clouds of these stars by an external, distant source. Models of the ISM in early galaxies should account for such mechanisms.\nIntroduction\n============\n[tbl-1]\n| Stellar ID | RA (J2000) | Dec (J2000) | *V* | *B* − *V* | [Fe/H] | [Eu/Fe] | Class |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| | 00:05:15 | − 27:16:18 | 9.24 | 0.54 | − 1.58 | + 0.80 | *r*-I |\n| | 00:28:07 | − 26:03:03 | 14.03 | 0.70 | − 2.81 | + 1.62 | *r*-II |\n| | 01:29:31 | − 16:00:48 | 11.67 | 0.77 | − 2.78 | + 1.66 | *r*-II |\n| | 04:34:26 | − 09:16:50 | 15.17 | 0.73 | − 3.19 | + 1.25 | *r*-II |\n| | 04:44:52 | − 12:28:46 | 12.91 | 1.08 | − 2.41 | + 0.52 | *r*-I |\n| | 05:27:04 | − 20:52:42 | 14.01 | 0.88 | − 2.58 | + 0.49 | *r*-I |\n| | 10:47:16 | − 25:25:17 | 14.35 | 0.67 | − 2.89 | + 0.94 | *r*-I |\n| | 11:07:49 | + 00:11:38 | 15.65 | 0.39 | − 2.42 | + 1.81 | *r*-II |\n| | 11:29:51 | − 12:00:13 | 15.89 | 0.68 | − 2.73 | + 1.08 | *r*-II |\n| | 12:21:34 | − 03:28:40 | 15.94 | 0.64 | − 2.81 | + 1.41 | *r*-II |\n| | 14:33:17 | + 00:40:49 | 13.69 | 0.59 | − 3.03 | + 0.72 | *r*-I |\n| | 15:26:01 | − 09:11:38 | 11.13 | 1.06 | − 2.95 | + 1.82 | *r*-II |\n| | 22:17:01 | − 16:39:26 | 13.21 | 0.80 | − 2.95 | + 1.54 | *r*-II |\n| | 22:27:23 | + 01:58:33 | 13.68 | 0.71 | − 2.58 | + 1.05 | *r*-II |\n| | 22:47:26 | − 14:47:30 | 15.35 | 0.66 | − 2.88 | + 0.95 | *r*-I |\n| | 23:29:29 | + 30:25:57 | 7.71 | 1.02 | − 2.14 | + 0.85 | *r*-I |\n| | 23:34:27 | − 26:42:19 | 13.66 | 0.92 | − 3.33 | + 0.72 | *r*-I |\nThe pioneering HK survey of Beers, Preston, & Shectman demonstrated the existence of substantial numbers of stars with metallicities well below those of globular clusters, and opened a new observational window on the epoch of early star formation in the Galactic halo. Modern surveys of metal-poor stars has greatly expanded the numbers of such stars known at present, and spectroscopy with 8-metre class telescopes have demonstrated a surprisingly uniform abundance pattern in the majority of them.\nSince very metal-poor (VMP; [Fe/H] < − 2.0) and extremely metal-poor (EMP; [Fe/H] < − 3.0) halo stars probe the earliest epochs of chemical evolution in the Galaxy, their elemental-abundance patterns reflect the products of the primary heavy-element synthesis and enrichment processes in the early Galaxy. However, significant samples of chemically peculiar stars have been identified, in particular among the VMP and EMP stars. Because anomalies affecting a single element or group of elements must be produced by a small number of nucleosynthesis events (including possible single events), the chemically peculiar stars provide the opportunity to characterize the progenitors and enrichment processes that produced the abundance patterns in the long-lived metal-poor stars seen today.\nHowever, in the relatively more metal-rich Population I and II stars, some chemical anomalies are known to be due to evolution in a close binary system (e.g. Ba and CH stars), in which a higher-mass binary companion has evolved to the asymptotic giant-branch (AGB) stage. Through processes involving Roche-lobe overflow and/or wind accretion, the higher-mass star is expected to have contaminated the envelope of its surviving companion with any elements that formed during its final evolutionary stages. This mechanism could explain the presence of elements produced in the early Universe by stars more massive than ∼ 0.7 M$\\_{\\sun}$ (i.e. with main-sequence lifetimes less than the Hubble time), but does of course require the presence of a binary companion, which by now is presumably a white dwarf or neutron star.\nHence, for the chemically peculiar VMP and EMP stars, a crucial first step is to establish whether any such putative binary companions actually exist, and to determine their main orbital parameters (i.e. period, semi-major axis, and eccentricity). This requires radial-velocity monitoring of adequate precision and duration for a sufficiently large sample of stars. High spectroscopic resolution ($R\n~\\ga30,000$) allows one to determine radial velocities efficiently by cross-correlating even a relatively low signal-to-noise ratio (S/N) spectrum with an appropriate template, while long-term thermal and mechanical stability of the spectrograph employed is needed in order to detect variations with amplitudes below 1 km s− 1. As we demonstrate below, the FIES spectrograph at the 2.56-m Nordic Optical Telescope (NOT), which we have used to carry out our programme, meets these requirements.\nThis paper is the first in a series that explores the nature of three broad classes of chemically peculiar metal-poor halo stars. Here we consider the nature of VMP and EMP stars that exhibit moderate-to-large enhancements of elements associated with the rapid neutron-capture process, the so-called *r*-I and *r*-II stars, respectively. Other papers in this series consider samples of carbon-enhanced metal-poor (CEMP) stars : the CEMP-no stars (which exhibit no over-abundances of neutron-capture process elements; Paper II), and the CEMP-*s* stars (which exhibit over-abundances of elements originating in the slow neutron-capture process; Paper III).\nThis paper is structured as follows: Section 2 describes the sample of *r*-process-enhanced stars explored in our programme. Sections 3 and 4 describe in detail the observations, and the reduction and analysis procedures employed in the papers of this series. Section 5 summarizes our results, while Section 6 discusses these results and their implications for the progenitors of the *r*-I and *r*-II stars and the associated enrichment processes. Section 7 presents our final conclusions.\nSample stars\n============\nAll but two of our sample stars were selected from the HK survey of Beers and colleagues and the Hamburg/ESO survey of Christlieb and collaborators. The sample includes a number of the canonical examples of the *r*-process-element enhancement phenomenon;, the first EMP star with detected Th,, the first EMP star with detected U, and, the most extremely *r*-process enhanced giant known. All except HE 1523 − 0901 were analysed in the Hamburg/ESO R-process Enhanced Star (HERES) survey. Accordingly, most of our programme stars are in the Southern Hemisphere (but north of declination $\\delta\n\\sim -25\\degr$) and have $V\\lesssim16$, which is the practical limit for 1-hour integrations with the NOT.\nThe sample stars are listed in Table [tbl-1], including their *V* magnitudes and *B* − *V* colours, and reported [Fe/H] and [Eu/Fe] abundances. The last column of this table indicates whether a given star is considered to be a member of the moderately *r*-process-enhanced class (*r*-I; + 0.3 ≤ [r/Fe] ≤ + 1.0) or the highly *r*-process-enhanced class (*r*-II; [r/Fe] > + 1.0), according to the definitions of.\nObservations\n============\nSpectra for our radial-velocity monitoring programme were obtained with the NOT, in service mode, using the FIES spectrograph[1](#fn1), which has been used successfully for exo-planet research. The spectra cover the wavelength range 3640Å − 7360Å in 78 orders, at a resolving power of *R* ∼ 46,000. The S/N of the spectra is ∼ 10 on average, but ranges from ∼ 2 to ∼ 20. A S/N of ∼ 10 is obtained in ∼ 20 min for a star of *V* = 14.5, so a typical clear night yielded ∼ 10-15 spectra of the stars in Table [tbl-1]. Integrations of 900 s or longer were split into three exposures, in order to enable effective cosmic ray rejection.\nThe observing strategy was based on the assumed analogy with the Ba II and CH binaries found by and to have periods of the order of ∼ 300-3000 days and amplitudes of ∼ 3-10 km s− 1. Accordingly, spectra have been obtained at roughly monthly intervals since June 2007, and reduced immediately, so that follow-up of any variable objects could be planned efficiently. As we demonstrate, this strategy has worked well.\nReduction and analysis\n======================\nThe observations were reduced with pipeline software originally developed by Lars Buchhave to deliver high-precision radial velocities of exo-planet host stars from echelle spectrographs, in particular the FIES instrument.\nThis reduction procedure includes all the normal steps, such as bias subtraction, division by a flat-field exposure, cosmic ray removal, and 2-D order extraction. Sky background is only significant for the faintest stars if observed close to the full Moon, which was avoided in the nightly planning. Moreover, even substantial amounts of scattered moonlight are harmless if the two cross-correlation peaks are well separated, which is virtually always the case for our high-velocity programme stars.\nFor the wavelength calibration, a separate wavelength solution is created for each target spectrum, using Th-Ar calibration spectra taken just before each science frame. This procedure has been found to yield adequate velocity stability, as we demonstrate below.\nMulti-order cross-correlation\n-----------------------------\nWith the reduced spectra in hand, multi-order cross-correlation against an optimized template spectrum is then performed, using software also developed by L. Buchhave. The radial velocity from each individual order is determined by a Gaussian fit to the peak of the cross-correlation function (CCF); their mean value, weighted by the total photon count in each order, is taken as the final radial velocity from the observation.\nPerforming the cross-correlation order-by-order enables us to hand-pick the spectral regions to be used in the correlation, including regions with strong absorption lines and excluding regions with only a few and/or weak lines, which is a significant advantage when dealing with spectra of stars of such peculiar chemical compositions. Filtering is also applied to the spectrum before the cross-correlation to remove unwanted frequencies. The filters are carefully optimized for each star to remove noise while retaining even the narrowest stellar absorption lines.\nOptimization of the template spectra\n------------------------------------\nThe choice of the template spectrum for the cross-correlation is crucial for the accuracy of the resulting radial velocities, especially in these spectra where the usual iron-peak elements only show weak lines, but strong lines exist from the normally rare neutron-capture elements.\nFour different recipes have been used to construct the template spectra, depending on the quality of the target spectra: “Strongest”, “Co-add”, CS 31082 − 001, and “Delta”. The Strongest template is the spectrum of a given star with the maximum signal level for that star. The advantage of using a spectrum of the same star as a template is the perfect match to the observed spectrum; the disadvantage is that a template with relatively low S/N will introduce noise into the correlation. The Co-add template is constructed by shifting a selection of the best spectra of the star to a common radial velocity and co-adding them. This results in a template with high S/N (which is also a perfect match for the target spectra), and will generally allow more orders to be included in the correlation, compared to a correlation with the strongest single spectrum as template. However, when creating the Co-add template, an initial correlation with the strongest spectrum as template is used to determine the shift of the other spectra. Any small residual shift will then broaden the spectral lines in the Co-add template spectrum and yield less precise results in the final correlation.\nFor the fainter stars, the Strongest template may introduce too much noise into the correlation, and a good Co-add template cannot then be constructed. Instead, a Co-add template from a bright star with a very similar spectrum may then be used as template; for these stars we have used CS 31082 − 001 in this manner. The final template option, used here for the faintest stars, is the Delta template, a synthetic spectrum consisting of *δ* functions at the (solar) wavelengths of selected lines. Sample spectra of the faint stars have been inspected for strong lines to be included in the Delta template spectrum. Correlation with this template yields velocities on an absolute scale, and has thus also been used to determine the absolute velocity of the other templates and to convert all velocities to an absolute scale.\n[tbl-2]\n| | | | | | | | | |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| Stellar ID | RA (J2000) | Dec (J2000) | *B* | *V* | RV mean | *σ* | Nobs | ΔT |\n| | | | | | (km s− 1) | (km s− 1) | | (days) |\n| | 00:40:49 | + 40:11:14 | 8.30 | 7.36 | − 63.35 | 0.033 | 60 | 2672 |\n| | 05:46:02 | + 37:17:05 | 8.19 | 7.36 | − 29.14 | 0.035 | 46 | 2771 |\n| | 09:14:23 | + 52:41:12 | 9.07 | 7.63 | + 10.45 | 0.048 | 29 | 2746 |\n| | 13:16:51 | + 17:01:02 | 7.46 | 6.52 | + 7.67 | 0.036 | 34 | 2891 |\n| | 16:42:39 | + 68:11:18 | 8.32 | 7.56 | + 9.44 | 0.028 | 30 | 2857 |\n| | 19:23:34 | + 33:13:19 | 7.15 | 6.36 | − 21.65 | 0.035 | 61 | 2072 |\n| | 20:40:45 | + 19:56:08 | 7.06 | 6.44 | − 35.37 | 0.052 | 26 | 2857 |\n[tbl-3]\n| | | | | | | |\n| --- | --- | --- | --- | --- | --- | --- |\n| Stellar ID | Nobs | Template | RV mean | *σ* | ΔT | Binary |\n| | | | (km s− 1) | (km s− 1) | (days) | |\n| | 14 | Strongest | − 57.914 | 0.041 | 2603 | No |\n| | 12 | Co-add | + 105.008 | 0.366 | 2583 | No |\n| | 24 | Co-add | + 139.068 | 0.105 | 2642 | No |\n| | 18 | Delta | − 64.800 | 0.988 | 2737 | No |\n| | 28 | Co-add | + 237.805 | 8.294 | 2618 | Yes |\n| | 13 | CS 31082 − 001 | + 255.425 | 0.195 | 2338 | No |\n| | 14 | Delta | + 365.789 | 17.110 | 1887 | Yes |\n| | 9 | Delta | + 76.197 | 0.496 | 1573 | No |\n| | 7 | Delta | + 229.157 | 0.454 | 1998 | No |\n| | 5 | Delta | + 162.416 | 1.094 | 2171 | No |\n| | 20 | Co-add | − 107.749 | 0.426 | 2493 | No |\n| | 34 | Co-add | − 163.271 | 0.284 | 2594 | Yes |\n| | 19 | Co-add | + 13.549 | 0.164 | 2174 | No |\n| | 24 | Co-add | − 113.085 | 0.190 | 2420 | No |\n| | 14 | Delta | + 147.928 | 0.246 | 2207 | No |\n| | 30 | Strongest | − 121.201 | 0.105 | 2174 | No |\n| | 14 | Co-add | − 169.346 | 0.352 | 2672 | No |\nStandard stars\n--------------\nA few well-established radial-velocity standard stars from Table 2 of were observed on every usable observing night in order to monitor any zero-point variations in the velocities. These are listed in Table [tbl-2], along with our mean (heliocentric) velocities and dispersions for these stars. The mean difference of our velocities from the standard values is 73 m s− 1, with a standard deviation of 69 m s− 1. This demonstrates that our results for the target stars are on a system consistent with that of, and that the accuracy of our results is not limited by the stability of the spectrograph.\nError estimates\n---------------\nThe internal error on the mean velocity from each spectrum is calculated as the standard deviation of the velocities from the individual orders used in the correlation. It is listed along with each observed velocity in Appendix A, and is used to plot the error bars of the data in the orbital plots shown in Fig. [fig:orbit]. For each star, the average internal error on the velocities is computed as the mean of these internal standard deviations, and is used to assess the quality of the correlation and the order selection.\nThe *external* standard deviation, *σ*, of the radial-velocity observations for each star, given in Table [tbl-2] for the radial-velocity standards and in Table [tbl-3] for the programme stars, is computed as\n \n$$\\begin{aligned}\n\\sigma = \\sqrt{\\frac{1}{N-1}\\sum^{N-1}\\_{j-0}(v\\_j-\\bar{v})^2}.\\end{aligned}$$ \nImperfect guiding and centring of the star on the fibre end along with imperfect cancellation of changes in ambient temperature and atmospheric pressure, etc., contributes to small variations in the derived radial velocities, which are reflected in the standard deviations of the constant programme and radial-velocity standard stars.\nFor the three detected binaries in our sample, the above value for *σ* is inflated due to the orbital motion, and the relevant uncertainty estimate is the *σ* from the orbital solutions given in Table [tbl-4].\nResults\n=======\n[tbl-4]\n| Parameter | | | |\n| --- | --- | --- | --- |\n| Period (days) | 2515.2 ± 5.5 | 36.561 ± 0.009 | 303.05 ± 0.25 |\n| *T*0 (BJD) | 2457918.3 ± 1.0 | 2455737.1 ± 0.1 | 2455068.0 ± 4.1 |\n| *K* (km s− 1) | 12.541 ± 0.016 | 27.024 ± 0.96 | 0.350 ± 0.003 |\n| *γ* (km s− 1) | 236.35 ± 0.01 | 360.22 ± 0.57 | -163.23 ± 0.003 |\n| *e* | 0.767 ± 0.001 | 0.000 | 0.163 ± 0.010 |\n| *ω* $\\degr$ | 316.6 ± 0.1 | 90.00 | 81.7 ± 4.7 |\n| *a*sin *i* ($R\\_{\\sun}$) | 400.2 ± 1.2 | 19.6 ± 0.7 | 2.07 ± 0.02 |\n| *f*(*m*) ($M\\_{\\sun}$) | 0.136 ± 0.001 | 0.075 ± 0.009 | 1.3E-5 ± 4E-8 |\n| *σ* (km s− 1) | 0.28 | 1.88 | 0.11 |\n| *R**R**o**c**h**e* ($R\\_{\\sun}$, *M*1 = 0.8 $M\\_{\\sun}$, *M*2 = 0.6 $M\\_{\\sun}$) | 59 | 13 | 53 |\n| *R**R**o**c**h**e* ($R\\_{\\sun}$, *M*1 = 0.8 $M\\_{\\sun}$, *M*2 = 1.4 $M\\_{\\sun}$) | 770 | 46 | 220 |\nThe results of our radial-velocity monitoring of the sample stars are summarized in Table [tbl-3], which lists the star name, the number of observations (Nobs), the template used for each star, the mean (heliocentric) radial velocity and standard deviation over the observed time span (ΔT), and the binary status for each star. The individual observed heliocentric radial velocities are listed in Appendix A, together with the Julian dates of the observations and their internal errors.\nAs can be seen from a glance at Table [tbl-3], fourteen of our stars exhibit no variation in their radial velocities at the level of a few hundred metres per second over the eight years of monitoring. For the brighter targets, the standard deviations of the observed velocities are ∼ 100 m s− 1 (dominated by centring and guiding errors), rising to ∼ 1 km s− 1 for the fainter targets, due to the lower S/N of their spectra. Moreover, least-squares fits of the velocities vs. time reveal no net trends over the observing period.\nComparison with literature data\n-------------------------------\nTable [tbl-B1] in appendix B lists mean radial velocities (based on high-resolution spectroscopy) from the literature for the single stars in our sample, along with the complete time span covered by the combined data (including our own measurements). Most of the stars have only been observed once earlier by, for which they estimate an error of a few km s− 1, and we find excellent agreement between their results and ours. For HD 20 and HD 221170, radial velocities were also reported by, who list 13 observations for HD 20 and 18 for HD 221170, spanning 4641 and 5145 days, with (external) standard deviations of 0.41 and 0.61 km s− 1, respectively. Their mean radial velocities of − 57.18 km s− 1 and − 121.77 km s− 1 are consistent with our results, given the slight offset of the CfA velocities from the system of.\nFor the faint star HE 1219 − 0312, our five measurements with the 2.5-m NOT span 2171 days from 2007, while the five epochs of the earlier VLT/UVES observations span ∼ 400 days. The standard deviations ( ∼ 1 km s− 1) and their mean radial velocity (163.1 km s− 1) are similar to ours, and consistent with our own conclusion that this star is single.\nFrom inspection of Table [tbl-B1] it is clear that, while the coverage of the listed spans for most of the single stars is rather sparse, the spans themselves are from two to five times longer than those obtained during the course of our own radial-velocity monitoring programme. The lack of observed variations beyond what can be accounted for by the expected errors across multiple spectrograph/telescope combinations strengthens our claim that the stars we classify as single are indeed so.\nBinary orbits\n-------------\nThree of our stars are spectroscopic binaries, as first reported by. HE 0442 − 1234 was found already by P. Bonifacio et al. (private comm.) to be a long-period binary, while HE 1044 − 2509 and HE 1523 − 0901 are new discoveries from our programme. All three orbits have now been fully completed, with particular attention being paid to assessing the reality of the very low-amplitude velocity variations of HE 1523 − 0901. After rejecting early observations made under poor conditions (strong moonlight, poor seeing), and with now almost nine orbital revolutions completed, we have satisfied ourselves that this nearly face-on orbit is real and the orbital parameters are reliable (see below).\nIn contrast to this result, our own (far more accurate) velocity data clearly disprove the putative orbit suggested by for CS 22892-052, with *P* ∼ 128 days and *K* ∼ 1.0 km s− 1. Both this star and CS 31082 − 001 are clearly single stars, as can be seen from Table [tbl-3] and Figure [fig:single], where the derived radial velocities for the two stars are plotted as a function of time. Our results for stars observed with similar time spans and accuracies in parallel parts of our overall programme show that binaries with orbital periods of 20-30 years are detected with certainty even after the first couple of years with ∼ monthly observations.\n[fig:single]\nThe observed *K* ∼ 350 m s− 1 of HE 1523 − 0901 is the smallest measured with certainty in our programme, highlighting the null result for the other, single stars. The corresponding tiny mass function implies either a companion of mass in the brown-dwarf range (for $i \\sim\n90\\degr$), or a very low orbital inclination, or a combination of both. Assuming $i \\la2.5\\degr$ leads to a secondary mass in the late M-dwarf range, 0.25 $M\\_\\sun$ – still plausible within the statistics for a single case in a sample of several tens of potential (southern) *r*-I and *r*-II targets.\nLong-period envelope pulsations are a potential alternative origin of radial-velocity variations for this and other cool giant stars, and we have also found some strongly carbon-enhanced VMP and EMP stars showing similar low-level radial-velocity fluctuations. The frequency of real spectroscopic binaries with very low inclinations is far too low to ascribe all such low-amplitude velocity variations to binary orbital motion, and most marginal variations are, in the end, found not to be strictly periodic. Thus, it may be that low-level pulsations, rather than velocity accuracy, may set the ultimate limit to the length of the binary periods that can be reliably detected by the radial-velocity technique, as is the case for exo-planet orbits.\nOur current knowledge of the pulsational characteristics of late-type evolved stars is derived from the systematic microlensing surveys of MACHOs towards the Magellanic Clouds. The late-type pulsators with periods of 200-1000 days are typically C-type AGB stars or Mira variables, but stars below the tip of the red-giant branch have shorter periods, and the cause of their light variation is not known. Precise light and colour curves for field VMP/EMP stars with equally well-determined distances, or at least log *q*, periods or pseudo-periods of the order of a year, and phasing consistent with pulsations would be needed to settle the issue definitively. However, obtaining them with sufficient accuracy is not an easy task, and none has been reported for HE 1523 − 0901.\nThe final orbital elements for the three binary systems among our programme stars are listed in Table [tbl-4], and velocity curves with all available observations are shown in Figure [fig:orbit].\n[fig:orbit]\nDiscussion\n==========\nBinary frequency of *r*-I and *r*-II stars\n------------------------------------------\nThe salient result of our study is that only 3 of our 17 programme stars are binaries, while 14 are confirmed to be single stars, a binary frequency of ∼ 18 ± 6%.\nOur sample is relatively small, since HERES stars south of the NOT limit could not be observed by us, but only two other *r*-process-enhanced stars, the *r*-II star and the *r*-I star, have been reported in the literature to show variable radial velocities. was discovered by ; their data cover a range of ∼ 4.3 years, during which the radial velocity of the star varied by ∼ 20 km s− 1. The other star,, is included in the sample of, who reported on the identification of nine new *r*-process-enhanced stars. The radial-velocity data for neither of these stars is sufficient to derive an orbital solution for the systems. The bright *r*-I star was not observed in our programme, but literature data confirm (with variations of no more than 1 km s− 1 over a sparsely-sampled range of 24 years; see Table [tbl-B2]) that it too is vey likely a single star, as is its *r*-process-*poor* counterpart (with variations of no more than 1.5 km s− 1 over a sparsely-sampled range of 59 years; see Table [tbl-B2]).\nIn summary, the binary frequency of 18 ± 6% found here for the *r*-I and *r*-II stars is completely consistent with the 16 ± 4% of binaries with periods up to 6,000 days found by in their survey of 91 metal-poor field giants, and the ∼ 22% binaries with periods up to ∼ 15,000 days found by in their sample of ∼ 1,300 Population I giants in Galactic open clusters. A binary frequency of 100% for this class of VMP and EMP stars is clearly ruled out, a conclusion that would only be reinforced if HE 1523 − 0901 were eventually proved to be a single pulsating star rather than a binary with a nearly face-on orbit.\nOverall, we can thus conclude that the observed dramatic excess of *r*-process elements in our sample of stars is not just a surface effect produced locally by a binary companion, but rather was produced by a remote source and imprinted on the parent clouds across interstellar distances. It also seems hard to imagine how the local binary scenario could produce stars like, with a *deficit* of *r*-process elements relative to the standard abundance pattern.\n[Peplotr]\nFrequency and properties of *r*-I and *r*-II binaries\n-----------------------------------------------------\nThe distribution of periods and eccentricities of our three binaries is also fully as expected for binary systems with normal giant primary stars. This is illustrated most simply in Figure [Peplotr], which shows a period – eccentricity diagram constructed from the orbital data for 141 giant binary members of (Population I) Galactic open clusters by and, plotted as dots, while our three binaries are shown as red plus symbols. As seen, their orbital eccentricities are completely normal for giant binaries, which are typically tidally circularized for periods up to ∼ 150 days, depending on age and stellar mass, and show no sign of tidal or other processes that could be connected to their outstanding chemical peculiarity. In contrast, the transition between circular and eccentric orbits seems to occur at periods of ∼ 4 − 800 days for binaries with former AGB companions.\nFurther information on the presently unseen companions can be derived by considering the volume available to them during their evolution; i.e. their Roche-lobe radii, which can be computed from the observed orbital separation (assuming no exchange or loss of mass and angular momentum) and a range of assumed stellar masses. We first adopt a common mass of 0.8 $M\\_\\sun$ for the observed EMP giants and assume that the companion is a (sub)dwarf (i.e. unevolved) star of mass 0.6 $M\\_\\sun$, at least three magnitudes fainter than the star we do see. The present radius of the latter can be estimated from the adopted mass and a typical log *g* ∼ 1.5 dex, i.e. R ∼ 30 $R\\_\\sun$.\nWith the observed orbital elements, notably *a*1sin*i* and the mass function, we then adjust *i* until the computed *M*2 reaches 0.6 $M\\_\\sun$, and calculate the corresponding Roche-lobe radii, which are given in Table [tbl-4]. For HE 0442 − 1234, the minimum secondary mass is 0.67 $M\\_\\sun$ already for *i* = 90$\\degr$; for HE 1044 − 2509 and HE 1523 − 0901, we find *i* = 62$\\degr$ and 1.3$\\degr$, respectively, for a secondary mass of 0.6 $M\\_\\sun$.\nHowever, we might alternatively assume that the companions were initially *more* massive than 0.8 $M\\_\\sun$, i.e. in the range of 1-8 $M\\_\\sun$, where they would have gone through the AGB phase and likely have evolved into now-invisible white dwarfs (WDs) with typical masses of 0.6 $M\\_\\sun$, and thus identical R*R**o**c**h**e* to the ones found above. It is possible that the WD might also have the maximum WD mass of 1.4 $M\\_\\sun$, in which case the entire system would be much larger and the Roche-lobe radius of the companion a larger share of that again. Nominal Roche-lobe radii for this case are also given in Table [tbl-4], and illustrate the dramatic change in the space available to the star during its evolution, depending on its assumed mass.\nFor HE 1044 − 2509, even this R*R**o**c**h**e* is still too small to accommodate a typical AGB star of ∼ 200 $R\\_\\sun$. In any case, the observed absence of any *s*-process signatures indicates that mass transfer from a putative AGB companion did not happen in any of these systems, while a supernova explosion of a putative, even more massive companion would likely have disrupted the binary system.\nIn summary, binary systems seem to occur as a normal part of the formation of *r*-process-enhanced metal-poor stars, and are unrelated to the process by which they acquired their outstanding chemical anomalies.\nOrigin of *r*-I and *r*-II stars\n--------------------------------\nThe separation of *r*-process-enhanced stars into the *r*-I and *r*-II classes was originally a matter of convenience, since the heavy-element abundance patterns of the most enhanced stars could be more easily observed due to their relatively stronger lines at low metallicity. As the sample of such stars has grown, it has become clear that these classes also exhibit rather different behaviour with metallicity; the *r*-II stars are found in a relatively narrow range of metallicity near [Fe/H] ∼ − 3.0, while the *r*-I stars cover a larger range of metallicity, − 3.5 < [Fe/H] < − 1.5. It remains unclear whether this constraint implies that different classes of astrophysical progenitors might be responsible for the *r*-I and *r*-II stars, or whether the initial *r*-process content in the natal clouds of the *r*-I stars has simply been diluted by the elemental mix of standard chemical evolution.\nAll of our programme stars are giants. However, it is important to note that have shown that the star is a cool (T$\\_{\\rm{eff}} \\sim $5000 K) main-sequence dwarf with [Fe/H] = − 3.4, and [Eu/Fe] = + 2.0, making it simultaneously the lowest metallicity and most Eu-enriched *r*-II star yet found. This star is of particular interest, as all previous *r*-process-enhanced stars identified to date have been in more evolved stages of evolution. Since dwarfs with the temperature of SDSS J2357 − 0052 do not have convective atmospheres, it can be reasonably concluded that the *r*-process-enhancement phenomenon is not due to some chemical peculiarity arising from the presence of a convective envelope in such stars. Unfortunately, although this star is sufficiently bright (*V* ∼ 15.6) for the programme described above, it was discovered too late to be included in our target list, but the results of radial-velocity monitoring over about a year by Aoki et al. did not reveal any evidence of significant variation.\nThe nine newly recognized *r*-process-enhanced stars (based on high-resolution spectroscopic follow-up of HK survey stars) by include subgiants and the field equivalents of red horizontal-branch stars, reaffirming that *r*-process enhancement is not an evolutionary effect in the stars. also compare the abundance pattern of the light elements in their *r*-process-enhanced stars with that of *non-* *r*-process-enhanced stars with similar stellar parameters. No evidence was found to indicate that the *r*-process-enhanced stars have different abundance patterns for the light elements than for the comparison sample, leading the authors to conclude that the event(s) producing the high levels of *r*-process material seen in these stars do not produce a distinct light-element abundance pattern. Neither does the large *r*-process enhancement seem to be coupled to the carbon and nitrogen abundances in the stars, although a few of the *r*-II stars are found to also be enhanced in carbon, the most well-known example being CS 22892 − 052.\n explored the Sr, Ba and Eu abundances for a number of *r*-I and *r*-II stars. They found very similar [Ba/Eu] abundance ratios for the two groups ([Ba/Eu] ∼ + 0.60), but the mean [Sr/Eu] ratio differed by 0.36 dex between the two groups, with the *r*-II stars having the lowest ratio, [Sr/Eu] = − 0.93. The authors argued that elements from the first and second *r*-process peak are of common origin in the *r*-II stars, whereas for the *r*-I stars the picture is less clear.\nCompeting scenarios for the origin of the *r*-I and *r*-II stars invoke non-spherical, jet-producing supernova explosions or neutrino winds from merging neutron star binaries. Detailed observations of the abundance patterns predicted from the two competing scenarios may be the best guide to identifying the production site(s), but the frequency of these stars ( ∼ 3% among VMP and EMP stars; ) provides another clue; any jets in such scenarios must be highly collimated in order to selectively enrich only a small fraction of molecular clouds in the early ISM.\nOne circumstance is worthy of note, although its interpretation is currently unclear: Of the two *r*-II stars with secure detections of uranium, exhibits the so-called “actinide boost”, and has now been shown to be a single star, while the detailed *r*-process abundance pattern of the newly identified binary does not. While this may just be a result of small-number statistics on these very rare objects, it should be kept in mind as the sample grows; the details of their chemical-abundance patterns would appear to offer our most reliable clue to the origin of this difference.\nConclusions\n===========\nIn order to detect a possible link between the nature of the binary population among stars that exhibit moderate to strong enhancements in their *r*-process-element abundances and the origin of their peculiar abundance patterns, we have monitored the radial velocities of 17 such stars with high precision ( ∼ 100 m s− 1) over a period of eight years.\nOf the 17 programme stars, 14 exhibit no radial-velocity variations during this period, and are thus presumably single, while three are binaries with normal orbital periods and eccentricities, yielding a normal binary frequency of ∼ 18% among these stars. Hence, there is no evidence that the *r*-process enhancement seen in the *r*-I and *r*-II stars is causally linked to the binary nature of the stars. Furthermore, the detection of *r*-process-element rich stars in dwarf, subgiant, giant, and horizontal-branch evolutionary phases shows that the enhancement phenomenon is also not linked to the evolution of the stars.\nWe conclude that the moderate-to-high *r*-process-element abundances derived for these stars must be an imprint of the clouds from which the stars were formed. Detailed abundance analysis of larger samples of *r*-I and *r*-II stars is required to better constrain the nature of the objects and the chain of events that polluted the natal clouds of these stars. Moreover, the mechanism(s) for such sporadic, inhomogeneous enrichment of the ISM in early galaxies, and their implication for understanding the observed chemical composition of high-redshift DLA systems, should be considered in the next generation of models for the formation and early chemical evolution of galaxies.\nThis paper is based on observations made with the Nordic Optical Telescope, operated by the Nordic Optical Telescope Scientific Association at the Observatorio del Roque de los Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias.\nWe thank several NOT staff members and students for obtaining most of the observations for us in service mode.\nThe work of T.H. was supported by Sonderforschungsbereich SFB 881 “The Milky Way System” (subproject A4) of the German Research Foundation (DFG). 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E., & Joy, A. 1950,, 111, 221\nWood, P.R., 2000,, 17, 18\nWoolley, R., & Harding, G.A., 1965, Royal Observatory Bulletin, 93, 363\nYong, D., Norris, J.E., Bessell, M.S., et al., 2013,, 762, 26\nIndividual heliocentric radial velocities measured for the programme stars\n==========================================================================\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.661653 | − 57.892 | 0.033 |\n| 2454338.615723 | − 57.891 | 0.022 |\n| 2454373.612652 | − 57.900 | 0.024 |\n| 2454819.293270 | − 57.983 | 0.030 |\n| 2455126.580295 | − 57.946 | 0.041 |\n| 2455175.394755 | − 57.978 | 0.023 |\n| 2455503.480513 | − 57.855 | 0.070 |\n| 2455776.652777 | − 57.940 | 0.022 |\n| 2455796.657395 | − 57.927 | 0.025 |\n| 2455859.514789 | − 57.862 | 0.034 |\n| 2456139.703103 | − 57.863 | 0.027 |\n| 2456241.446457 | − 57.899 | 0.064 |\n| 2456529.679820 | − 57.934 | 0.023 |\n| 2456917.579578 | − 57.924 | 0.027 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454373.642892 | 104.329 | 0.116 |\n| 2454705.631179 | 104.873 | 0.065 |\n| 2454780.500993 | 105.409 | 0.292 |\n| 2454819.320354 | 105.561 | 0.084 |\n| 2455175.419215 | 105.010 | 0.063 |\n| 2455415.686599 | 104.941 | 0.061 |\n| 2455439.606743 | 105.124 | 0.060 |\n| 2455796.671082 | 105.434 | 0.077 |\n| 2455858.545060 | 105.060 | 0.060 |\n| 2456191.589163 | 105.050 | 0.045 |\n| 2456530.687786 | 104.829 | 0.054 |\n| 2456956.535485 | 104.471 | 0.094 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.681742 | 138.985 | 0.044 |\n| 2454338.656269 | 139.075 | 0.036 |\n| 2454396.548819 | 139.095 | 0.050 |\n| 2454480.372487 | 139.271 | 0.050 |\n| 2454705.645953 | 139.049 | 0.048 |\n| 2454780.529216 | 139.152 | 0.052 |\n| 2454819.336946 | 139.207 | 0.036 |\n| 2455059.684154 | 139.053 | 0.031 |\n| 2455149.486943 | 139.131 | 0.047 |\n| 2455207.337039 | 139.026 | 0.034 |\n| 2455232.321077 | 139.059 | 0.036 |\n| 2455415.734837 | 139.058 | 0.032 |\n| 2455439.622078 | 138.995 | 0.027 |\n| 2455485.577689 | 138.927 | 0.032 |\n| 2455503.470979 | 138.867 | 0.139 |\n| 2455776.741087 | 138.965 | 0.034 |\n| 2455821.588602 | 138.997 | 0.025 |\n| 2455882.542281 | 139.101 | 0.047 |\n| 2456139.725335 | 138.991 | 0.040 |\n| 2456276.485443 | 139.163 | 0.039 |\n| 2456519.736731 | 138.956 | 0.024 |\n| 2456574.672228 | 139.197 | 0.047 |\n| 2456956.570319 | 139.252 | 0.078 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454338.702068 | − 64.948 | 0.983 |\n| 2454373.722874 | − 64.876 | 1.097 |\n| 2454396.715153 | − 66.294 | 1.602 |\n| 2454406.637662 | − 64.942 | 1.710 |\n| 2454459.590478 | − 62.264 | 2.232 |\n| 2454480.431485 | − 65.026 | 1.202 |\n| 2454516.372208 | − 64.832 | 0.592 |\n| 2454780.591463 | − 64.462 | 1.896 |\n| 2454819.450079 | − 64.460 | 1.351 |\n| 2455075.713863 | − 65.352 | 1.544 |\n| 2455176.552628 | − 64.543 | 0.684 |\n| 2455232.419859 | − 65.290 | 1.106 |\n| 2455531.609740 | − 64.654 | 1.303 |\n| 2455620.393797 | − 63.236 | 2.281 |\n| 2455944.449119 | − 66.678 | 2.308 |\n| 2456191.699846 | − 64.856 | 1.104 |\n| 2456545.689792 | − 64.892 | 1.340 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454338.746568 | 231.895 | 0.043 |\n| 2454373.755033 | 231.272 | 0.022 |\n| 2454396.675349 | 231.387 | 0.045 |\n| 2454406.673221 | 231.488 | 0.047 |\n| 2454480.469787 | 231.625 | 0.037 |\n| 2454496.400729 | 231.149 | 0.135 |\n| 2454516.413321 | 231.777 | 0.020 |\n| 2454705.704003 | 230.516 | 0.030 |\n| 2454780.549248 | 230.670 | 0.046 |\n| 2454793.526401 | 230.398 | 0.173 |\n| 2454819.485376 | 231.059 | 0.041 |\n| 2454852.550506 | 231.299 | 0.045 |\n| 2455059.709842 | 254.627 | 0.039 |\n| 2455079.721105 | 255.868 | 0.030 |\n| 2455126.674591 | 252.221 | 0.032 |\n| 2455149.614597 | 250.514 | 0.037 |\n| 2455171.540400 | 248.975 | 0.021 |\n| 2455207.416924 | 246.988 | 0.029 |\n| 2455232.382885 | 245.710 | 0.030 |\n| 2455439.745696 | 240.146 | 0.032 |\n| 2455478.744628 | 239.853 | 0.067 |\n| 2455531.646433 | 239.862 | 0.033 |\n| 2455821.705466 | 236.600 | 0.024 |\n| 2455944.486938 | 236.352 | 0.046 |\n| 2455971.360048 | 236.071 | 0.040 |\n| 2456191.737737 | 234.847 | 0.023 |\n| 2456574.706961 | 233.444 | 0.020 |\n| 2456956.723997 | 231.920 | 0.055 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454396.640800 | 255.070 | 0.185 |\n| 2454406.717056 | 255.306 | 0.110 |\n| 2454480.496860 | 255.712 | 0.098 |\n| 2454820.481773 | 255.614 | 0.142 |\n| 2455149.688274 | 255.624 | 0.084 |\n| 2455207.491478 | 255.546 | 0.104 |\n| 2455503.608363 | 255.614 | 0.551 |\n| 2455531.544536 | 255.478 | 0.171 |\n| 2455882.598085 | 255.358 | 0.108 |\n| 2456005.376504 | 255.236 | 0.360 |\n| 2456207.754219 | 255.340 | 0.082 |\n| 2456603.661353 | 255.443 | 0.088 |\n| 2456956.750015 | 255.185 | 0.196 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454909.489857 | 342.628 | 3.089 |\n| 2454930.426948 | 369.639 | 1.671 |\n| 2455174.767340 | 379.567 | 2.399 |\n| 2455207.625989 | 362.229 | 1.939 |\n| 2455620.563564 | 384.868 | 1.939 |\n| 2455704.385256 | 343.075 | 2.077 |\n| 2455712.405032 | 335.836 | 1.872 |\n| 2455718.397062 | 358.699 | 2.243 |\n| 2455722.397011 | 375.517 | 1.896 |\n| 2455725.396840 | 386.965 | 3.807 |\n| 2455730.395197 | 385.629 | 2.338 |\n| 2455733.391004 | 378.210 | 3.132 |\n| 2455738.401973 | 354.412 | 1.072 |\n| 2456796.438819 | 363.768 | 1.346 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454459.682587 | 76.353 | 1.799 |\n| 2454464.668667 | 76.746 | 1.762 |\n| 2454516.585694 | 75.947 | 1.514 |\n| 2454909.650314 | 75.405 | 0.794 |\n| 2455232.610390 | 76.608 | 0.991 |\n| 2455344.429621 | 76.620 | 1.860 |\n| 2455531.735132 | 76.372 | 1.831 |\n| 2455662.477811 | 76.280 | 1.239 |\n| 2456033.449115 | 75.442 | 1.403 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454459.748479 | 229.676 | 2.402 |\n| 2454481.662602 | 229.183 | 1.172 |\n| 2454964.473253 | 228.850 | 1.277 |\n| 2455232.656932 | 228.583 | 1.005 |\n| 2455620.602643 | 229.334 | 1.314 |\n| 2455662.541359 | 228.736 | 1.308 |\n| 2456458.454152 | 229.734 | 0.725 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454481.728699 | 163.210 | 3.750 |\n| 2454625.460187 | 162.919 | 1.680 |\n| 2455620.657226 | 163.474 | 2.343 |\n| 2456090.413526 | 161.166 | 2.464 |\n| 2456652.702430 | 161.310 | 2.342 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454219.667720 | − 107.225 | 0.276 |\n| 2454314.443374 | − 108.199 | 0.203 |\n| 2454459.780220 | − 107.456 | 0.161 |\n| 2454464.791534 | − 107.738 | 0.217 |\n| 2454480.776288 | − 107.848 | 0.177 |\n| 2454625.420967 | − 107.371 | 0.217 |\n| 2454930.698058 | − 107.699 | 0.173 |\n| 2454951.700884 | − 107.748 | 0.256 |\n| 2454987.473997 | − 107.662 | 0.181 |\n| 2455232.782505 | − 107.709 | 0.145 |\n| 2455344.579755 | − 107.138 | 0.179 |\n| 2455620.687317 | − 107.719 | 0.243 |\n| 2455662.702207 | − 107.560 | 0.173 |\n| 2455704.614991 | − 107.359 | 0.157 |\n| 2455738.545590 | − 108.383 | 0.218 |\n| 2455776.455326 | − 108.773 | 0.204 |\n| 2456005.755704 | − 108.120 | 0.622 |\n| 2456033.623529 | − 108.201 | 0.206 |\n| 2456078.504562 | − 107.171 | 0.187 |\n| 2456712.726996 | − 107.900 | 0.651 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454219.705449 | − 163.546 | 0.016 |\n| 2454254.642226 | − 163.593 | 0.039 |\n| 2454285.507316 | − 163.348 | 0.015 |\n| 2454314.462351 | − 163.385 | 0.012 |\n| 2454625.508366 | − 163.091 | 0.010 |\n| 2454930.677427 | − 163.244 | 0.016 |\n| 2454951.718880 | − 162.971 | 0.017 |\n| 2454964.675531 | − 162.951 | 0.022 |\n| 2454987.509060 | − 163.041 | 0.014 |\n| 2455344.542474 | − 162.849 | 0.013 |\n| 2455415.386096 | − 163.549 | 0.010 |\n| 2455439.363821 | − 163.724 | 0.016 |\n| 2455620.738975 | − 162.875 | 0.017 |\n| 2455662.681656 | − 163.110 | 0.012 |\n| 2455704.633870 | − 163.372 | 0.016 |\n| 2455738.564258 | − 163.723 | 0.020 |\n| 2455776.438140 | − 163.732 | 0.016 |\n| 2456005.728789 | − 163.530 | 0.020 |\n| 2456078.539561 | − 163.442 | 0.015 |\n| 2456307.786409 | − 163.457 | 0.016 |\n| 2456351.717323 | − 163.347 | 0.014 |\n| 2456372.687769 | − 163.477 | 0.014 |\n| 2456399.619896 | − 163.371 | 0.015 |\n| 2456426.543882 | − 163.298 | 0.021 |\n| 2456474.451216 | − 163.108 | 0.011 |\n| 2456488.426825 | − 162.960 | 0.024 |\n| 2456489.500440 | − 163.000 | 0.015 |\n| 2456520.424407 | − 162.818 | 0.020 |\n| 2456529.377789 | − 162.938 | 0.017 |\n| 2456664.770980 | − 163.461 | 0.012 |\n| 2456685.790500 | − 163.626 | 0.011 |\n| 2456712.759468 | − 163.414 | 0.020 |\n| 2456796.646910 | − 162.880 | 0.011 |\n| 2456813.610868 | − 162.988 | 0.015 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.576915 | 13.178 | 0.053 |\n| 2454338.541838 | 13.553 | 0.136 |\n| 2454373.414188 | 13.559 | 0.053 |\n| 2454390.364640 | 13.579 | 0.079 |\n| 2454625.684458 | 13.594 | 0.058 |\n| 2454665.611641 | 13.917 | 0.063 |\n| 2454705.568239 | 13.747 | 0.051 |\n| 2455009.693164 | 13.480 | 0.098 |\n| 2455059.511399 | 13.695 | 0.076 |\n| 2455174.314885 | 13.374 | 0.050 |\n| 2455415.522393 | 13.471 | 0.061 |\n| 2455531.326532 | 13.467 | 0.108 |\n| 2455704.706380 | 13.742 | 0.090 |\n| 2455776.612557 | 13.367 | 0.049 |\n| 2455822.600538 | 13.505 | 0.103 |\n| 2455882.349889 | 13.624 | 0.051 |\n| 2456139.644767 | 13.413 | 0.087 |\n| 2456191.399426 | 13.585 | 0.099 |\n| 2456488.673600 | 13.584 | 0.066 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.626987 | − 113.002 | 0.070 |\n| 2454338.455766 | − 113.041 | 0.061 |\n| 2454373.470335 | − 113.043 | 0.113 |\n| 2454625.646394 | − 112.867 | 0.147 |\n| 2454665.630399 | − 112.981 | 0.118 |\n| 2454705.522460 | − 113.104 | 0.115 |\n| 2455009.711530 | − 113.138 | 0.099 |\n| 2455059.529868 | − 113.020 | 0.056 |\n| 2455070.548978 | − 112.989 | 0.159 |\n| 2455175.444551 | − 113.221 | 0.079 |\n| 2455344.685686 | − 112.945 | 0.119 |\n| 2455415.497735 | − 112.930 | 0.102 |\n| 2455439.426753 | − 112.968 | 0.103 |\n| 2455503.356818 | − 113.402 | 0.352 |\n| 2455531.410900 | − 113.220 | 0.136 |\n| 2455704.677515 | − 113.216 | 0.095 |\n| 2455776.507942 | − 112.924 | 0.093 |\n| 2455796.580352 | − 112.817 | 0.094 |\n| 2455882.328613 | − 113.245 | 0.106 |\n| 2456078.720195 | − 113.064 | 0.081 |\n| 2456140.610920 | − 113.674 | 0.186 |\n| 2456191.379978 | − 113.152 | 0.059 |\n| 2456488.693610 | − 113.205 | 0.084 |\n| 2456956.553882 | − 112.876 | 0.303 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454338.580931 | 147.980 | 1.346 |\n| 2454373.501898 | 148.051 | 1.007 |\n| 2454390.393739 | 148.275 | 1.573 |\n| 2454665.675576 | 147.791 | 1.192 |\n| 2455059.563356 | 147.724 | 1.446 |\n| 2455149.413670 | 148.224 | 1.470 |\n| 2455415.573319 | 148.194 | 1.123 |\n| 2455439.540189 | 147.532 | 1.306 |\n| 2455478.531528 | 147.567 | 0.899 |\n| 2455738.686456 | 147.855 | 0.872 |\n| 2455776.556273 | 147.947 | 1.562 |\n| 2455822.637317 | 148.094 | 1.569 |\n| 2456191.476165 | 148.111 | 0.922 |\n| 2456545.578750 | 147.649 | 1.362 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.636048 | − 121.161 | 0.010 |\n| 2454338.524441 | − 121.179 | 0.009 |\n| 2454373.528280 | − 121.107 | 0.009 |\n| 2454390.345759 | − 121.253 | 0.010 |\n| 2454406.586097 | − 121.114 | 0.011 |\n| 2454480.397616 | − 121.164 | 0.013 |\n| 2454625.656022 | − 121.267 | 0.012 |\n| 2454705.500584 | − 121.138 | 0.017 |\n| 2454780.452621 | − 121.111 | 0.012 |\n| 2454793.471559 | − 121.182 | 0.011 |\n| 2454820.348668 | − 121.321 | 0.014 |\n| 2455009.719853 | − 121.078 | 0.011 |\n| 2455059.695625 | − 121.185 | 0.009 |\n| 2455071.639351 | − 121.199 | 0.017 |\n| 2455171.399245 | − 121.340 | 0.008 |\n| 2455344.693948 | − 121.230 | 0.011 |\n| 2455415.506520 | − 121.259 | 0.012 |\n| 2455439.446714 | − 121.227 | 0.008 |\n| 2455503.346059 | − 121.358 | 0.014 |\n| 2455531.470241 | − 121.412 | 0.011 |\n| 2455704.717098 | − 121.256 | 0.009 |\n| 2455738.710786 | − 121.253 | 0.018 |\n| 2455776.728437 | − 121.333 | 0.012 |\n| 2455796.566301 | − 121.342 | 0.011 |\n| 2455859.532089 | − 121.249 | 0.009 |\n| 2455892.424873 | − 121.011 | 0.029 |\n| 2455915.320942 | − 121.166 | 0.077 |\n| 2456140.740348 | − 121.075 | 0.010 |\n| 2456241.340390 | − 121.056 | 0.013 |\n| 2456488.721460 | − 121.000 | 0.010 |\n[ht]\n| | | |\n| --- | --- | --- |\n| HJD | RV | RV*e**r**r* |\n| | (km s− 1) | (km s− 1) |\n| 2454314.742671 | − 168.937 | 0.074 |\n| 2454373.581743 | − 169.408 | 0.066 |\n| 2454705.601838 | − 169.668 | 0.074 |\n| 2455059.649134 | − 169.963 | 0.071 |\n| 2455175.377253 | − 168.867 | 0.130 |\n| 2455415.665073 | − 169.830 | 0.078 |\n| 2455776.636347 | − 169.582 | 0.071 |\n| 2455796.640879 | − 169.184 | 0.055 |\n| 2455821.555408 | − 169.165 | 0.065 |\n| 2455859.497621 | − 169.214 | 0.092 |\n| 2455882.472080 | − 169.087 | 0.100 |\n| 2456139.682491 | − 169.008 | 0.242 |\n| 2456530.669987 | − 169.196 | 0.120 |\n| 2456987.357305 | − 169.738 | 0.077 |\nLiterature data for the programme stars\n=======================================\n[ht]\n[tbl-B1]\n| | | | | | |\n| --- | --- | --- | --- | --- | --- |\n| Star ID | ΔT Total | $\\bar{RV}$ (this work) | $\\bar{RV}$ (lit) | N | Ref |\n| | (days) | (km s− 1) | (km s− 1) | | |\n| | 10011 | − 57.914 | − 57.4 | 3 | |\n| | | | − 57.2 | 13 | |\n| | | | − 57.5 | 1 | |\n| | 4742 | + 105.008 | + 105.1 | 1 | |\n| | 5193 | + 139.068 | + 139.1 | 8 | |\n| | | | + 138.9 | 2 | |\n| | | | + 138.2 | 1 | |\n| | | | + 138.9 | 1 | |\n| | | | + 139.4 | 1 | |\n| | | | + 139.0 | 1 | |\n| | | | + 138.4 | 1 | |\n| | | | + 138.9 | 1 | |\n| | 3582 | − 64.800 | − 66.6 | 1 | |\n| | 4032 | + 255.425 | + 255.3 | 1 | |\n| | 3267 | + 76.197 | + 77.0 | 1 | |\n| | 3785 | + 229.157 | + 228.5 | 1 | |\n| | 3885 | + 162.416 | + 163.6 | 1 | |\n| | | | + 163.1 | 5 | |\n| | 3942 | − 107.749 | − 107.4 | 1 | |\n| | 8788 | + 13.549 | + 13.1 | 10 | |\n| | | | + 13.6 | 1 | |\n| | | | + 12.5 | 15 | |\n| | | | + 13.2 | 1 | |\n| | | | + 12.7 | 1 | |\n| | | | + 14.5 | 1 | |\n| | | | + 13.3 | 1 | |\n| | | | + 13.0 | 2 | |\n| | 3968 | − 113.085 | − 112.3 | 1 | |\n| | 3960 | + 147.928 | + 148.1 | 1 | |\n| | 10921 | − 121.201 | − 119.0 | 4 | |\n| | | | − 123.7 | 3 | |\n| | | | − 121.8 | 18 | |\n| | | | − 120.7 | 1 | |\n| | | | − 121.7 | 1 | |\n| | | | − 121.8 | 1 | |\n| | 4741 | − 169.346 | − 169.2 | 1 | |\n[ht] [tbl-B2]\n| | | | | |\n| --- | --- | --- | --- | --- |\n| Star ID | ΔT Total | $\\bar{RV}$ | N | Ref |\n| | (days) | (km s− 1) | | |\n| | 8812 | − 27.6 | 3 | |\n| | | − 26.2 | 2 | |\n| | | − 27.1 | 2 | |\n| | | − 27.2 | 1 | |\n| | 21575 | − 26.1 | 3 | |\n| | | − 26.5 | 6 | |\n| | | − 24.9 | 6 | |\n| | | − 26.0 | 1 | |\n| | | − 26.0 | 1 | |\n| | | − 26.4 | 1 | |\n| | | − 25.6 | 2 | |\n| | | − 26.1 | 3 | |\n---\n1. http://www.not.iac.es/instruments/fies/[↩](#fnref1)\n"},"data_id":{"kind":"string","value":"arxiv_0182371"}}},{"rowIdx":45591,"cells":{"content":{"kind":"string","value":"Hitting Sets and Reconstruction for Dense Orbits in VPe and ΣΠΣ Circuits\n========================================================================\nIn this paper we study polynomials in VPe (polynomial-sized formulas) and in ΣΠΣ (polynomial-size depth-3 circuits) whose orbits, under the action of the affine group GL*n*aff(F),[2](#fn2) are *dense* in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. Specifically, we obtain the following results:\n1. For ${\\text{C}}\\_n{\\ensuremath{\\left( \\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_n({\\bm{x}}) \\right) }}\\triangleq {\\ensuremath{\\text{Trace}}}\\left(\n \\begin{pmatrix}\n \\ell\\_1({\\bm{x}}) & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot \n \\ldots \\cdot\n \\begin{pmatrix}\n \\ell\\_n({\\bm{x}}) & 1 \\\\\n 1 & 0\n \\end{pmatrix} \n \\right)$, where the ℓ*i*s are linearly independent linear functions, we construct a polynomial-sized interpolating set, and give a polynomial-time reconstruction algorithm. By a result of Bringmann, Ikenmeyer and Zuiddam, the set of all such polynomials is dense in VPe, thus our construction gives the first polynomial-size interpolating set for a dense subclass of VPe.\n2. For polynomials of the form ${\\ensuremath{\\text{ANF}\\_{\\Delta}}}{\\ensuremath{\\left( \\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_{4^\\Delta}({\\bm{x}}) \\right) }}$, where ${\\ensuremath{\\text{ANF}\\_{\\Delta}}}({\\bm{x}})$ is the canonical read-once formula in *alternating normal form*, of depth 2Δ, and the ℓ*i*s are linearly independent linear functions, we provide a quasipolynomial-size interpolating set. We also observe that the reconstruction algorithm of works for *all* polynomials in this class. This class is also dense in VPe.\n3. Similarly, we give a quasipolynomial-sized hitting set for read-once formulas (not necessarily in alternating normal form) composed with a set of linearly independent linear functions. This gives another dense class in VPe.\n4. We give a quasipolynomial-sized hitting set for polynomials of the form $f{\\ensuremath{\\left( \\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_{m}({\\bm{x}}) \\right) }}$, where *f* is an *m*-variate *s*-sparse polynomial and the ℓ*i*s are linearly independent linear functions in *n* ≥ *m* variables. This class is dense in ΣΠΣ.\n5. For polynomials of the form $\\sum\\_{i=1}^{s}\\prod\\_{j=1}^{d}\\ell\\_{i,j}({\\bm{x}})$, where the ℓ*i*, *j*s are linearly independent linear functions, we construct a polynomial-sized interpolating set. We also observe that the reconstruction algorithm of works for *every* polynomial in the class. This class is dense in ΣΠΣ.\nAs VP = VNC2, our results for VPe translate immediately to VP with a quasipolynomial blow up in parameters.\nIf any of our hitting or interpolating sets could be made *robust* then this would immediately yield a hitting set for the superclass in which the relevant class is dense, and as a consequence also a lower bound for the superclass. Unfortunately, we also prove that the kind of constructions that we have found (which are defined in terms of *k*-independent polynomial maps) do not necessarily yield robust hitting sets.\nIntroduction\n============\nProving lower bounds on the size of algebraic circuits (also called arithmetic circuits), is an outstanding open problem in algebraic complexity. In spite of much effort, only a handful of lower bounds are known (a detailed account of most known lower bounds can be found in the excellent survey of Saptharishi ). One common theme of most known lower bounds is that they are proved using *algebraic arguments*. That is, a proof of a lower bound for a class of circuits C, usually has the following structure: one comes up with a set of (nonzero) polynomials *F*1, …, *F**m*, in $N={n+d \\choose d}$ many variables, such that the coefficient vector of every *n*-variate, degree-*d* polynomial that can be computed in C, is a common zero of all the *F**i*s (such *F**i*s are called *separating polynomials*). Then, one exhibits a polynomial *f* whose coefficient vector is not a common zero, thus proving *f* ∉ C. As an example one can immediately see that the well known partial derivative technique, and its predecessor, shifted partial derivative technique, are algebraic. Grochow demonstrated this for most of the known lower bound proofs. As the set of common zeros of a set of polynomials is closed,[3](#fn3) this immediately implies that if we prove that *f* ∉ C using an algebraic argument, then the same argument also implies that $f\\not \\in \\overline{\\mathcal{C}}$, the closure of C. Recall that, in characteristic zero, the closure of a class C is the set of all polynomials that are limit points of sequences of polynomials from C, where convergence is coefficient-wise (see for a general definition over arbitrary characteristic). As most known techniques are algebraic, we see that for proving a lower bound for a class C one actually has to consider the larger, and less structured class, $\\overline{\\mathcal{C}}$.\nGeometric Complexity Theory (GCT for short), which was initiated by Mulmuley and Sohoni, approaches the lower bound question from a different angle. GCT also looks for an algebraic lower bound proof, but rather than exhibiting an algebraic argument, it aims to prove the existence of a separating polynomial. Specifically, GCT attempts to prove Valiant’s hypothesis, that VP ≠ VNP, over C, via *representation theory*. Valiant’s hypothesis is, more or less, equivalent to showing that the permanent of a symbolic *n* × *n* matrix is not a *projection* of the symbolic *m* × *m* determinant for any *m* = *m*(*n*) polynomial in *n*.[4](#fn4) Recall that a projection of a polynomial is a restriction of the polynomial to an affine subspace of its inputs. Observe that a restriction of an *n*-variate polynomial $f({\\bm{x}})$ to a subspace of its inputs, is equivalent to considering the polynomial $f(A{\\bm{x}}+{\\bm{b}})$, where *A* is an *n* × *n* matrix and ${\\bm{b}}\\in {\\mathbb{C}}^n$. As any matrix is a limit point of a sequence of invertible matrices, an algebraic proof that the permanent is not a projection of the *m* × *m* determinant, over C, is equivalent to an algebraic proof showing that the permanent is not in the closure of the set of polynomials $\\{{\\ensuremath{\\text{Det}}}(AX+{\\bm{b}}) \\mid A\\in \\GLm{m}{{\\mathbb{C}}},\\; {\\bm{b}}\\in{\\mathbb{C}}^m\\}$, where $\\GLm{m}{{\\mathbb{C}}}$ is the group of invertible *m* × *m* matrices (this is true for every field of characteristic ≠ 2). The set $\\{{\\ensuremath{\\text{Det}}}(AX+{\\bm{b}}) \\mid A\\in \\GLm{m}{{\\mathbb{C}}},\\; {\\bm{b}}\\in{\\mathbb{C}}^m\\}$ is called the *orbit* of the determinant under the action of the affine group (we denote the affine group over C*m* with GL*m*aff(C)). GCT considers the linear space of polynomials that vanish on every coefficient vector in the orbit of the determinant, and similarly the linear space of polynomials that vanish on every coefficient vector in the orbit of the permanent. There is a natural action of on those linear spaces, thus defining two representations of. GCT wishes to find a separating polynomial by showing that some irreducible representation of has strictly larger multiplicity when considering the representation corresponding to the determinant. This approach bypasses the barrier given in as it does not exhibit any efficiently computable separating polynomial but rather just proves the existence of one. However, the representation theory questions arising in this program are quite difficult, even when considering the analog questions for restricted classes. For an introduction to GCT see the lecture notes of Bläser and Ikenmeyer.\nAnother possible approach for proving lower bounds against a class of polynomials C, is via the construction of a *hitting set* for C. Recall that a hitting set $\\cal H$ for a class C is a set of points such that for any nonzero polynomial *f*, that can be computed by a circuit from C, there is ${\\bm{v}}\\in{\\cal H}$ such that $f({\\bm{v}})\\neq 0$. In Heintz and Schnorr observed that if we have such a hitting set $\\cal H$ then any nonzero polynomial *g* that vanishes on $\\cal H$ cannot be computed in C. It is also not hard to see that this way of obtaining lower bounds also bypasses the natural proof barrier of. The problem is that in most cases we obtained a hitting set for a class only after proving a lower bound for it.\nIn Forbes and Shpilka defined the notion of a *robust* hitting set for a circuit class C. Over fields of characteristic zero, a hitting set H for a class C is *c*-robust if it also satisfies that for every *f* ∈ C there is ${\\bm{v}}\\in{\\mathcal{H}}$ such that $|f({\\bm{v}})|\\geq c \\cdot {\\left\\lVertf\\right\\rVert}$, where ∥ ⋅ ∥ is some fixed norm on ${\\mathbb{C}}[{\\bm{x}}]$ (see for a definition over arbitrary fields). It is not hard to see that if H is a robust hitting set for a class C then it also hits the closure of C.\nIn this work we focus on depth-3 algebraic circuits, known as ΣΠΣ, and on VPe, the class of algebraic formulas, two classes for which we lack strong lower bounds, and in particular we do not have hitting sets for them. For ΣΠΣ circuits the best lower bound is the near cubic lower bound of Kayal, Saha and Tavenas, and for VPe the best lower bound is the quadratic lower bound of Kalarkoti. Recall that by the result of Valiant et al., a super-quasipolynomial lower bound against VPe implies a super-polynomial lower bound against VP. Similarly, a hitting set for VPe implies a hitting set for VP. We also note that by a result of Gupta et al., a strong enough lower bound or a hitting set for ΣΠΣ imply both a lower bound for general circuits and a hitting set for them. This result also implies that a polynomial-time reconstruction algorithm for ΣΠΣ circuits would give rise to a sub-exponential time *reconstruction algorithm* for general circuits. Recall that a reconstruction algorithm for a class C is an algorithm that, given black-box access to a circuit from C, outputs a circuit in C that computes the same polynomial.\nInstead of viewing robust hitting sets as a way to obtain hitting sets for the closure of circuit classes, we suggest to find subclasses of interesting classes, $\\tilde{{\\mathcal{C}}}\\subset {\\mathcal{C}}$, such that C is contained in the closure of $\\tilde{{\\mathcal{C}}}$, and aim to construct a robust hitting set for the subclass $\\tilde{{\\mathcal{C}}}$. This offers a new approach for constructing hitting sets for known classes and for obtaining lower bounds. Specifically, we consider subclasses of ΣΠΣ and VPe that are dense in their superclasses. Each of these subclasses is the orbit of some simple polynomial under the group of invertible affine transformations.\nFor VPe, we first consider a subclass that was defined by Bringmann, Ikenmeyer and Zuiddam –the orbit of the so called *continuant* polynomial (see ). We give a polynomial-sized interpolating set[5](#fn5) for this subclass as well as a polynomial-time deterministic reconstruction algorithm that uses as oracle a *root-finding algorithm*.[6](#fn6) In particular, this implies a polynomial-time randomized reconstruction algorithm, and, in some cases, a polynomial-time deterministic algorithm.\nIn addition, we exhibit two other subclasses that are dense in VPe. The first class is defined as the orbit of read-once formulas (ROF for short, see ) and the second as the orbit of read-once formulas in *alternating normal form* (ROANF for short, see ). We obtain hitting sets for both classes and an interpolating set for the second. We also observe that the reconstruction algorithm of works for the polynomials in the orbit of ROANFs. Although the results that we obtain for the subclass defined by the continuant polynomial are stronger, we think that every such dense subclass can shed more light on VPe and may eventually be used in order to obtain new lower bounds.\nFor ΣΠΣ we consider two subclasses. One is based on orbits of *sparse* polynomials (polynomials having polynomially many monomials) and the other on orbits of *diagonal* tensors (see ). We give a hitting set for the first, an interpolation set for the second, and we also observe that a slight modification of the randomized reconstruction algorithm of applies for the second class.\nIn particular, our results give the first dense subclasses inside VPe and ΣΠΣ for which a polynomial-size interpolating set is known as well as a polynomial-time reconstruction algorithm. By our result immediately translate to VP, giving a dense subclass of for which a quasipolynomial-sized interpolating set is known as well as a quasipolynomial-time reconstruction algorithm.\nIf we could transform the interpolating sets that we have found to *robust hitting sets* for the orbits, then this will immediately give hitting sets for the closure of the orbits, i.e. for ΣΠΣ and VPe, which, by gives a lower bound for the class. Thus, our work raises an intriguing problem:\n[prob:robust] Given an interpolating set for a class C construct a robust hitting set for C.\nWe stress that by our results, solving this problem would lead to hitting sets, and lower bounds, for VPe and VP.\nAnother advantage for having small interpolating sets for dense subclasses is the following: One approach for searching for separating polynomials for a class, is by considering the map from circuits in the class to the coefficient vectors of the polynomials that they compute. That is, once we fix a computation graph, an assignment to the constants appearing in the circuit determines the output polynomial. Each coefficient is a polynomial in those constants, and as there are “few” constants (polynomially many for polynomially sized circuits), and there are exponentially many coefficients, there should be many polynomials vanishing on the closure of the image of this map. If we could get a good understanding of this map then perhaps we could use it to construct a polynomial that vanishes on all such coefficient vectors. This polynomial will vanish on all coefficient vectors of the superclass in which the subclass is dense. A different approach is to find a coefficient vector that is not in the closure of the image of this map (this is the approach of Raz in ). Now, assume that H is an interpolating set for a dense subclass $\\tilde{{\\mathcal{C}}}\\subset {\\mathcal{C}}$. We know that the map $f\\to {{ \\left.\\kern-\\nulldelimiterspace f \\vphantom{\\big|} \\right|\\_{{\\mathcal{H}}} }}$ is one-to-one on $\\tilde{{\\mathcal{C}}}$. Thus, the list of values ${{ \\left.\\kern-\\nulldelimiterspace f \\vphantom{\\big|} \\right|\\_{{\\mathcal{H}}} }}$ can be viewed as an efficient encoding that is given in terms of values of the computed polynomial. This provides a different encoding of a circuit – instead of the constants in it, use the evaluations on H. Thus, by studying the closure of this map (i.e. the closure of the set of points on F∣H∣ that can be obtained as evaluation vectors of polynomials in the subclass) we may be able to find a separating polynomial, or, as in Raz’s approach, find an evaluation vector that is not obtained by any polynomial in the superclass. It is clear that one can also try this approach even if $\\cal H$ is not an interpolating set, however, as interpolating sets “preserve information” of a dense set, we believe that such sets are better suited for this approach.\nTo conclude, focusing on dense subclasses and studying their properties could lead to better understanding of their superclasses and perhaps to breakthrough results in algebraic complexity.\nTo formally state our results we need some definitions that we give next.\nBasic definitions\n-----------------\n### Circuit classes\n[defFormula] An algebraic formula (also called arithmetic formula) over a field F, is a rooted tree whose leaves are labeled with either variable or scalars from F, and whose root and internal nodes (called gates) are labeled with either “ + ” (addition) or “ × ” (multiplication). An algebraic formula computes a polynomial in the natural way. Each leaf computes the polynomial that labels it, and each gate computes either the sum or product of its children, depending on its label. The output of the formula is the polynomial computed at its root. The size of a formula is the number of wires in it. The depth of a formula is the length of the longest simple leaf-root path in it. The formula size of a polynomial *f* is defined as the smallest size of a formula that outputs *f*.\nA sequence *m*(*n*) of natural numbers is called polynomially bounded if there exists a univariate polynomial *q* such that *m*(*n*) ≤ *q*(*n*) for all *n*.\nThe complexity class VPe is defined as the set of all families of polynomials (*f**n*)*n*, with *f**n* ∈ F[*x*1, …, *x**n*], whose formula size is polynomially bounded.\n[defDepthTow] An arithmetic circuit Φ is a Σ[*s*]Π[*d*] circuit if it is a layered graph of depth-2, has a top gate labeled + with fan-in ≤ *s* and its second layer is comprised entirely of × gates with fan-in ≤ *d*. In other words, Σ[*s*]Π[*d*] compute polynomials of degree *d* with at most *s* monomials.\n[defDepthThree] An arithmetic circuit Φ in *n* variables is a Σ[*s*]Π[*d*]Σ circuit if it is a layered graph of depth-3, has a top gate labeled + with fan-in ≤ *s*, its second layer is comprised entirely of × gates with fan-in ≤ *d*, and its bottom layer is comprised of linear functions in *x*1, …, *x**n*. In other words, Σ[*s*]Π[*d*]Σ circuit compute polynomials of the form \n$$f({\\bm{x}})=\\sum\\_{i=1}^s\\prod\\_{j=1}^d(\\alpha\\_{i,j,0}+\\sum\\_{k=1}^n\\alpha\\_{i,j,k}x\\_k) \\;.$$ \nGiven a family of circuits C, we will sometime denote it as C(F) to stress that we allow coefficients to come from the field F. Observe that the definitions of the classes above do not depend on the field and so we can define them over any field of our choice.\n### Approximate complexity\nThe following definition gives sense to the notion of approximation over arbitrary fields. In what follows we let *ɛ* be a new formal variable.[7](#fn7) For a field F we denote with F[*ɛ*] the ring of polynomial expressions in *ɛ* over F, and with F(*ɛ*) the fraction field of F[*ɛ*], i.e. the field of rational expressions in *ɛ*.\n[def:approx] Let C(F) be a circuit class over a field F. The closure of C, denoted $\\overline{{\\mathcal{C}}({{\\mathbb{F}}})}$, is defined as follows: A family of functions (*f**n*)*n*, where *f**n* ∈ F[*x*1, …, *x**n*], is in $\\overline{{\\mathcal{C}}({{\\mathbb{F}}})}$ if there is a polynomially bounded function *m* : N → N, and a family of functions (*g**m*(*n*))*n* ∈ C(F(*ɛ*)), with *g**m*(*n*) ∈ F[*ɛ*][*x*1, …, *x**m*(*n*)], such that for all *n* ∈ N,\n \n*g**m*(*n*)(*x*1, …, *x**m*(*n*)) = *f**n*(*x*1, …, *x**n*) + *ɛ* ⋅ *g**n*, 0(*x*1, …, *x**m*(*n*)) , \nfor some polynomial *g**n*, 0 ∈ F[*ɛ*][*x*1, …, *x**m*(*n*)]. Whenever an equality as in holds we say that \n*g**m*(*n*) = *f**n* + *O*(*ɛ*) or *f**n* = *g**m*(*n*) + *O*(*ɛ*) . \n In that case we think of *g**m*(*n*) as an “approximation” of *f**n*, and we say that the family (*g**m*(*n*))*n* approximates the family (*f**n*)*n*.\nAlder have shown that over C it holds that $(f\\_n) \\in \\overline{{\\mathcal{C}}({\\mathbb{C}})}$, in the sense of, if and only if it is in the closure of C(C) in the usual sense. That is, if for every *n* there exists a sequence of polynomials *g**n*, *k* ∈ C(C) such that lim*k* → ∞*g**n*, *k* = *f**n*, where convergence is taken coefficient wise. This result holds over R as well, see.\nFinally, we note that every matrix is approximable (in the sense of ) by a non-singular matrix (which is equivalent to being a limit of a sequence of non-singular matrices, in characteristic zero).\n[matLimits] For every *A* ∈ F*n* × *n* there exists a non-singular matrix *B* ∈ F(*ɛ*)*n* × *n* such that *A* = *B* + *O*(*ɛ*).\n### Hitting and interpolating sets\n[def:hitting] A set of points $\\mathcal H\\subseteq{{\\mathbb{F}}}^n$ is called a *hitting set* for a circuit class $\\mathcal C$ (we also say that $\\mathcal H$ *hits* $\\mathcal C$) if for every circuit $\\Phi\\in\\mathcal C$, computing a non-zero polynomial, there exists some ${\\bm{a}}\\in\\mathcal H$ such that $\\Phi({\\bm{a}})\\neq 0$.\nWe next give the definition of a robust hitting set, a notion first defined in. Here we extend the definition for arbitrary characteristic. We start by giving the definition of, over characteristic zero (and focus on C) and then the more general definition.\n[Following Definition 5.1 of ][def:robust-0] Let ∥ ⋅ ∥ be some norm on ${\\mathbb{C}}[{\\bm{x}}]$. A hitting set $\\mathcal H$ for a circuit class $\\mathcal C\\subseteq{\\mathbb{C}}[{\\bm{x}}]$ is called *robust* if there exists some constant *c* > 0 such that, for every $0\\neq f\\in\\mathcal C$,[8](#fn8) there exists some ${\\bm{a}}\\in\\mathcal H$ such that $|f({\\bm{a}})|\\geq c\\cdot {\\left\\lVertf\\right\\rVert}$.\nFor arbitrary characteristic we use the same approach as in.\n[def:robust] Let F be a field of arbitrary characteristic. A hitting set $\\mathcal H\\subset {{\\mathbb{F}}}^n$ for a circuit class C(F) is called *robust* if for every circuit Φ ∈ C(F(*ɛ*)) computing a polynomial $f({\\bm{x}})= h({\\bm{x}})+\\varepsilon \\cdot g({\\bm{x}})$, where $h({\\bm{x}})\\in{{\\mathbb{F}}}[{\\bm{x}}]$ and $g({\\bm{x}})\\in{{\\mathbb{F}}}[\\varepsilon][{\\bm{x}}]$, there exists some ${\\bm{a}}\\in\\mathcal H$ such that $f({\\bm{a}}) \\not\\in \\varepsilon \\cdot {{\\mathbb{F}}}[\\varepsilon]$.\nIt is not hard to prove using the result of that for F = C, are equivalent.\n[tm:robust-hits] If H is a finite robust hitting set for C(F), then H hits $\\overline{{\\mathcal{C}}({{\\mathbb{F}}})}$ as well.\nConsider $0\\neq f \\in \\overline{{\\mathcal{C}}({{\\mathbb{F}}})}$. By there is *g* ∈ C(F(*ɛ*)), such that *f* = *g* + *O*(*ɛ*). Clearly *g* ≠ 0. Let ${\\bm{a}}\\in {\\mathcal{H}}$ be such that $g({\\bm{a}})\\not \\in \\varepsilon\\cdot{{\\mathbb{F}}}[\\varepsilon]$. It follows that $f({\\bm{a}})\\not \\in \\varepsilon\\cdot{{\\mathbb{F}}}[\\varepsilon]$. In particular, $f({\\bm{a}})\\neq 0$.\nWe next define the notion of an interpolating set.\n[defInterSet] Let C be a class of *n*-variate polynomials. A set $\\mathcal H\\subseteq{{\\mathbb{F}}}^n$ is called an *interpolating set* for C if, for every *f* ∈ C, the evaluations of *f* on $\\mathcal H$ uniquely determine *f*.\n[obs:inter] If $\\mathcal H$ is a hitting set for ${\\mathcal{C}}({{\\mathbb{F}}})+{\\mathcal{C}}({{\\mathbb{F}}})\\triangleq \\{\\alpha f+\\beta g:f,g\\in{\\mathcal{C}},\\alpha,\\beta\\in{{\\mathbb{F}}}\\}$, then $\\mathcal H$ is an interpolating set for C.\nA common method for designing hitting and interpolating sets is via hitting set generators.\nA polynomial mapping G : F*k* → F*n* is called a *hitting set generator* (or simply a generator) for a circuit class C(F) if for any non-zero *n*-variate polynomial *f* ∈ C, the *k*-variate polynomial *f* ∘ G is non-zero.\nSimilarly, we call G : F*k* → F*n* an *interpolating set generator* for a circuit class C(F) if for any two different *n*-variate polynomials *f*1, *f*2 ∈ C, the *k*-variate polynomial (*f*1 − *f*2) ∘ G is non-zero.\nGenerators immediately give rise to hitting sets.\n[obsHitSetGen] Let G : F*k* → F*n* be a generator for C(F) such that the individual degree of each coordinate of G is at most *r*. Let *W* ⊂ F be any set of size ∣*W*∣ = *d* ⋅ *r* + 1. Let H = G(*W**k*). Then H hits every *n*-variate polynomial *f* ∈ C of degree at most *d*.\nAs G is a generator, the *k*-variate polynomial *f* ∘ G is nonzero. As its individual degrees are bounded by *d* ⋅ *r* it follows that at least one of the values in (*f* ∘ G)(*W**k*) = *f*(H) is not zero.\n### *k*-independent maps\nOur constructions rely on polynomial mappings G*k*, parameterized by some integer *k* ≤ *n*, with the property that the image of *f* ∘ G*k* contains all projections of *f* to *k* variables. We call such a map a *k*-independent map.\n[def:indPolyMap] We call a polynomial mapping G(*y*1, …, *y**t*, *z*1) : F*t* + 1 → F*n* a 1 if for every index *i* ∈ [*n*] there exists an assignment ${\\bm{a}}\\_i\\in{{\\mathbb{F}}}^t$ to *y*1, …, *y**t* such that the *i*th coordinate of ${{\\ensuremath{\\mathcal{G}}}}({\\bm{a}}\\_i,z\\_1)$ is *z*1, and the rest of the coordinates are 0. For *k* > 1, a polynomial mapping G(*y*1, …, *y**t**k*, *z*1, …, *z**k*) : F*k*(*t* + 1) → F*n* is called a *k* (or a *k*-independent map) if G is a sum of *k* variable-disjoint 1-independent polynomial maps. We denote *k*-independent polynomial maps as ${{\\ensuremath{\\mathcal{G}}}}(\\bm y,\\bm z)$ when *k*, *t* are implicit. The $\\bm y$ variables are called *control variables*.\nA *k*-independent polynomial map G is called *uniform* if all *n* coordinates of G are homogeneous polynomials of the same degree.\n### The linear and affine groups and their actions\nGiven a matrix *A* ∈ F*n* × *n* and a tuple of variables ${\\bm{x}}= (x\\_1,\\ldots,x\\_n)$, we denote \n$$A{\\bm{x}}={\\ensuremath{\\left( \\sum\\_{i=1}^n A\\_{1,i}x\\_i,\\sum\\_{i=1}^n A\\_{2,i}x\\_i,\\ldots,\\sum\\_{i=1}^n A\\_{n,i}x\\_i \\right) }}\\;.$$ \n Let *n* ≥ *m* ∈ N. For an *m*-variate polynomial *f*(*x*1, …, *x**m*) ∈ F[*x*1, …, *x**m*], a matrix *A* = (*A**i*, *j*)*i*, *j* = 1*n* ∈ F*n* × *n* and a vector ${\\bm{b}}=(b\\_1,\\ldots,b\\_n)\\in{{\\mathbb{F}}}^n$, we define the *n*-variate polynomial $f{\\ensuremath{\\left( A{\\bm{x}}+{\\bm{b}}\\right) }}$ to be\n \n$$\\label{eq:compose}\n f{\\ensuremath{\\left( A{\\bm{x}}+{\\bm{b}}\\right) }}\\triangleq f\\left(\\sum\\_{i=1}^n A\\_{1,i}x\\_i+b\\_1,\\sum\\_{i=1}^n A\\_{2,i}x\\_i+b\\_2,\\ldots,\\sum\\_{i=1}^n A\\_{m,i}x\\_i+b\\_m\\right)\\;.$$ \nNote that we ignored the last *n* − *m* coordinates of $A{\\bm{x}}+{\\bm{b}}$.\nWe denote with $\\GLm{n}{{{\\mathbb{F}}}}$ the group of invertible *n* × *n* matrices over F, and with $\\GLplus[n][{{\\mathbb{F}}}]$ the group of invertible affine transformation, i.e. all the maps ${\\bm{x}}\\to A{\\bm{x}}+{\\bm{b}}$, where $A\\in \\GLm{n}{{{\\mathbb{F}}}}$ and ${\\bm{b}}\\in{{\\mathbb{F}}}^n$.\nFor an *m*-variate polynomial *f* over F, and *n* ≥ *m* we denote with $f^{\\GLplus[n][{{\\mathbb{F}}}]}$ the orbit of *f* under the natural action of $\\GLplus[n][{{\\mathbb{F}}}]$:[9](#fn9) \n$$f^{\\GLplus[n][{{\\mathbb{F}}}]}\\triangleq \\left\\{ f(A{\\bm{x}}+{\\bm{b}}) \\mid A\\in \\GLm{n}{{{\\mathbb{F}}}},\\; {\\bm{b}}\\in{{\\mathbb{F}}}^n\\right\\}\\;.$$ \n We similarly define $f^{\\GLm{n}{{{\\mathbb{F}}}}}$. More generally, for a class of *m*-variate polynomials ${\\mathcal C}({{\\mathbb{F}}})$, we denote the *orbit* of $\\mathcal C$ under $\\GLplus[n][{{\\mathbb{F}}}]$ by \n$${\\mathcal C}^{\\GLplus[n][{{\\mathbb{F}}}]}\\triangleq{\\ensuremath{\\left\\{ f(A{\\bm{x}}+{\\bm{b}})\\mid f\\in{\\mathcal C},\\; A\\in \\GLm{n}{{{\\mathbb{F}}}},\\; {\\bm{b}}\\in{{\\mathbb{F}}}^n \\right\\} }}\\;.$$ \n We similarly define ${\\mathcal C}^{\\GLm{n}{{{\\mathbb{F}}}}}$. When we want to speak about orbits of families of polynomials from C(F), with arbitrary number of variables, we use the notation ${\\mathcal{C}}^{\\GLm{}{{{\\mathbb{F}}}}}$ or ${\\mathcal C}^{\\GLplus[][{{\\mathbb{F}}}]}$.\nOur results\n-----------\nWe first give our results for the class VPe and then for the class of depth-3 circuits, for which it may be easier to obtain a robust hitting set, or prove super-polynomial lower bounds.\n### The continuant polynomial\nBringmann, Ikenmeyer and Zuiddam defined the following polynomial (in Remark 3.14 of their paper), which they called the continuant polynomial:\n[def:cont] The continuant polynomial on *n* variables, C*n*(*x*1, …, *x**n*), is defined as the *trace* of the following matrix product:\n \n$${\\text{C}}\\_n(x\\_1,\\ldots,x\\_n)\\triangleq {\\ensuremath{\\text{Trace}}}\\left(\n \\begin{pmatrix}\n x\\_1 & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot \n \\begin{pmatrix}\n x\\_2 & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot \\ldots \\cdot\n \\begin{pmatrix}\n x\\_n & 1 \\\\\n 1 & 0\n \\end{pmatrix} \n \\right) \\;.$$ \nWe denote with CGLaff(F) the class of families of polynomials (*f**n*)*n* such that *f**n* ∈ F[*x*1, …, *x**n*] and for some *m* ≤ *n*, *f**n* ∈ C*m*GL*n*aff(F).\nA result of Allender and Wang implies that the polynomial *x*1 ⋅ *y*1 + ⋯ + *x*8 ⋅ *y*8 is not in CGLaff(F). Thus, as a computational class it is very weak. However, Theorem 3.12 of states that for every field F of characteristic different than 2, it holds that\n \n$$\\overline{{{\\text{C}}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}} = \\overline{{\\ensuremath{\\text{VP}\\_{\\text{e}}}}}\\;.$$ \nWe give a polynomial-size interpolating set for the class CGLaff(F) as well as a polynomial-time reconstruction algorithm for it. We first state a simple result that gives a hitting set for the class.\ntheoremthmhitcont[thmhitcont] Let *f*(*x*1, …, *x**n*) ∈ C*m*GL*n*aff(F), for *m* ≤ *n*, and arbitrary F. Then, for any uniform 1-independent polynomial map G over F, *f* ∘ G ≠ 0.\nAs immediate corollary we get a hitting set for the class.\n[cor:hit-cont] For every field F, there is an explicit hitting set H ⊂ F*n*, of size ∣H∣ = *O*(*n*6), that hits every 0 ≠ *f* ∈ C*m*GL*n*aff(F). If ∣F∣ < *n*2 then H is defined over a polynomial-sized extension field of F, K such that ∣K∣ ≥ *n*2.\n[thm:cont-int] For every field F, there is an explicit interpolating set H ⊂ F*n*, of size ∣H∣ = *O*(*n*10), for ⋃*m* = 1*n*C*m*GL*n*aff(F). If ∣F∣ < *n*2 then H is defined over a polynomial-sized extension field of F, K such that ∣K∣ ≥ *n*2.\n[thm:cont-recon] There is a deterministic algorithm that given F, an integer *n*, oracle access to a root-finding algorithm over F, and black-box access to a polynomial *f*(*x*1, …, *x**n*) ∈ C*m*GL*n*aff(F) (for any *m* ≤ *n*), runs in polynomial-time and outputs linear functions (ℓ1(*x*1, …, *x**n*), …, ℓ*m*(*x*1, …, *x**n*)) such that \n$$f(x\\_1,\\ldots,x\\_n) = {\\text{C}}\\_m\\left(\\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_m({\\bm{x}})\\right)\\;.$$ \n If ∣F∣ < *n*3 then the algorithm will make queries from a polynomial-sized extension field of F, K, such that ∣K∣ ≥ *n*3, and it also requires oracle access to a root-finding algorithm over K.\n### Orbits of read-once formulas\nRoughly, a read-once formula (ROF) is a formula in which every variable labels at most one leaf. However, following we also allow gates of the formula to pass on their output wire a linear function of their polynomial (see ). We denote with ${\\text{ROF}}^{\\GLm{}{{{\\mathbb{F}}}}}$ the class of families of polynomials (*f**n*)*n*, such that for every *n* there exists a ROF Φ, on *m* ≤ *n* variables, such that $f\\_n(x\\_1,\\ldots,x\\_n) \\in \\Phi^{\\GL[n][{{\\mathbb{F}}}]}$.\nA ROF is in *alternating normal form* (ROANF) if it is a full binary tree of depth 2Δ with alternating layers of addition and multiplication gates. In particular, it is a ROF on 4Δ many variables (see ).\nWe denote with ANFΔ the canonical ROANF of depth 2Δ in which the leaves are labeled with the variables *x*1, …, *x*4Δ according to their order (see ). We denote with ${\\ensuremath{\\text{ANF}\\_{}}}^{\\GLaff[{{\\mathbb{F}}}]}$ the class of families of polynomials (*f**n*)*n*, such that for every *n* there exists Δ such that 4Δ ≤ *n* and *f**n*(*x*1, …, *x**n*) ∈ ANFΔGL*n*aff(F).\nWe first make the following simple observation.\ntheoremROANFisDense [ROANFisDense] For every field F, it holds that\n \n$${\\ensuremath{{\\ensuremath{\\text{ANF}\\_{}}}^{\\GLaff({{\\mathbb{F}}})}}}\\subsetneq{\\text{ROF}}^{{\\text{GL}\\_{}({{\\mathbb{F}}})}}\\subsetneq {\\ensuremath{\\text{VP}\\_{\\text{e}}}}({{\\mathbb{F}}})\\;.$$ \nHowever, when taking closures we get\n \n$$\\overline{{\\ensuremath{\\text{ANF}\\_{}}}^{\\GLaff({{\\mathbb{F}}})}}=\\overline{{\\text{ROF}}^{\\GLm{}{{{\\mathbb{F}}}}}}=\\overline{{\\ensuremath{\\text{VP}\\_{\\text{e}}}}({{\\mathbb{F}}})}\\;.$$ \nOur main results for ROFs and ROANFs are a construction of a hitting set for the orbit of ROFs, and an interpolating set for the orbit of ROANFs. Both constructions are obtained using independent polynomial maps ().\ntheoremPITROPINV [thm:PITROPINV] Let 0 ≠ *f* ∈ ROFGL*n*aff(F) where the underlying depends on 2*t* variables, for 2*t* ≤ *n*. Then, for any (*t* + 1)-independent polynomial map G, over F, *f* ∘ G ≠ 0.\n[cor:PITROPINV] For every field F, there is a hitting set H ⊂ F*n*, of size ∣H∣ = *n**O*(log*n*), that hits every 0 ≠ *f* ∈ ROFGL*n*aff(F). If ∣F∣ < *n*2 then H is defined over a polynomial-sized extension field of F, K such that ∣K∣ ≥ *n*2.\nSince a hitting set for all polynomials of the form *g* − *h* where *g*, *h* ∈ C is the same as an interpolating set for C, the following theorem gives an interpolating set for the orbit of ROANFs.\ntheorempitSumOfRoanfThm[thm:pitSumOfRoanfThm] Let $f\\_1={\\ensuremath{\\text{ANF}\\_{\\Delta\\_1}}}(A\\_1{\\bm{x}}+{\\bm{b}}\\_1),f\\_2={\\ensuremath{\\text{ANF}\\_{\\Delta\\_2}}}(A\\_2{\\bm{x}}+{\\bm{b}}\\_2)\\in{\\ensuremath{{\\ensuremath{\\text{ANF}\\_{}}}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{n}({{\\mathbb{F}}})}}}}}}$ and *f* = *f*1 − *f*2. Set $k\\triangleq 2\\max\\{\\Delta\\_1,\\Delta\\_2\\}+7$ and let G be any uniform *k*-independent polynomial map, over F. If *f* ≠ 0 then *f* ∘ G ≠ 0.\n[cor:pitSumOfRoanf] For any field F, the class ANFΔGL*n*aff(F), for 4Δ ≤ *n*, admits an interpolating set H ⊂ F*n*, of size ∣H∣ = *n**O*(Δ). If ∣F∣ < *n*2 then H is defined over a polynomial-sized extension field of F, K, such that ∣K∣ ≥ *n*2.\nFinally, we observe that the randomized algorithm of Gupta, Kayal And Qiao, for reconstructing *random algebraic formula* (for a natural definition of a random formula), yields a randomized reconstruction algorithm for ${{\\orbitofgroup{{\\ensuremath{\\text{ANF}\\_{}}}}{\\GLplus[]}}}$. Naturally, the reconstruction is up to the symmetry group of ROANFs.\n[A special case of Theorem 1.1 of ]theoremreconstructROANF [thm:reconstructROANF] Let *T* be a finite subset of C. Let *n*, Δ ≥ 1 be integers such that $s\\triangleq 4^\\Delta\\leq n$. Given black-box access to the output *f* of a circuit Φ ∈ ANFGL*n*aff(C), with probability at least $1-\\frac{n^2s^{O(1)}}{|T|}$ (on internal randomness), Algorithm 6.9 of successfully computes a tuple of *s* linearly independent linear functions $L=(\\ell\\_1,\\ldots,\\ell\\_s)\\in({\\mathbb{C}}[{\\bm{x}}])^s$ such that *f* = ANFΔ(ℓ1, …, ℓ*s*), and the ℓ*i*s are identical to the labels of the leaves of Φ up to $\\TS$-equivalence (see ). Moreover, the running time of the algorithm is poly(*n*, *s*, log(∣*T*∣)).\n[rem:gupta2014-char] Theorem 1.1 of is stated only for characteristic zero fields. However, in Remark 6.10 they explain how to make the algorithm work over any characteristic, for a large enough field. Thus, also holds over large enough fields in arbitrary characteristic.\n[rem:ze-anf] As a direct implication of, the reconstruction algorithm of can be converted into a zero-error algorithm, with expected quasipolynomial running time: Given black-box access to some $f\\_1\\in{\\ensuremath{{\\ensuremath{\\text{ANF}\\_{}}}^{\\GLaff({{\\mathbb{F}}})}}}$, we define *f*2 to be the output of the algorithm of on input *f*1, and then verify *f*1 = *f*2 using.\n### Dense subclasses of ΣΠΣ\nWe start by defining the canonical diagonal tensor of degree *d* and rank *s*, $\\sdmpol[s][d]\\in{{\\mathbb{F}}}[x\\_{1,1},\\ldots,x\\_{s,d}]$, and the resulting class of polynomials ${\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}$.\n[defT] Let $\\sdmpol[s][d]\\triangleq \\sum\\_{i=1}^{s}\\prod\\_{j=1}^{d}x\\_{i,j}$. I.e., it is a sum of *s* variable-disjoint monomials. For *n* ≥ *s* ⋅ *d*, we denote with $\\sdmpol[s][d]^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{n}({{\\mathbb{F}}})}}}}$ the orbit of $\\sdmpol[s][d]$ over F, under the action of the affine group. Finally, we denote with ${\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}$ the class of families of polynomials (*f**n*)*n*, such that for every *n* there exist *s* and *d* such that *n* ≥ *s* ⋅ *d* and $f\\_n(x\\_1,\\ldots,x\\_n) \\in \\sdmpol[s][d]^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{n}({{\\mathbb{F}}})}}}}$.\nClearly, $\\sdmpol[s][d]^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{n}({{\\mathbb{F}}})}}}}\\subset \\Sigma^{[s]}\\Pi^{[d]}\\Sigma$. We next define the class consisting of orbits of sparse polynomials.\nLet ΣΠGLaff(F) denote the class of families of polynomials that are computed by orbits of depth-2 circuits, of polynomially bounded size, over F. I.e., it is all families (*f**n*)*n*, of polynomially bounded degree, such that for some polynomially bounded *m*(*n*), there exist Σ*m*(*n*)Πdeg(*f**n*) circuits Φ*m*, in *k* ≤ *n*, many variables, such that *f**n* ∈ Φ*m*GL*n*aff(F).\nAs before we first give the basic observation connecting all three classes.\n[thm:denseInDepthThree] For every field F it holds that\n \n$${\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}\\subsetneq {\\ensuremath{{\\ensuremath{\\Sigma\\Pi}}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}} \\subseteq{\\ensuremath{\\Sigma\\Pi\\Sigma}}({{\\mathbb{F}}})\\;,$$ \nand for fields of size ∣F∣ ≥ *n* + 1 \n$${\\ensuremath{{\\ensuremath{\\Sigma\\Pi}}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}} \\subsetneq{\\ensuremath{\\Sigma\\Pi\\Sigma}}({{\\mathbb{F}}})\\;.$$ \n In addition,\n \n$$\\label{eq:Tdense}\n \\overline{{\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}} = \\overline{{\\ensuremath{{\\ensuremath{\\Sigma\\Pi}}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}}}= \\overline{{\\ensuremath{\\Sigma\\Pi\\Sigma}}({{\\mathbb{F}}})} \\;.$$ \nOur main results for this section are a quasipolynomial-size hitting set for the class ΣΠGLaff(F), and a polynomial-size interpolating set for ${\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}$.\ntheoremPITSINV[thm:PITSINV] Let $0\\neq g\\in{{\\mathbb{F}}}[{\\bm{x}}]$ have sparsity ≤ 2*t*. Let $(A,{\\bm{b}})\\in{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{n}({{\\mathbb{F}}})}}}$, and $f({\\bm{x}})=g(A{\\bm{x}}+{\\bm{b}})$. Then, for any (*t* + 1)-independent polynomial map G, *f* ∘ G ≠ 0.\n[cor:hs-sparse] For any integers *s*, *d*, *n*, there exists an explicit hitting set ${\\cal H}\\subset {{\\mathbb{F}}}^n$, of size $|{\\cal H}|= (nd)^{O(\\log s)}$, such that $\\cal H$ hits every nonzero polynomial *f* ∈ (Σ[*s*]Π[*d*])GL*n*aff(F). If ∣F∣ ≤ *n* ⋅ *d* then we let H be defined over an extension field K of F of size ∣K∣ > *n* ⋅ *d*.\nWe next state our result concerning an interpolating set for ${\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}$.\ntheoremPITsumSDMinv[thm:PITsumSDMinv] Let *n*, *s*1, *s*2, *d*1, *d*2 ∈ N be such that *n* ≥ *s*1 ⋅ *d*1, *s*2 ⋅ *d*2. For *i* ∈ {1, 2} let $f\\_i\\in \\sdmpol[s\\_i][d\\_i]^{{\\text{GL}\\_{n}({{\\mathbb{F}}})}}$, and let *f* = *f*1 − *f*2. If *f* ≠ 0, then any uniform 6-independent polynomial map G satisfies *f* ∘ G ≠ 0.\nFinally we note that the randomized reconstruction algorithm of Kayal and Saha, which works for (as it is termed in their paper) “non-degenerate” homogeneous depth-3 circuits, works for ${\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}$. This follows from the observation that ${\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}$ circuits are always non-degenerate.\n[special case of Theorem 1 of ]theoremreconstructSDM [thm:reconstructSDM] Let *n*, *d*, *s* ∈ N, *n* ≥ (3*d*)2 and $s\\leq(\\frac{n}{3d})^\\frac{d}{3}$. Let F be a field of characteristic zero or greater than *d**s*2. There is a randomized poly(*n*, *d*, *s*) = poly(*n*, *s*) time algorithm which takes as input black-box access to a polynomial *f* that is computable by a $\\sdmpol[s][d]^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{n}({{\\mathbb{F}}})}}}}$ circuit, and outputs a $\\sdmpol[s][d]^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{n}({{\\mathbb{F}}})}}}}$ circuit Φ computing *f* with high probability. Furthermore, Φ is unique up to $\\TPS[s,d][{{\\mathbb{F}}}]$-equivalence (see ).\n[rem:zeroErrRecon] As in, enables us to convert the reconstruction algorithm of to a zero-error algorithm, with expected polynomial running time. Given black-box access to some $f\\_1\\in{\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}$, we define *f*2 to be the output of the algorithm of on input *f*1, and then verify *f*1 ≡ *f*2 by applying to *f* = *f*1 − *f*2.\n### Robust hitting sets?\nAs we showed in, if a hitting set $\\mathcal H$ for a circuit class $\\mathcal C$ is *robust*, then $\\mathcal H$ hits $\\overline{\\mathcal C}$ as well. It is thus natural to ask whether our interpolating sets are already robust. Our next result shows that the property of being a *t*-independent map, which was sufficient for the constructions in (for the appropriate values of *t*), by itself is not sufficient for obtaining robust hitting sets. We prove this by constructing an independent polynomial map which gives rise to a provably non-robust hitting set. Our construction is the same as the one given by Forbes et al. (Construction 6.3 in the full version).\n[thm:tIndNotDense] Let F be of characteristic zero. For every *t*, there exists a uniform *t*-independent polynomial map G and a nonzero polynomial *f* such that *f* ∘ G ≡ 0, and *f* can be computed by a ΣΠΣ formula of size $t^{O(\\sqrt t)}$. If F has a positive characteristic then *f* can be computed by a ΣΠΣ formula of size *t**t*, or by a general formula of size *t**O*(log*t*). Furthermore, for a certain arrangement of the variables in a $\\sqrt{n}\\times \\sqrt{n}$ matrix, *f* can be taken to be the determinant of any (*t* + 1) × (*t* + 1) minor.\nPolynomial Identity Testing\n---------------------------\nSo far we discussed our work from the perspective of dense subclasses of classes for which no strong lower bounds are known. Here we put our work in the context of the polynomial identity testing problem.\nPolynomial Identity Testing (PIT for short) is the problem of designing efficient deterministic algorithms for deciding whether a given arithmetic circuit computes the identically zero polynomial. PIT has many applications, e.g. deciding primality, finding a perfect matching in parallel etc., and strong connection to circuit lower bounds. See for surveys on PIT and for a survey of algebraic hardness-randomness tradeoffs.\nPIT is considered both in the white-box model, in which we get access to the graph of computation of the circuit, and in the black-box model in which we only get query access to the polynomial computed by the circuit. Clearly, a deterministic PIT algorithm in the black-box model is equivalent to a hitting set for the circuit class. In this work we only focus on the black-box model.\n#### The continuant polynomial and algebraic branching programs:\nThe continuant polynomial is trivially computed by width-2 *Algebraic Branching Programs* (ABPs). Recall that an ABP of depth-*d* and width-*w* computes polynomials of the form ${\\ensuremath{\\text{Trace}}}{\\ensuremath{\\left( M\\_1({\\bm{x}})\\cdot \\ldots\\cdot M\\_d({\\bm{x}}) \\right) }}$, where each *M**i* is a *w* × *w* matrix whose entries contain variables or field elements. Ben-Or and Cleve proved that every polynomial in VPe can be computed by a width-3 ABP of polynomial-size.\nRaz and Shpilka gave the first polynomial-time white-box PIT algorithm for read-once ABPs (ABPs in which every variable can appear in at most one matrix). Forbes, Saptharishi and Shpilka gave the first quasipolynomial-sized hitting set for read-once ABPs (ROABPs). This result was slightly improved in for the case where the width of the ROABP is small. Anderson et al. gave a subexponential hitting set for read-*k* ABPs. We note that none of these models is strong enough to contain the orbit CGLaff(F). For ABPs that are not constant-read we do not have sub-exponential time PIT algorithms. Thus, the following is an interesting open problem (recall that by the result of Ben-Or and Cleve a PIT algorithm for width-3 ABPs works for VPe as well).\nGive a sub-exponential time PIT algorithm for ABPs of width-2.\nAlthough we do not have a PIT algorithm for general branching programs, in Kayal et al. gave an average-case reconstruction algorithm for low width ABPs. Kayal, Nair and Saha obtained a significantly better algorithm in. Their algorithm succeeds w.h.p, provided the ABP satisfies four non-degeneracy conditions (these conditions are defined in Section 4.3 of ). However, the ABP computing the continuant polynomial does not satisfy the non-degeneracy conditions that are required for their algorithm to work. Thus, does not follow from.\nTo the best of our knowledge, CGLaff(F) is the first natural[10](#fn10) computational class that is dense in VPe for which a polynomial (or even sub-exponential)-sized interpolating set (or a hitting set) is known.\n#### Read-Once formulas:\nHitting sets for read-once formulas were first constructed by Volkovich and Shpilka, who gave quasipolynomial-sized hitting set for the model, as well as a deterministic reconstruction algorithm of the same running time (earlier randomized reconstruction algorithms were known ). Minahan and Volkovich obtained a polynomial-sized hitting set for the class, which led to a similar improvement in the running time of the reconstruction algorithm. Anderson, van Melkebeek and Volkovich constructed a hitting set of size *n**k**O*(*k*) + *O*(*k*log*n*) for read-*k* formulas. All these results work in a slightly stronger model in which we allow to label leaves with univariate polynomials, of polynomial degree, such that every variable appears in at most one polynomial, or with sparse polynomials on disjoint sets of variables.\nThe read-once models that we consider here, ${\\ensuremath{{\\ensuremath{\\text{ANF}\\_{}}}^{\\GLaff({{\\mathbb{F}}})}}}$ and ROFGL(F), can be viewed as read-once formulas composed with a layer of addition gates with the restriction that the bottom layer of additions computes linearly independent linear functions. We note that these models do not fall into any of the previously studied models, as a variable can appear in all the linear functions.\nAs is the case with CGLaff(F), our hitting sets for ${\\ensuremath{{\\ensuremath{\\text{ANF}\\_{}}}^{\\GLaff({{\\mathbb{F}}})}}}$ and ROFGL(F) are the first sub-exponential-sized hitting sets for natural dense subclasses of VPe.\n#### Small depth circuits:\nThe class of ΣΠ circuits was considered in many works, see e.g. and polynomial-sized hitting sets were constructed. The class of ΣΠΣ circuits also received a lot of attention but with lesser success. Dvir and Shpilka and Karnin and Shpilka gave the first quasipolynomial-time white-box and black-box PIT algorithms for Σ[*k*]Π[*d*]Σ circuits, respectively. Currently, the best result is by Saxena and Seshadhri who gave a hitting set of size (*n**d*)*O*(*k*) for such circuits. In a subexponential-size hitting set for *multilinear* ΣΠΣ circuits was given. In, Agrawal et al. gave a hitting set of size *n**O*(1) ⋅ (*k**d*)*O*(*r*) for Σ[*k*]Π[*d*]Σ circuits, where *r* is an upper bound on the *algebraic rank* of the multiplication gates in the circuit. Thus, known quasipolynomial-size hitting sets for subclasses of ΣΠΣ circuits are known when the fan-in of the top gate is poly-logarithmic, or when the algebraic rank of the set of multiplication gates is poly-logarithmic. In contrast, polynomials in ${\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{n}({{\\mathbb{F}}})}}}}}$ and ΣΠGLaff(F), when viewed as ΣΠΣ circuits, can have polynomially many multiplication gates and their algebraic rank can be *n*. On the other hand, the corresponding ΣΠΣ circuits are such that the *different* linear functions that are computed at their bottom layer are linearly independent (when we view linear functions that are a constant multiple of each other as the same function). Thus, our provides a hitting set for a new subclass of ΣΠΣ circuits.\nTo the best of our knowledge, our results for ${\\ensuremath{\\mathcal T}^{{{\\ensuremath{\\text{GL}^{\\text{aff}}\\_{}({{\\mathbb{F}}})}}}}}$ and ΣΠGLaff(F) give the first sub-exponential size hitting sets for natural subclasses that are dense in ΣΠΣ.\nMore related work\n-----------------\nApproximations in algebraic complexity were first studied by Bini et al. in the context of algorithms for matrix multiplication. For more on the history of border rank in the context of matrix multiplication see notes of chapter 15 in. More recently, influenced by the GCT program, a lot of research was invested in trying to find polynomials characterizing tensors of small rank. See for a discussion on this approach. More recently, Kumar proved that *every* polynomial over C can be approximated by a Σ[2]ΠΣ circuit (of exponential degree).\nVery little is known about the closure of circuit classes. Forbes observed that the class of ROABPs is closed. I.e. $\\text{ROABP}=\\overline{\\text{ROABP}}$. We are not aware of other collapses or separation between general “natural” classes and their closures.\nBeside the reconstruction algorithms mentioned earlier, reconstruction algorithms are known for ΣΠ circuits ; for random depth three *powering* circuits ; for set-multilinear ΣΠΣ and ROABPs ; for ΣΠΣ circuits with bounded top fan-in ; and for multilinear depth-4 circuits with a constant top fan-in.\nIn general, we do not expect the reconstruction problem to be solvable efficiently, as the problem of finding the minimal circuit computing a given polynomial is a notoriously hard problem. A detailed discussion on the hardness of reconstruction can be found in.\nProof technique\n---------------\nOur proofs are based on the following simple yet important, and as far as we know novel, observations concerning *k*-independent polynomial maps. Specifically, our proofs are based on the following two claims:\n1. If we have a hitting-set generator *H* for nonzero polynomials of the form ${\\frac{\\partial f}{\\partial x\\_1}}$, for *f* ∈ C, and if G is a 1-independent map then *H* + G hits every nonzero *f* ∈ C. This is proved in.\n2. Similarly, we prove that if we have a hitting-set generator *H* for nonzero polynomials of the form ${{ \\left.\\kern-\\nulldelimiterspace f \\vphantom{\\big|} \\right|\\_{\\ell=0} }}(A{\\bm{x}}+{\\bm{b}})$, for *f* ∈ C, a linear function ℓ, and an invertible affine transformation $(A,{\\bm{b}})$, and if G is a 1-independent map then *H* + G hits every nonzero *f* ∈ C. This follows from.\nBy applying these claims *k* + *r* times we get that composition with a (*k* + *r*)-independent map allows to reduce the problem of hitting a class C to hitting polynomials of the form ${{ \\left.\\kern-\\nulldelimiterspace {\\frac{\\partial ^k f}{\\partial x\\_{i\\_1}\\partial x\\_{i\\_2}\\cdots\\partial x\\_{i\\_k}}} \\vphantom{\\big|} \\right|\\_{\\ell\\_1=\\ldots=\\ell\\_r=0} }}$. Thus, if we could prove that for a class C, there is such a sequence of derivatives and restrictions that simplifies the polynomials in it to a degree that they can be easily hit by some map *H*, then we conclude that *H* + G*k* + *r*, for a (*k* + *r*)-independent map G*k* + *r*, is a hitting set generator for C.\nIt seems that all that is left to do is prove that for each of the orbits that we consider in Section [secResults] that is such small *k* and *r*. However, a potential problem is that a partial derivative of the polynomial $g({\\bm{x}})=f(A{\\bm{x}}+{\\bm{b}})$ gives ${\\frac{\\partial g}{\\partial x\\_1}}=\\sum\\_{i=1}^{n} {\\frac{\\partial f}{\\partial y\\_i}}\\cdot {\\frac{\\partial \\ell\\_i}{\\partial x\\_1}}$, where ℓ*i* is the *i*th coordinate of $A{\\bm{x}}+{\\bm{b}}$. Thus, it is no longer a derivative composed with an affine transformation but rather a sum of such derivatives, which could lead to polynomials outside of our class. For example, it is not hard ot prove that if we compose the ROF *y*1 ⋅ *y*2 ⋅ *y*3 with (*x*1, *x*1 + *x*2, *x*1 + *x*3) and then take a derivative according to *x*1, then the resulting polynomial, ${\\frac{\\partial {\\ensuremath{\\left( x\\_1\\cdot (x\\_1+x\\_2)\\cdot (x\\_1+x\\_3) \\right) }}}{\\partial x\\_1}}=3x\\_1^2 + 2x\\_1\\cdot (x\\_2+x\\_3) + x\\_2\\cdot x\\_3$, is not in the orbit of any ROF. The solution to this problem is to take a *directional derivative* in a direction coming from a *dual basis*. For example if $\\ell\\_i({\\bm{v}}\\_j)=\\delta\\_{i,j}$ then ${\\frac{\\partial g}{\\partial {\\bm{v}}\\_1}}={\\frac{\\partial f}{\\partial x\\_1}}{\\ensuremath{\\left( A{\\bm{x}}+{\\bm{b}}\\right) }}$ (see ). Now, comes another important observation: If *H* is a hitting-set generator for nonzero polynomials of the form ${\\frac{\\partial f}{\\partial {\\bm{v}}}}$, for *f* ∈ C and a direction ${\\bm{v}}$, and if G is a 1-independent map then *H* + G hits every nonzero *f* ∈ C. The point is that if ${\\frac{\\partial f}{\\partial {\\bm{v}}}}\\circ H \\neq 0$ then for some *i*, ${\\frac{\\partial f}{\\partial x\\_i}}\\circ H \\neq 0$ and the claim follows from the first claim above. Thus, composition with (*k* + *r*)-independent maps allows us to reduce the problem of hitting a class C to finding a generator for polynomials that are obtained as a restriction to a subspace of co-dimension *r* of a directional partial derivative of order *k* of polynomials in C.\nLet us demonstrate this idea for the case of orbits of sparse polynomials. I.e. to polynomials of the form $g({\\bm{x}})=f(A{\\bm{x}}+{\\bm{b}})$, where the number of monomials in *f* is at most 2*t*. It is not hard to see that there is a variable *x**i* such that if we consider ${{ \\left.\\kern-\\nulldelimiterspace f \\vphantom{\\big|} \\right|\\_{x\\_i=0} }}$ and ${\\frac{\\partial f}{\\partial x\\_i}}$ then one of these polynomials has at most 2*t* − 1 monomials.[11](#fn11) Thus, after a a sequence of at most *t* partial derivatives and restrictions, we get to a polynomial with only one monomial that we can easily hit. Hence after at most *t* directional derivatives and restrictions to a subspace, we get that *g* is a product of linear forms, which we can easily hit. This proves that any (*t* + 1)-independent map hits such nonzero polynomials *g*.\nTo obtain interpolating sets for our classes (and also a reconstruction algorithm for the orbit of the continuant polynomial), we prove that if two polynomials in the orbit, of any of the classes that we consider, are different, then there is a sequence of a few (directional) partial derivatives and restrictions that makes one of them zero while keeping the other nonzero. Using this and the ideas from above we construct our interpolating sets.\nDiscussion\n----------\nAs shows, our hitting sets are not necessarily robust. It is thus an outstanding open problem to find a way to convert a hitting set to a robust one (recall Problem [prob:robust]).\nThe following toy example demonstrates that converting a hitting set for a class C to a robust hitting set for C, cannot be done in a black-box manner and one has to use information about C for that: let C(F) be the class of all polynomials with non-zero free term. A trivial hitting set for C would simply be the singleton set $\\mathcal H=\\{{\\bm{0}}\\}$. On the other hand, it is clear that $\\overline{\\mathcal C}={{\\mathbb{F}}}[{\\bm{x}}]$, so making H robust would yield a hitting set for *all* polynomials. Note, however, that this is not a “computational class.”\nAnother potential approach for obtaining robust hitting sets follows from the observation that the set of queries made by a non-adaptive deterministic black-box reconstruction algorithm, $\\cal A$, for C, which is *continuous* at 0 (i.e. at the identically zero polynomial) is a robust hitting set for C. The reason is, that if $0\\neq f\\in\\overline{\\mathcal C}$ and $\\{f\\_k\\}\\_{k=1}^\\infty\\subseteq\\mathcal C$ converges to *f*, then for large enough *k*: ${\\left\\lVertf\\_k\\right\\rVert}\\_2\\geq\\frac{1}{2}{\\left\\lVertf\\right\\rVert}\\_2>0$. As the *f**k* sequence converges and polynomial evaluation is continuous (and their evaluation vectors are bounded), the sequence ${\\bm{v}}\\_k={{ \\left.\\kern-\\nulldelimiterspace f\\_k \\vphantom{\\big|} \\right|\\_{\\mathcal H} }}\\subseteq{\\mathbb{C}}^{|\\mathcal H|}$ must also converge to some vector ${\\bm{v}}={{ \\left.\\kern-\\nulldelimiterspace f \\vphantom{\\big|} \\right|\\_{\\mathcal H} }}\\in {\\mathbb{C}}^{|\\mathcal H|}$. If ${\\bm{v}}={\\bm{0}}$ then the continuity of $\\mathcal A$ at ${\\bm{0}}$ implies the coefficients of the polynomials $f\\_k({\\bm{x}})$ must also converge to zero, as $\\mathcal A({\\bm{0}})= 0$. This would contradict ${\\left\\lVertf\\_k\\right\\rVert}\\_2\\geq\\frac{1}{2}{\\left\\lVertf\\right\\rVert}\\_2>0$ for large enough *k*, so ${\\bm{v}}\\neq {\\bm{0}}$ and thus $\\mathcal H$ hits $\\overline{\\mathcal C}$.\nThus, an interesting challenge is to derandomize the reconstruction algorithms given in, hoping that the resulting algorithms are continuous at ${\\bm{0}}$. We note however, that currently we do not even have efficient deterministic root-finding algorithms over C. It is also known that in general, finding the minimal circuit for a polynomial can be very difficult. E.g., in it was shown that the question of computing, or even approximating, tensor rank, for degree 3 tensors, is NP hard, over any field.\nIn, we have seen that any uniform *O*(log(*s**n*))-independent polynomial map G is an interpolating set generator for ${{\\orbitofgroup{\\ensuremath{\\mathcal T}}{\\GLplus[]}}}$; i.e, G induces an interpolating set $\\mathcal H$ for ${{\\orbitofgroup{\\ensuremath{\\mathcal T}}{\\GLplus[]}}}$. On the other hand, in, we constructed such a map G, with the additional property that G is *not* a hitting set generator for ΣΠΣ circuits. In particular, this implies that the induced (non-efficient) reconstruction map $\\mathcal A$ (that takes *f*(H) and returns a circuit computing *f*) is not continuous at ${\\bm{0}}$.\nWe conclude this section with a somewhat vague question.\nFind a “computational” class of polynomials $\\mathcal C$ with a known hitting set $\\mathcal H$, such that $\\overline{\\mathcal C}\\neq\\mathcal C$, and convert $\\mathcal H$ to a robust hitting set.\nWe note that the closure of Σ⋀Σ circuits (i.e. circuits computing polynomials of the form $\\sum\\_i \\ell\\_i({\\bm{x}})^d$, for linear functions ℓ*i*) is contained in the class of commutative read-once algebraic branching programs (see ). Thus, the hitting set for the latter class gives a robust hitting set for the former. However, we seek an example in which there is an “interesting” conversion of a hitting set to a robust one.\nOrganization\n------------\nThe paper is organized as follows. Section [sec:prelim] contains some more basic notations and definitions as well as characterization of the groups of symmetries of ANFΔ and of $\\sdmpol$. In Section [sec:k-ind] we give properties and constructions of *k*-independent polynomial maps and prove. In Section [sec:cont] we study the continuant polynomial and prove. In Section [sec:rof] we study orbits of ROFs and ROANFs and prove. Section [sec:sps] contains our results for subclasses of ΣΠΣ circuits (). The appendix contains missing definitions that are required for explaining the reconstruction algorithm of.\nPreliminaries\n=============\nNotation\n--------\nFor *k* ∈ N, we denote $[k]\\triangleq \\{1,2,3,\\ldots,k\\}$ and $[k]\\_0\\triangleq \\{0,1,2,\\ldots,k-1\\}$. We use boldface lowercase letters to denote tuples of variables or vectors, as in ${\\bm{x}}=(x\\_1,\\ldots,x\\_n)$, ${\\bm{a}}=(a\\_1,\\ldots,a\\_m)$, when the dimension is clear from the context. For any two elements *i*, *j* coming from some set *S* (usually *i* and *j* will be numbers), *δ**i*, *j* equals 1 when *i* = *j* and 0 otherwise. For every *m* ∈ N we denote with *I**m* the *m* × *m* identity matrix. When we wish to treat the entries of a matrix *A* as formal variables, we use boldface $\\bm A$. We will note use capital bold face letters other than to denote such matrices.\nFor an exponent vector ${\\bm{a}}=(a\\_1,\\ldots,a\\_n)\\in{\\mathbb{N}}^n$, we denote ${\\bm{x}}^{{\\bm{a}}}\\triangleq \\prod\\_{i=1}^n x\\_i^{a\\_i}$. In some cases we shall consider “monomials” with respect to set of linear functions {ℓ*i*}*i* = 1*m*: for an exponent vector ${\\bm{e}}=(e\\_1,\\ldots,e\\_m)\\in {{\\mathbb{N}}}^m$ we denote ${\\bm \\ell}^{{\\bm{e}}} = \\prod\\_{i=1}^m\\ell\\_i^{e\\_i}$ and refer to it as an *{ℓ*i*}-monomial*. For a polynomial $f({\\bm{x}})$ we define the *monomial support* of *f*, denoted \\normalfont mon(*f*), as the set of monomials with non-zero coefficient in *f*. The *variable set* of *f*, denoted var(*f*), is the set of variables that *f* depends on. I.e., all variables that appear in \\normalfont mon(*f*). The individual degree of a variable *x**i* in $f({\\bm{x}})$ is the degree of *f* as a polynomial in *x**i*. A polynomial $f\\in{{\\mathbb{F}}}[{\\bm{x}}]$ of deg(*f*) ≤ 1 is called a linear function, and if *f* is homogeneous then it is called a *linear form*. For a polynomial $f\\in{{\\mathbb{F}}}[{\\bm{x}}]$ and an integer *k* ∈ N we denote by *f*[*k*] the degree-*k* homogeneous part of $f({\\bm{x}})$,i.e. the sum of all monomials of *f* of degree exactly *k*. In particular, \n$$f({\\bm{x}})=f^{[0]}({\\bm{x}})+f^{[1]}({\\bm{x}})+\\ldots+f^{[\\deg(f)]}({\\bm{x}}) \\;.$$ \n Note that for a linear function *f*, *f*[1] is a linear form. We say that a polynomial *f* is homogeneous of degree *k* or that *f* is *k*-homogeneous if *f* = *f*[*k*]. We say a set of linear functions ${\\ensuremath{\\left\\{ \\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_n({\\bm{x}}) \\right\\} }} \\subset {{\\mathbb{F}}}[{\\bm{x}}]$ is *linearly independent* if the set {ℓ*i*[1]} is linearly independent.[12](#fn12) Given a polynomial $f({\\bm{x}})$, a subset of variables $\\bm y\\subseteq\\{x\\_1,\\ldots,x\\_n\\}$ and an assignment to those variables ${\\bm{a}}\\in{{\\mathbb{F}}}^{|\\bm y|}$, we denote by ${{ \\left.\\kern-\\nulldelimiterspace f \\vphantom{\\big|} \\right|\\_{\\bm y={\\bm{a}}} }}\\in{{\\mathbb{F}}}[{\\bm{x}}\\setminus\\bm y]$ the polynomial resulting from assigning the values of ${\\bm{a}}$ to the variables of $\\bm y$ in $f({\\bm{x}})$. We sometimes abuse notation and write $\\bm y\\subseteq[n]$ to indicate the indices of the assigned variables instead of the variables themselves.\nGiven an arithmetic circuit Φ, we frequently denote by $\\Phi({\\bm{x}})$ or, abusing notation, by Φ, the polynomial computed at the output node of Φ. Given a class of arithmetic circuits C and a polynomial $f\\in{{\\mathbb{F}}}[{\\bm{x}}]$, we say *f* ∈ C if *f* can be computed by some circuit from C. For a circuit class C(F) we denote by $\\overline{\\mathcal C}({{\\mathbb{F}}})$ the *closure* of C(F), as in.\nGroups of matrices and their action\n-----------------------------------\nWe first list some simple properties of composition with a linear (or affine) transformation that we shall use implicitly.\n[obsTrivProps] For any *m* variate polynomial *f*(*x*1, …, *x**m*) and *n* ≥ *m*:\n* For any *A* ∈ GL*n*(F) and *d* ∈ N, $f^{[d]}(A{\\bm{x}})$ is the *d*-homogeneous part of $f(A{\\bm{x}})$.\n* For any *A* ∈ GL*n*aff(F), $f({\\bm{x}})$ is irreducible if and only if $f{\\ensuremath{\\left( A{\\bm{x}}\\right) }}$ is irreducible.\n* The set of matrices *A* for which $f({\\bm{x}})=f(A{\\bm{x}})$ forms a multiplicative subgroup of GL*n*(F) and a similar claim holds for GL*n*aff(F).\nWe next define some special groups that serve as group of symmetries of some of the models that we consider. We first define the group of symmetries of ${\\ensuremath{\\text{ANF}\\_{\\Delta}}}({\\bm{x}})$.\n[defTR] For *m*, Δ ∈ N such that *m* = 2Δ, the *tree-symmetry group* $\\TR[m][{{\\mathbb{F}}}]$ denotes the automorphisms of a rooted complete binary tree of depth Δ. It is defined recursively as follows.\n* For *m* = 1, $\\TR[1][{{\\mathbb{F}}}]$ consists only of the identity matrix.\n* For *m* > 0, $\\TR[m][{{\\mathbb{F}}}]$ is generated by matrices of the form \n$$\\begin{pmatrix}\n A&0\\\\0&B\n \\end{pmatrix}\\;\\;\\;and\\;\\;\\;\\begin{pmatrix}\n 0&I\\_{\\frac{m}{2}}\\\\I\\_{\\frac{m}{2}}&0\n \\end{pmatrix}$$ \n where $A,B\\in \\TR[\\frac{m}{2}][{{\\mathbb{F}}}]$.\n[defTS] For any *m* = 4Δ, the *tree-scale group* $\\TS[m][{{\\mathbb{F}}}]$ is the group generated by elements of $\\TR[m][{{\\mathbb{F}}}]$ and matrices of the form \n$$\\begin{pmatrix}\n \\alpha I\\_{\\frac{m}{4}}&0&0&0\\\\\n 0&\\alpha^{-1}I\\_{\\frac{m}{4}}&0&0\\\\\n 0&0&\\beta I\\_{\\frac{m}{4}}&0\\\\\n 0&0&0&\\beta^{-1}I\\_{\\frac{m}{4}}\n \\end{pmatrix}$$ \n where 0 ≠ *α*, *β* ∈ F.\nThe importance of the group $\\TS[m][{{\\mathbb{F}}}]$ stems from the fact that it is the symmetry group of ANFΔ. To intuitively see why this is the case, notice that in any representation of an ANF one may swap children of any node without changing the output polynomial. We call such symmetries “tree-symmetries” and they are captured by the group $\\TR[n][{{\\mathbb{F}}}]$. A second source of ambiguity comes from the fact that we can rescale the formula. Recall that the output polynomial is of the form *f*1 ⋅ *f*2 + *f*3 ⋅ *f*4 (). Clearly, the output does not change if we replace *f*1 by, say, 2*f*1 and *f*2 by *f*2/2. Such rescaling symmetries are captured by the group $\\TS[n][{{\\mathbb{F}}}]$. Finally, another source for ambiguity comes from the fact that the quadratic polynomials computed at the bottom two layers of the ANF may have different representations. For example, \n4*x**y* + 4*w**z* = (*x* + *y* + *w* − *z*) ⋅ (*x* + *y* − *w* + *z*) + (*w* + *z* + *x* − *y*) ⋅ (*w* + *z* − *x* + *y*) . \n As there is an infinite number of representations for each quadratic polynomial (over infinite fields), we can expect to characterize the symmetries in term of the quadratics computed at the bottom two layers of the ANF.\n[Special case of Theorem 5.43(iii) of ][fact:anf-sym] Let *m*, Δ, *n* ∈ N such that *m* = 4Δ − 1 ≤ *n*/4. Let *f* = ANFΔ(ℓ1, …, ℓ4*m*) ∈ ANFΔGL*n*aff(F). Let *Q* = (*q*1, …, *q**m*) be the list of quadratic polynomials that are computed at the bottom two layers of the formula ANFΔ(ℓ1, …, ℓ4*m*). In particular, *f* = ANFΔ − 1(*q*1, …, *q**m*). If *Q*ʹ = (*q*ʹ1, …, *q*ʹ*m*) is any other *m*-tuple of quadratic polynomials for which *f* = ANFΔ − 1(*q*ʹ1, …, *q*ʹ*m*) then *Q* is $\\TS[m][{{\\mathbb{F}}}]$-equivalent to *Q*ʹ.\nNext, we define the group of symmetries of $\\sdmpol({\\bm{x}})$.\nFor any *n* ∈ N the *permutation-scale group*, denoted $\\PS[n][{{\\mathbb{F}}}]$, is the set of all matrices $A\\in\\GL[n][{{\\mathbb{F}}}]$ which are row-permutations of non-singular diagonal matrices with determinant one.\nFor example, $\\begin{pmatrix}\n 0 & -2 & 0 \\\\\n 0 & 0 & -1 \\\\\n 1/2 & 0 & 0 \\end{pmatrix}\\in \\PS[3][{\\mathbb{C}}]$.\n[sdmSymGroup] Let *s*, *d*, *n* ∈ N such that *n* = *s* ⋅ *d*. A matrix $A\\in\\GL[n][{{\\mathbb{F}}}]$ is a member of the *tensor permutation-scale group*, denoted $\\TPS[s,d][{{\\mathbb{F}}}]$, if *A* = (*P* ⊗ *I**d*) ⋅ *B*, where *P* is an *s* × *s* permutation matrix and $B=\\begin{pmatrix}\n B\\_1 & 0 & \\ldots & 0 \\\\\n 0 & B\\_2 & \\ldots & 0 \\\\\n \\vdots & & \\ddots & \\vdots \\\\\n 0 & \\ldots & 0 & B\\_d\n \\end{pmatrix}$ is a block diagonal matrix such that each block *B**i* of *B* satisfies $B\\_i\\in\\PS[d][{{\\mathbb{F}}}]$.\nFor example, for *s* = *d* = 2 the matrix $A= \\begin{pmatrix}\n 0 & 0 & 0 & 2 \\\\\n 0 & 0 & 1/2 & 0 \\\\\n -1 & 0 & 0 & 0 \\\\\n 0 & -1 & 0 & 0\n\\end{pmatrix}$ is in $\\TPS[2,2][{\\mathbb{C}}]$, as for $P= \\begin{pmatrix}\n 0 & 1\\\\\n 1 & 0\n\\end{pmatrix}$ and $B=\\begin{pmatrix}\n -1 & 0 & 0 & 0 \\\\\n 0 & -1 & 0 & 0\\\\\n 0 & 0 & 0 & 2 \\\\\n 0 & 0 & 1/2 & 0\n\\end{pmatrix}$, we have *A* = (*P* ⊗ *I*2) ⋅ *B*, and clearly each block of *B* is in $\\PS[2][{\\mathbb{C}}]$. \nAnother way of defining the group is as follows: index rows and columns of *A* with pairs (*i*, *j*) ∈ [*s*] × [*d*]. Then, $A \\in \\TPS[s,d][{{\\mathbb{F}}}]$ if and only if there exists a permutation *π* : [*s*] → [*s*], and for all *i* ∈ [*s*] permutations *θ**i* : [*d*] → [*d*] and constants *α**i*, *j* satisfying ∏*j* = 1*d**α**i*, *j* = 1, such that *A*(*i*, *j*), (*i*ʹ, *j*ʹ) = *δ**π*(*i*), *i*ʹ ⋅ *δ**θ**i*(*j*), *j*ʹ ⋅ *α**i*, *j* for all *i*, *j*.\nWe next prove that $\\TPS[s,d][{{\\mathbb{F}}}]$ is the group of symmetries of $\\sdmpol({\\bm{x}})$. In other words, we show that $\\sdmpol({\\bm{x}})=\\sdmpol(A{\\bm{x}})$ if and only if $A\\in \\TPS[s,d][{{\\mathbb{F}}}]$. Intuitively, $\\sdmpol$ admits no symmetries other than the trivial ones: permutations on the product gates, and internal permutation-scale of each product gate such that the product of the scale coefficients is 1. This is exactly captured by the group $\\TPS[s,d][{{\\mathbb{F}}}]$, which is therefore contained in the group of symmetries of $\\sdmpol({\\bm{x}})$.\nlemmasdmSymmetry[sdmSymmetry] Let *s*, *d*, *n* ∈ N, such that *d* > 2 and *n* = *s* ⋅ *d*. If *A* ∈ GL*n*(F) satisfies $\\sdmpol({\\bm{x}})=\\sdmpol(A{\\bm{x}})$, then $A\\in\\TPS[s,d][{{\\mathbb{F}}}]$.\n[Proof of ]\nFix linear forms ℓ1, 1, …, ℓ*s*, *d* such that the (*i*, *j*)th coordinate of $A{\\bm{x}}$ (using the indexing [*n*] = [*s*] × [*d*]) is $\\ell\\_{i,j}({\\bm{x}})$, and $\\sdmpol(A{\\bm{x}})=\\sum\\_{i=1}^s\\prod\\_{j=1}^d\\ell\\_{i,j}({\\bm{x}})$. By the discussion above, our goal is to prove that there exists a permutation *π* : [*s*] → [*s*], and for all *i* ∈ [*s*] permutations *θ**i* : [*d*] → [*d*] and constants *α**i*, *j* satisfying ∏*j* = 1*d**α**i*, *j* = 1, such that $\\ell\\_{i,j}({\\bm{x}})=\\alpha\\_{i,j}\\cdot x\\_{\\pi(i),\\theta\\_i(j)}$ for all *i*, *j*. Fix some *i* ∈ [*s*] and take a derivative of the equation $\\sdmpol({\\bm{x}})=\\sdmpol(A{\\bm{x}})$ by *x**i*, 1:\n \n$$\\label{eq:der-tsd}\n \\prod\\_{j\\in\\{2,\\ldots,d\\}}x\\_{i,j}={\\frac{\\partial \\sdmpol({\\bm{x}})}{\\partial x\\_{i,1}}}={\\frac{\\partial \\sdmpol(A{\\bm{x}})}{\\partial x\\_{i,1}}}=\\sum\\_{r=1}^s{\\frac{\\partial }{\\partial x\\_{i,1}}}\\left(\\prod\\_{j=1}^d\\ell\\_{r,j}({\\bm{x}})\\right)\\;.$$ \nFor *r* ∈ [*s*], denote $h\\_{i,r}({\\bm{x}})\\triangleq {\\frac{\\partial }{\\partial x\\_{i,1}}}\\left(\\prod\\_{j=1}^d\\ell\\_{r,j}({\\bm{x}})\\right)$. As *d* > 2, the LHS of Equation is a reducible polynomial, so $\\sum\\_{r=1}^sh\\_{i,r}({\\bm{x}})$ is also reducible. Composition with a non-singular matrix preserves reducibility, so $\\sum\\_{r=1}^sh\\_{i,r}(A^{-1}{\\bm{x}})$ is also reducible. However, $h\\_{i,1}(A^{-1}{\\bm{x}}),\\ldots,h\\_{i,s}(A^{-1}{\\bm{x}})$ are *s* variable-disjoint, multilinear polynomials, each of which is either (*d* − 1)-homogeneous or zero. Thus, by below, at most one *h**i*, *r*(*A*− 1*x*) can be non-zero. Accordingly, for every variable *x**i*, *j* there exists a unique *i*ʹ such that $x\\_{i,j}\\in\\text{var}\\left(\\prod\\_{j'=1}^d\\ell\\_{i',j'}({\\bm{x}})\\right)$. Thus, for some *i*ʹ we have\n \n$$\\begin{aligned}\n \\prod\\_{j\\in\\{2,\\ldots,d\\}}x\\_{i,j}={\\frac{\\partial }{\\partial x\\_{i,1}}}\\left(\\prod\\_{j=1}^d\\ell\\_{i',j}({\\bm{x}})\\right)\\;.\\label{eqRefOneDeriv}\\end{aligned}$$ \nFor any *j* > 1, if we take a derivative of by *x**i*, *j* then the LHS is clearly non-zero. Thus, both *x**i*, 1 and *x**i*, *j* exist in $\\text{var}\\left(\\prod\\_{j'=1}^d\\ell\\_{i',j'}({\\bm{x}})\\right)$, proving variables in the same product gate of $\\sdmpol({\\bm{x}})$ are mapped to the same product gate of $\\sdmpol(A{\\bm{x}})$. A similar argument shows that variables from distinct product gates of $\\sdmpol({\\bm{x}})$ are mapped to different product gates of $\\sdmpol(A{\\bm{x}})$. It follows that product gates of $\\sdmpol(A{\\bm{x}})$ are variable-disjoint and that there exists a permutation *π* : [*s*] → [*s*] satisfying \n$$\\forall i\\in[s]:\\qquad\\text{var}\\left(\\prod\\_{j=1}^d\\ell\\_{i,j}({\\bm{x}})\\right)=\\{x\\_{\\pi(i),1},\\ldots,x\\_{\\pi(i),d}\\} \\;.$$ \n In particular, there can be no cancellations between different product gates of $\\sdmpol(A{\\bm{x}})$. Therefore, by multilinearity, for every *i* ∈ [*s*], the linear forms $\\ell\\_{i,1}({\\bm{x}}),\\ldots,\\ell\\_{i,d}({\\bm{x}})$ must be variable-disjoint. Exactly *d* variables appear in $\\prod\\_{j=1}^d\\ell\\_{i,j}({\\bm{x}})$, so for every *i* ∈ [*s*] and *j* ∈ [*d*] there exists a permutation *θ**i* : [*d*] → [*d*] and a non-zero constant *α**i*, *j* ∈ F such that $\\ell\\_{i,j}({\\bm{x}})=\\alpha\\_{i,j}x\\_{\\pi(i),\\theta\\_i(j)}$. As ∏*j* = 1*d**α**i*, *j* is the coefficient of ∏*j* = 1*d**x**π*(*i*), *j* in $\\sdmpol(A{\\bm{x}})$, this product must be 1, which completes the proof.\n[irreducibleMultilinear] If *f*, *g* are non-constant, variable-disjoint, multilinear polynomials, then for every *c* ∈ F the polynomial $f({\\bm{x}})+g({\\bm{x}})+c$ is irreducible.\n*k*-independent polynomial maps and their properties\n====================================================\nAll the hitting and interpolating sets that we construct are based on *k*-independent polynomial maps (). We next give some simple properties of independent polynomial maps, that follow immediately from the definition.\n[obs:kwise] It holds that\n1. [reduceIndPolyMap] If ${{\\ensuremath{\\mathcal{G}}}}(\\bm y,\\bm z)$ is a (*k* + 1)-independent polynomial map, then there exists a subset of variables *S* and an assignment ${\\bm{\\alpha}}\\in{{\\mathbb{F}}}^{|S|}$ such that ${{ \\left.\\kern-\\nulldelimiterspace {{\\ensuremath{\\mathcal{G}}}}\\vphantom{\\big|} \\right|\\_{S={\\bm{\\alpha}}} }}$ is a *k*-independent polynomial map.\n2. [item:coordsGenInd] For any *k* ≥ 1, the *n* coordinates of any *k*-independent polynomial map are F-linearly independent.\n3. [item:ind-lin] Let $\\ell\\_1({\\bm{x}})$ and $\\ell\\_2({\\bm{x}})$ be linearly independent linear functions in ${{\\mathbb{F}}}[{\\bm{x}}]$. Let ${{\\ensuremath{\\mathcal{G}}}}({\\bf y},z\\_1,z\\_2)$ be any 2-independent polynomial map. Consider ℓ1 ∘ G and ℓ2 ∘ G as polynomials in *z*1, *z*2 over ${{\\mathbb{F}}}({\\bf y})$. Then, (ℓ1 ∘ G)[1] and (ℓ2 ∘ G)[1] are linearly independent, as linear forms in *z*1, *z*2 over ${{\\mathbb{F}}}({\\bf y})$.\nWe next give the construction of of a *k*-independent polynomial map (denoted *G**k* in ).\n[defSVGEN] Fix *n* and a set of *n* distinct field elements $\\mathcal A=\\{\\alpha\\_1,\\ldots,\\alpha\\_n\\}\\subseteq{{\\mathbb{F}}}$.[13](#fn13) For every *i* ∈ [*n*] let *L**i*(*w*) : F → F be the *i*th Lagrange Interpolation polynomial for the set $\\mathcal A$. That is, each *L**i*(*w*) is polynomial of degree *n* − 1 that satisfies *L**i*(*α**j*) = *δ**i*, *j*. We define GSV1(*y*1, *z*1) : F2 → F*n* as: \n$${\\ensuremath{{{\\ensuremath{\\mathcal{G}}}}^{\\text{SV}}}}\\_1(y\\_1,z\\_1)\\triangleq\\left(L\\_1(y\\_1)\\cdot z\\_1,L\\_2(y\\_1)\\cdot z\\_1,\\ldots,L\\_n(y\\_1)\\cdot z\\_1\\right),$$ \n and for any *k* ≥ 1, we define GSV*k* : F2*k* → F*n* as: \n$${\\ensuremath{{{\\ensuremath{\\mathcal{G}}}}^{\\text{SV}}}}\\_k(\\bm y,\\bm z)\\triangleq{\\ensuremath{{{\\ensuremath{\\mathcal{G}}}}^{\\text{SV}}}}\\_1(y\\_1,z\\_1)+{\\ensuremath{{{\\ensuremath{\\mathcal{G}}}}^{\\text{SV}}}}\\_1(y\\_2,z\\_2)+\\ldots+{\\ensuremath{{{\\ensuremath{\\mathcal{G}}}}^{\\text{SV}}}}\\_1(y\\_k,z\\_k)=\\left(\\sum\\_{j=1}^kL\\_1(y\\_j)\\cdot z\\_j,\\sum\\_{j=1}^kL\\_2(y\\_j)\\cdot z\\_j,\\ldots,\\sum\\_{j=1}^kL\\_n(y\\_j)\\cdot z\\_j\\right).$$ \n[obs:SVgen] GSV*k* is a *k*-independent polynomial map, in which each variable has degree at most *n* − 1.\nThe generator GSV*k* can be converted to a uniform *k*-independent polynomial map by adding another *k* control variables *y**k* + 1, …, *y*2*k*, and swapping out the *L**i*(*y**j*)s for their homogenizations $y\\_{j+k}^{n-1}L\\_i{\\ensuremath{\\left( \\frac{y\\_j}{y\\_{j+k}} \\right) }}$:\n[defsvgenhom] With the notation used in, define the *uniform SV-generator* with *k* independence GSV-hom*k* : F3*k* → F*n* as:\n \n$$\\begin{aligned}\n{\\ensuremath{{{\\ensuremath{\\mathcal{G}}}}^{\\text{SV-hom}}}}\\_k{\\ensuremath{\\left( y\\_1,\\ldots,y\\_{2k},z\\_1,\\ldots,z\\_k \\right) }}&\\triangleq y\\_{1+k}^{n-1}\\cdot{\\ensuremath{{{\\ensuremath{\\mathcal{G}}}}^{\\text{SV}}}}\\_1{\\ensuremath{\\left( \\frac{y\\_1}{y\\_{1+k}},z\\_1 \\right) }}+y\\_{2+k}^{n-1}\\cdot{\\ensuremath{{{\\ensuremath{\\mathcal{G}}}}^{\\text{SV}}}}\\_1{\\ensuremath{\\left( \\frac{y\\_2}{y\\_{2+k}},z\\_2 \\right) }}+\\ldots +y\\_{2k}^{n-1}\\cdot{\\ensuremath{{{\\ensuremath{\\mathcal{G}}}}^{\\text{SV}}}}\\_1{\\ensuremath{\\left( \\frac{y\\_k}{y\\_{2k}},z\\_k \\right) }}\\\\\n&=\\left(\\sum\\_{j=1}^ky\\_{j+k}^{n-1}L\\_1{\\ensuremath{\\left( \\frac{y\\_j}{y\\_{j+k}} \\right) }}\\cdot z\\_j,\\sum\\_{j=1}^ky\\_{j+k}^{n-1}L\\_2{\\ensuremath{\\left( \\frac{y\\_j}{y\\_{j+k}} \\right) }}\\cdot z\\_j,\\ldots,\\sum\\_{j=1}^ky\\_{j+k}^{n-1}L\\_n{\\ensuremath{\\left( \\frac{y\\_j}{y\\_{j+k}} \\right) }}\\cdot z\\_j\\right).\\end{aligned}$$ \nGSV-hom*k* is a uniform *k*-independent polynomial map, with individual degrees at most *n* − 1.\nWe next show how we can use *k*-independent polynomial maps in order to, roughly, simulate a *k*th order directional derivative or, project a polynomial to a subspace of co-dimension *k*. We first need to define the notion of a directional derivative.\n[def:DeriveLinear] For an *n*-variate polynomial $f\\in{{\\mathbb{F}}}[{\\bm{x}}]$ and ${\\bm{v}}=(v\\_1,\\ldots,v\\_n)\\in{{\\mathbb{F}}}^n$, the *derivative of $f({\\bm{x}})$ in the direction ${\\bm{v}}$* is defined as: \n$${\\frac{\\partial f}{\\partial {\\bm{v}}}}=\\sum\\_{i=1}^n v\\_i\\cdot{\\frac{\\partial f}{\\partial x\\_i}}.$$ \nIf F has positive characteristic then by $\\frac{\\partial F}{\\partial x\\_i}$ we refer to the formal derivative (which in the case of fields of characteristic zero is equal to the analytical definition). Observe that we still have that \n$$\\frac{\\partial^2 f}{\\partial y \\partial x} = \\frac{\\partial^2 f}{\\partial x \\partial y} \\text{,} \\quad \n\\frac{\\partial (fg)}{\\partial x} = \\frac{\\partial f}{\\partial x} \\cdot g+ \\frac{\\partial g}{\\partial x} \\cdot f \\quad \\text{and} \\quad \\frac{\\partial f{\\ensuremath{\\left( g\\_1({\\bm{x}}),\\ldots,g\\_m({\\bm{x}}) \\right) }}}{\\partial x\\_k} =\\sum\\_{i=1}^{m}\\frac{\\partial f}{\\partial y\\_i}{\\ensuremath{\\left( g\\_1({\\bm{x}}),\\ldots,g\\_m({\\bm{x}}) \\right) }} \\cdot \\frac{\\partial g\\_i}{\\partial x\\_k}\\;,$$ \n where in the last expression *f* is an *m* variate polynomial, and *g*1, …, *g**m* are *n* variate polynomials.\nWe shall often take derivatives according to a *dual set* to a set of linearly independent linear functions:\n[def:dual] A dual set for *m* linearly independent linear functions (recall that we say that linear functions are linearly independent if and only if their degree-1 homogeneous parts are linearly independent) in *n* ≥ *m* variables, $\\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_m({\\bm{x}})$ is a set of *m* vectors ${\\ensuremath{\\left\\{ {{\\bm{v}}\\_i} \\right\\} }}\\subset {{\\mathbb{F}}}^n$ such that $\\ell\\_i^{[1]}({{\\bm{v}}\\_j})=\\delta\\_{i,j}$.\n[lem:dualder] Let ℓ1, …, ℓ*m* ∈ F[*x*1, …, *x**n*], for *n* ≥ *m*, be linearly independent linear functions. Let ${\\ensuremath{\\left\\{ {{\\bm{v}}\\_i} \\right\\} }}\\subset {{\\mathbb{F}}}^n$ be a dual set. Let *g* ∈ F[*y*1, …, *y**m*] be a polynomial. Then, for $f({\\bm{x}})=g{\\ensuremath{\\left( \\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_m({\\bm{x}}) \\right) }}$ it holds that \n$${\\frac{\\partial f}{\\partial {\\bm{v}}\\_i}}({\\bm{x}})={\\frac{\\partial g}{\\partial y\\_i}}{\\ensuremath{\\left( \\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_m({\\bm{x}}) \\right) }}\\;.$$ \n \n$$\\begin{aligned}\n {\\frac{\\partial f}{\\partial {\\bm{v}}\\_i}}({\\bm{x}}) &=\\sum\\_j v\\_{i,j}\\cdot{\\frac{\\partial f}{\\partial x\\_j}}({\\bm{x}})=\\sum\\_{j,k}v\\_{i,j}\\cdot\n {\\frac{\\partial \\ell\\_k}{\\partial x\\_j}}\\cdot{\\frac{\\partial g}{\\partial y\\_k}}{\\ensuremath{\\left( \\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_m({\\bm{x}}) \\right) }}\\\\\n &=\\sum\\_k\\ell\\_k^{[1]}({\\bm{v}}\\_i)\\cdot{\\frac{\\partial g}{\\partial y\\_k}}{\\ensuremath{\\left( \\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_m({\\bm{x}}) \\right) }}={\\frac{\\partial g}{\\partial y\\_i}}{\\ensuremath{\\left( \\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_m({\\bm{x}}) \\right) }}\\;. \\qedhere\n \\end{aligned}$$ \n[lem:indDerivLinear] Let $f\\in{{\\mathbb{F}}}[{\\bm{x}}]$ where ${\\bm{x}}=(x\\_1,\\ldots,x\\_n)$. Let $H(\\bm w):{{\\mathbb{F}}}^t\\to{{\\mathbb{F}}}^n$ be a polynomial map in variables $\\bm w$, and let ${{\\ensuremath{\\mathcal{G}}}}(\\bm y,\\bm z)$ be a *k*-independent polynomial map such that \\normalfont var(*H*) ∩ \\normalfont var(G) = ∅. Then, for any ${\\bm{v}}\\_1,\\ldots,{\\bm{v}}\\_k\\in{{\\mathbb{F}}}^n$: \n$${\\frac{\\partial ^k f}{\\partial {\\bm{v}}\\_1\\partial{\\bm{v}}\\_2\\cdots\\partial{\\bm{v}}\\_k}}\\circ H\\neq 0\\quad \\Rightarrow \\quad f\\circ({{\\ensuremath{\\mathcal{G}}}}+H)\\neq 0\\;.$$ \nBy definition of *k*-independent polynomial maps, ${{\\ensuremath{\\mathcal{G}}}}={{\\ensuremath{\\mathcal{G}}}}\\_1(\\bm{y\\_1},z\\_1)+\\ldots+{{\\ensuremath{\\mathcal{G}}}}\\_k(\\bm{y\\_k},z\\_k)$ for some variable-disjoint 1-independent polynomial maps G1, …, G*k*. It is therefore enough to prove the lemma for *k* = 1, as we can replace *f* with ${\\frac{\\partial ^{k-1} f}{\\partial {\\bm{v}}\\_2\\cdots\\partial{\\bm{v}}\\_k}}$, *H* with *H* + G2 + … + G*k* and G with G1; by iterative application of the result for *k* = 1, we will get the general result for an arbitrary *k* ∈ N.\nDenote *H* = (*H*1, *H*2, …, *H**n*). By, the condition ${\\frac{\\partial f}{\\partial {\\bm{v}}}}\\circ H\\neq 0$ implies that there exists some *i* ∈ [*n*] such that ${\\frac{\\partial f}{\\partial x\\_i}}\\circ H\\neq 0$. Assume, WLOG, ${\\frac{\\partial f}{\\partial x\\_1}}\\circ H\\neq 0$. As G is a 1-independent polynomial map, there exists some ${\\bm{\\alpha}}\\in{{\\mathbb{F}}}^{|\\bm{y\\_1}|}$ such that ${{ \\left.\\kern-\\nulldelimiterspace f\\circ({{\\ensuremath{\\mathcal{G}}}}+H) \\vphantom{\\big|} \\right|\\_{\\bm{y\\_1}={\\bm{\\alpha}}} }}=f(z\\_1+H\\_1,H\\_2,\\ldots,H\\_n)$; denote $g\\triangleq {{ \\left.\\kern-\\nulldelimiterspace f\\circ({{\\ensuremath{\\mathcal{G}}}}+H) \\vphantom{\\big|} \\right|\\_{\\bm{y\\_1}={\\bm{\\alpha}}} }}$. As no coordinate of *H* depends on *z*1: \n$${\\frac{\\partial g}{\\partial z\\_1}}= {\\frac{\\partial {\\ensuremath{\\left( z\\_1+H\\_1 \\right) }}}{\\partial z\\_1}}\\cdot {\\frac{\\partial f}{\\partial x\\_1}}(z\\_1+H\\_1,H\\_2,\\ldots,H\\_n)= 1\\cdot\\left({\\frac{\\partial f}{\\partial x\\_1}}\\right)(z\\_1+H\\_1,H\\_2,\\ldots,H\\_n)$$ \n and therefore: \n$${{ \\left.\\kern-\\nulldelimiterspace {\\frac{\\partial g}{\\partial z\\_1}} \\vphantom{\\big|} \\right|\\_{z\\_1=0} }}=1\\cdot\\left({\\frac{\\partial f}{\\partial x\\_1}}\\right)(0+H\\_1,H\\_2,\\ldots,H\\_n)=\\left({\\frac{\\partial f}{\\partial x\\_1}}\\right)\\circ H\\neq 0 \\;.$$ \n As *g* is a projection of *f* ∘ (G + *H*), it follows that *f* ∘ (G + *H*) ≠ 0.\nThe next lemma shows how to use *k*-independent maps in order to project a polynomial to a subset of its coordinates.\n[lem:indProjectZero] Let *m* ≤ *n* ∈ N and $g({\\bm{w}})\\in{{\\mathbb{F}}}[w\\_1,\\ldots,w\\_m]$. Let $f({\\bm{x}})=g(\\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_m({\\bm{x}}))$ for linearly independent linear functions $\\ell\\_1({\\bm{x}}),\\ldots,\\ell\\_m({\\bm{x}})$. Let ${{\\ensuremath{\\mathcal{G}}}}(\\bm y,\\bm z)$ be a *k*-independent polynomial map. For a set *S* ⊆ [*n*] of size *k* denote by $\\tilde g(x\\_i:i\\in[m]\\setminus S)={{ \\left.\\kern-\\nulldelimiterspace g \\vphantom{\\big|} \\right|\\_{S=0} }}$ the projection of *g* to the variables outside of *S*. Then, there exist linearly independent linear functions $\\{\\tilde\\ell\\_i({\\bm{x}}):i\\in[m]\\setminus S\\}$, additional linear functions ${\\bm{L}}({\\bm{x}})=(L\\_1({\\bm{x}}),\\ldots,L\\_k({\\bm{x}}))$ and an assignment ${\\bm{\\alpha}}\\in{{\\mathbb{F}}}^{|\\bm y|}$ such that: \n$$f({\\bm{x}}+{{\\ensuremath{\\mathcal{G}}}}({\\bm{\\alpha}},{\\bm{L}}({\\bm{x}}))) = \\tilde g(\\tilde\\ell\\_i({\\bm{x}}):i\\in[m]\\setminus S)\\;.$$ \nIt is enough to prove the lemma for the case *k* = 1, as we may then define $\\tilde f({\\bm{x}})\\triangleq f({\\bm{x}}+{{\\ensuremath{\\mathcal{G}}}}({\\bm{\\alpha}},L\\_1({\\bm{x}})))=\\tilde g(\\tilde\\ell\\_1({\\bm{x}}),\\ldots,\\tilde\\ell\\_{m-1}({\\bm{x}}))$ and apply the result iteratively. Thus, assume *k* = 1, and WLOG assume *S* = {*x*1} (thus, $\\tilde g(w\\_2,\\ldots,w\\_m)=g(0,w\\_2,\\ldots,w\\_m)$).\nLet *x**i* be some variable with a non-zero coefficient in $\\ell\\_1({\\bm{x}})$. Such a variable exists as the ℓ*j*s are linearly independent. For *j* ∈ [*m*], denote $\\beta\\_j= {\\frac{\\partial \\ell\\_j}{\\partial x\\_i}}$, i.e. *β**j* is the coefficient of *x**i* in ℓ*j*. By our choice of *i*, *β*1 ≠ 0. Choose some ${\\bm{\\alpha}}\\in{{\\mathbb{F}}}^{|\\bm y|}$ such that ${{\\ensuremath{\\mathcal{G}}}}({\\bm{\\alpha}},z\\_1)$ has *z*1 in the *i*th coordinate, and 0 in all other coordinates. Define $L({\\bm{x}})\\triangleq -\\frac{\\ell\\_1({\\bm{x}})}{\\beta\\_1}$, so we get: \n$$f({\\bm{x}}+{{\\ensuremath{\\mathcal{G}}}}({\\bm{\\alpha}},L({\\bm{x}}))=f{\\ensuremath{\\left( x\\_1,x\\_2,\\ldots,x\\_{i-1},x\\_i-\\frac{\\ell\\_1({\\bm{x}})}{\\beta\\_1},x\\_{i+1},\\ldots,x\\_n \\right) }}.$$ \n Observe that for every *i*, \n$$\\ell\\_i{\\ensuremath{\\left( {\\bm{x}}+{{\\ensuremath{\\mathcal{G}}}}({\\bm{\\alpha}},L({\\bm{x}}) \\right) }} = \\ell\\_i{\\ensuremath{\\left( x\\_1,x\\_2,\\ldots,x\\_{i-1},x\\_i-\\frac{\\ell\\_1({\\bm{x}})}{\\beta\\_1},x\\_{i+1},\\ldots,x\\_n \\right) }} = \\ell\\_i({\\bm{x}})-\\frac{\\beta\\_i}{\\beta\\_1}\\cdot \\ell\\_1({\\bm{x}}) \\;.$$ \n In particular, $\\ell\\_1{\\ensuremath{\\left( {\\bm{x}}+{{\\ensuremath{\\mathcal{G}}}}({\\bm{\\alpha}},L({\\bm{x}}) \\right) }} =0$. For *i* = 2, …, *m*, define: \n$$\\tilde\\ell\\_i({\\bm{x}})\\triangleq \\ell\\_i({\\bm{x}})-\\frac{\\beta\\_i}{\\beta\\_1}\\cdot \\ell\\_1({\\bm{x}}) \\;.$$ \n As ℓ1, …, ℓ*m* are linearly independent, it follows that $\\tilde\\ell\\_2,\\ldots,\\tilde\\ell\\_m$ are also linearly independent. We get that\n \n$$f({\\bm{x}}+{{\\ensuremath{\\mathcal{G}}}}({\\bm{\\alpha}},L({\\bm{x}})))=g(0,\\tilde\\ell\\_2({\\bm{x}}),\\ldots,\\tilde\\ell\\_m({\\bm{x}}))=\\tilde g(\\tilde\\ell\\_2({\\bm{x}}),\\ldots,\\tilde\\ell\\_m({\\bm{x}}))\\;. \\qedhere$$ \nProof of\n--------\nWe next prove that there are *k*-independent maps that are provably not robust. The proof is by giving a different construction of such maps that, for an appropriate arrangement of the *n* variables in a matrix, is guaranteed to output matrices of rank at most *k*. Thus, a determinant of any (*k* + 1) × (*k* + 1) minor, a polynomial that has small formulas for small values of *k*, vanishes on the output of any such map.\nThe fact that such a construction exists was already noticed in (Construction 6.3 of the full version of the paper). For completeness we repeat the construction here.\n*(of )* Fix the number of variables *n* and assume WLOG *n* is a perfect square, i.e., *n* = *m*2. We index the variables as *x**i*, *j* for *i*, *j* ∈ [*m*]. We let *f* = Det*t* + 1. By, over fields of characteristic zero, *f* has a $t^{O(\\sqrt t)}=O(n)$ sized ΣΠΣ formula, which is polynomial in *n* for *t* = *O*((log*n*/loglog*n*)2). Over fields of positive characteristic the formula size is quasipolynomial in *t*, and the ΣΠΣ complexity is at most *t*!, which is polynomial in *n* for *t* = *O*(log*n*/loglog*n*).\nDenote by $\\bm{M}$ the (*t* + 1) × (*t* + 1) symbolic matrix of variables $\\bm{M}\\_{i,j}=x\\_{i,j}$. We first construct a uniform 1-independent polynomial map G1 such that $\\bm{M}\\circ {{\\ensuremath{\\mathcal{G}}}}\\_1$ is of rank 1, and define G to be a sum of *t* variable-disjoint copies of G1. As $rank(\\bm{M}\\circ {{\\ensuremath{\\mathcal{G}}}}\\_1)=1$, we have $rank(\\bm{M}\\circ {{\\ensuremath{\\mathcal{G}}}})\\leq t$ so ${\\ensuremath{\\text{Det}}}\\_{t+1}(\\bm{M}\\circ {{\\ensuremath{\\mathcal{G}}}})=0$, as required. We now focus on G1.\nFix *n* distinct field elements {*α**i*, *j*}*i*, *j* = 1*m* ⊆ F and let *w*, *y*, *z* be new variables. Define two vectors of polynomials of degree *n* − 1, *R* = (*R*1, …, *R**m*), *C* = (*C*1, …, *C**m*) ∈ F[*y*]*m*, such that for every *k* ∈ [*m*] *R**k* and *C**k* satisfy\n \n$$\\begin{aligned}\n R\\_k(\\alpha\\_{i,j})&=\\delta\\_{i,k}\\quad \\text{and}\\quad C\\_k(\\alpha\\_{i,j})=\\delta\\_{j,k}.\n \\end{aligned}$$ \nDefine G1(*w*, *y*, *z*) as the *m* × *m* matrix $z\\cdot (w^{2n-2}R(\\frac{y}{w})\\cdot C(\\frac{y}{w})^T)$ (the (*i*, *j*) entry of G1 is $z\\cdot w^{2n-2}\\cdot R\\_i(\\frac{y}{w})\\cdot C\\_j(\\frac{y}{w})$). As every coordinate of G1 is a homogeneous polynomial of degree 2*n* − 1, G1 is a uniform polynomial map. For any *i*, *j* ∈ [*m*] we have that \nG1(1, *α**i*, *j*, *z*) = *z* ⋅ (*R**i*ʹ(*α**i*, *j*) ⋅ *C**j*ʹ(*α**i*, *j*))*i*ʹ, *j*ʹ ∈ [*m*] = *z* ⋅ (*δ**i*, *i*ʹ*δ**j*, *j*ʹ)*i*ʹ, *j*ʹ ∈ [*m*] . \n The above matrix has *z* in entry (*i*, *j*) and 0 everywhere else, so G1 is a uniform 1-independent polynomial map. The resulting matrix $\\bm{M}\\circ {{\\ensuremath{\\mathcal{G}}}}\\_1$ is of rank 1 since it is a product of vectors *R* ⋅ *C**T*, so the variable-disjoint sum G = ∑1*t*G1(*w**i*, *y**i*, *z**i*) is a uniform *t*-independent polynomial map satisfying *f* ∘ G = 0.\nInterpolation and reconstruction for orbits of the continuant polynomial\n========================================================================\nWe start by proving that any uniform 1-independent map hits CGLaff(F) ().\n[Proof of ] Let $f(x\\_1,\\ldots,x\\_n)={\\text{C}}\\_{m}{\\ensuremath{\\left( \\ell\\_1({\\bm{x}})+b\\_1,\\ldots,\\ell\\_m({\\bm{x}})+b\\_m \\right) }}$, where the ℓ*i*s are linear forms. Observe that C*m*(*y*1, …, *y**m*) is a multilinear polynomial that has a unique monomial of degree *m* and all other monomials are of smaller degree. Thus, \n$${\\text{C}}\\_{m}(y\\_1,\\ldots,y\\_m)= \\prod\\_{i=1}^{m}y\\_i + \\tilde{{\\text{C}}}\\_{m-1}(y\\_1,\\ldots,y\\_m)\\;,$$ \n where $\\deg{\\ensuremath{\\left( \\tilde{{\\text{C}}}\\_{m-1} \\right) }}\\leq m-1$. Hence, \n$$f({\\bm{x}})={\\text{C}}\\_{m}{\\ensuremath{\\left( \\ell\\_1({\\bm{x}})+b\\_1,\\ldots,\\ell\\_m({\\bm{x}})+b\\_m \\right) }} = \\prod\\_{i=1}^{m}\\ell\\_i + \\tilde{f}{\\ensuremath{\\left( \\ell\\_1,\\ldots,\\ell\\_m \\right) }}\\;,$$ \n where deg(*f̃*) ≤ *m* − 1.\nLet G1 be a uniform 1-independent polynomial map into F*n*. Let *d* be the degree of the different components of G1. ([item:coordsGenInd]) implies that (∏*i* = 1*m*ℓ*i*) ∘ G1 ≠ 0 and hence it is a nonzero homogeneous polynomial of degree *m* ⋅ *d*. As deg(*f̃* ∘ G1) ≤ (*m* − 1) ⋅ *d* < deg((∏*i* = 1*m*ℓ*i*) ∘ G1), we have that \n*f* ∘ G1 = (∏*i* = 1*m*ℓ*i*) ∘ G1 + *f̃* ∘ G1 ≠ 0 \n and the claim follows.\nfollows immediately from, and the construction of a uniform generator in.\nA similar argument would show that ${{\\ensuremath{\\mathcal{G}}}}(y,z)\\triangleq {\\ensuremath{\\left( y^{n-1},y^{n-2}z,\\ldots,z^{n-1} \\right) }}$ is a hitting set generator for C*m*GL*n*aff(F), which leads to a hitting set of size *n*4.\nWe now turn to giving a reconstruction algorithm for CGLaff(F). We start by proving some simple lemmas that will be used for constructing an interpolating set.\n[def:triplet] We call an ordered triplet (*i*, *j*, *k*) ∈ Z*m*3 a *consecutive triplet* if *j* = *i* + 1 and *k* = *i* + 2, or *j* = *k* + 1 and *i* = *k* + 2, where all equalities are taken modulo *m*.\n[lem:triplet] Let *m* ≥ 3. Then (*i*, *j*, *k*) is a consecutive triplet if and only if every monomial in C*m*(*x*0, …, *x**m* − 1) that contains both *x**i* and *x**k*, also contains *x**j*.\nObserve that a polynomial $f({\\bm{x}})$ has a monomial containing *x**i* and *x**k* but not *x**j*, if and only if this is also the case when we set *x**j* = 0. Assume that (*i*, *j*, *k*) is a consecutive triplet. Then,\n \n$$\\begin{aligned}\n {\\text{C}}\\_{m}(x\\_0,\\ldots,x\\_i,0,x\\_{i+2},\\ldots,x\\_{m-1}) = {\\ensuremath{\\text{Trace}}}{\\ensuremath{\\left( \\begin{pmatrix}\n x\\_0 & 1 \\\\\n 1 & 0\n \\end{pmatrix} \n \\cdot \\ldots \\cdot \n \\begin{pmatrix}\n x\\_{i} & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot \n \\begin{pmatrix}\n 0 & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot\\begin{pmatrix}\n x\\_{i+2} & 1 \\\\\n 1 & 0\n \\end{pmatrix} \n \\cdot \\ldots \\cdot \\begin{pmatrix}\n x\\_{m-1} & 1 \\\\\n 1 & 0\n \\end{pmatrix} \n \\right) }} \\\\\n = {\\ensuremath{\\text{Trace}}}{\\ensuremath{\\left( \\begin{pmatrix}\n x\\_0 & 1 \\\\\n 1 & 0\n \\end{pmatrix} \n \\cdot \\ldots \\cdot \n \\begin{pmatrix}\n x\\_{i-1} & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot \\begin{pmatrix}\n x\\_i+x\\_{i+2} & 1 \\\\\n 1 & 0\n \\end{pmatrix} \n \\cdot \\begin{pmatrix}\n x\\_{i+3} & 1 \\\\\n 1 & 0\n \\end{pmatrix} \n \\cdot \\ldots \\cdot \\begin{pmatrix}\n x\\_{m-1} & 1 \\\\\n 1 & 0\n \\end{pmatrix} \n \\right) }}\\;.\n \\end{aligned}$$ \nIt immediately follows that no monomial of C*m*(*x*0, …, *x**i*, 0, *x**i* + 2, …, *x**m* − 1) contains both *x**i* and *x**i* + 2.\nWe now prove the second direction in the claim. Since C*m* is a trace of a matrix product, by properties of trace we can assume WLOG that *i* < *j* < *k*, by first rotating the order of the matrices until we have *i* < *j* < *k* or *k* < *j* < *i* (where *a* < *b* means that the matrix corresponding to *a* comes before that of *b*). As both cases are equivalent we can assume that *i* < *j* < *k*. We next handle this case. Assume WLOG that *j* − *i* > 1. Set *x**r* = 0 for every *i* + 2 ≤ *r* < *k*, to 0. We get that the new polynomial has the form\n \n$$\\begin{aligned}\n & {\\ensuremath{\\text{Trace}}}{\\ensuremath{\\left( \\begin{pmatrix}\n x\\_{0} & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot \\ldots \\cdot \\begin{pmatrix}\n x\\_{i} & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot \\begin{pmatrix}\n x\\_{i+1} & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot \\begin{pmatrix}\n 0&1\\\\1&0\n \\end{pmatrix}^{k-i-2}\n \\cdot \\begin{pmatrix}\n x\\_{k} & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\cdot \\ldots \\begin{pmatrix}\n x\\_{m-1} & 1 \\\\\n 1 & 0\n \\end{pmatrix}\n \\right) }}\\\\ &=\\left\\{\\begin{array}{lr}\n {\\text{C}}\\_{m-k+i+2}{\\ensuremath{\\left( x\\_0,\\ldots,x\\_i,x\\_{i+1},x\\_k,\\ldots,x\\_{m-1} \\right) }}, & \\text{for }k-i \\text{ even}\\\\\n {\\text{C}}\\_{m-k+i+1}{\\ensuremath{\\left( x\\_0,\\ldots x\\_{i-1},x\\_i,x\\_{i+1}+x\\_k,\\ldots,x\\_{m-1} \\right) }}, & \\text{for }k-i \\text{ odd}\n \\end{array}\\right. \\;, \\end{aligned}$$ \nand a monomial of maximal degree in this polynomial contains both *x**i* and *x**k* (when *k* − *i* is even there is a unique monomial of maximal degree, and when *k* − *i* is odd there are two such monomials).\n[cor:triplet] Let *m* ≥ 3. Then (*i*, *j*, *k*) is a consecutive triplet if and only if ${{ \\left.\\kern-\\nulldelimiterspace \\frac{\\partial^2 {\\text{C}}\\_{m}}{\\partial x\\_i \\partial x\\_k} \\vphantom{\\big|} \\right|\\_{x\\_j=0} }}=0$.\nFor every list of three distinct indices (*i*, *j*, *k*) ∈ [*m*]03 denote \n$${\\text{C}}\\_{m}^{(i,j,k)}({\\bm{x}}) \\triangleq {{ \\left.\\kern-\\nulldelimiterspace \\frac{\\partial^2 {\\text{C}}\\_{m}}{\\partial x\\_i \\partial x\\_k} \\vphantom{\\big|} \\right|\\_{x\\_j=0} }}\\;.$$ \n[lem:cont-hit-deriv] Let *n* ≥ *m* ≥ 3 and *t* be integers. Assume $H({\\bm{w}}):{{\\mathbb{F}}}^t\\to {{\\mathbb{F}}}^n$ is a hitting-set generator for C*m*(*i*, *j*, *k*)GL*n*aff(F), for every list of three distinct indices (*i*, *j*, *k*) ∈ [*m*]03. Let ${{\\ensuremath{\\mathcal{G}}}}\\_3({\\bm{y}},{\\bm{z}})$ be a 3-independent polynomial map (into F*n*) that each of its coordinates is a homogeneous linear function in ${\\bm{z}}$, over ${{\\mathbb{F}}}({\\bm{y}})$ (for example, GSV*k* has this property, for every *k*). Then, for every *m*1, *m*2 and *n* and every two polynomials *f*1 ∈ C*m*1GL*n*aff(F) and *f*2 ∈ C*m*2GL*n*aff(F) it holds that *f*1 = *f*2 if and only if *f*1 ∘ (*H* + G3) = *f*2 ∘ (*H* + G3).\nRoughly, what the lemma claims is that if G3 is a 3-independent map and *H* hits C*m*(*i*, *j*, *k*)GL*n*aff(F), then *H* + G3 is an interpolating-set generator.\nDenote *f*1 = C*m*1(ℓ1, 0, …, �"},"data_id":{"kind":"string","value":"arxiv_0182375"}}},{"rowIdx":45592,"cells":{"content":{"kind":"string","value":" Achieving Budget-optimality with Adaptive Schemes in Crowdsourcing\n===================================================================\nCrowdsourcing platforms provide marketplaces where task requesters can pay to get labels on their data. Such markets have emerged recently as popular venues for collecting annotations that are crucial in training machine learning models in various applications. However, as jobs are tedious and payments are low, errors are common in such crowdsourced labels. A common strategy to overcome such noise in the answers is to add redundancy by getting multiple answers for each task and aggregating them using some methods such as majority voting. For such a system, there is a fundamental question of interest: how can we maximize the accuracy given a fixed budget on how many responses we can collect on the crowdsourcing system. We characterize this fundamental trade-off between the budget (how many answers the requester can collect in total) and the accuracy in the estimated labels. In particular, we ask whether adaptive task assignment schemes lead to a more efficient trade-off between the accuracy and the budget.\nAdaptive schemes, where tasks are assigned adaptively based on the data collected thus far, are widely used in practical crowdsourcing systems to efficiently use a given fixed budget. However, existing theoretical analyses of crowdsourcing systems suggest that the gain of adaptive task assignments is minimal. To bridge this gap, we investigate this question under a strictly more general probabilistic model, which has been recently introduced to model practical crowdsourced annotations. Under this generalized Dawid-Skene model, we characterize the fundamental trade-off between budget and accuracy. We introduce a novel adaptive scheme that matches this fundamental limit. We further quantify the fundamental gap between adaptive and non-adaptive schemes, by comparing the trade-off with the one for non-adaptive schemes. Our analyses confirm that the gap is significant.\nIntroduction\n============\nCrowdsourcing platforms provide labor markets in which pieces of micro-tasks are electronically distributed to any workers who are willing to complete them for a small fee. In typical crowdsourcing scenarios, such as those on Amazon’s Mechanical Turk, a requester first posts a collection of tasks, for example a set of images to be labelled. Then, from a pool of workers, whoever is willing can pick up a subset of those tasks and provide her labels for a small amount of payment. Typically, a fixed amount of payment per task is predetermined and agreed upon between the requester and the workers, and hence the worker is paid the amount proportional to the number of tasks she answers. Further, as the verification of the correctness of the answers is difficult, and also as the requesters are afraid of losing reputation among the crowd, requesters typically choose to pay for every label she gets regardless of the correctness of the provided labels. Hence, the budget of the total payments the requester makes to the workers is proportional to the total number of labels she collects.\nOne of the major issues in such crowdsourcing platforms is label quality assurance. Some workers are spammers trying to make easy money, and even those who are willing to work frequently make mistakes as the reward is small and tasks are tedious. To correct for these errors, a common approach is to introduce redundancy by collecting answers from multiple workers on the same task and aggregating these responses using some schemes such as majority voting. A fundamental problem of interest in such a system is how to maximize the accuracy of thus aggregated answers, while minimizing the cost. Collecting multiple labels per task can improve the accuracy of our estimates, but increases the budget proportionally. Given a fixed number of tasks to be labelled, a requester hopes to achieve the best trade-off between the accuracy and the budget, i.e. the total number of responses the requester collects on the crowdsourcing platform. There are two design choices the requester has in achieving this goal: *task assignment* and *inference algorithm*.\nIn typical crowdsourcing platforms, tasks are assigned as follows. Since the workers are fleeting, the requester has no control over who will be the next arriving worker. Workers arrive in an online fashion, complete the tasks that they are given, and leave. Each arriving worker is completely new and you may never get her back. Nevertheless, it might be possible to improve accuracy under the same budget, by designing better task assignments. The requester has the following control over the *task assignment*. At each point in time, we have the control over which tasks to assign to the next arriving worker. The requester is free to use all the information collected thus far, including all the task assignments to previous workers and the answers collected on those assigned tasks. By adaptively identifying tasks that are more difficult and assigning more (future) workers on those tasks, one hopes to be more efficient in the budget-accuracy trade-off. This paper makes this intuition precise, by studying a canonical crowdsourcing model and comparing the fundamental trade-offs between adaptive schemes and non-adaptive schemes. Unlike adaptive schemes, a non-adaptive scheme fixes all the task assignments before any labels are collected and does not allow future assignments to adapt to the labels collected thus far for each arriving worker. Precise definitions of adaptive and non-adaptive task assignments are provided in Section [sec:model].\nWhile adaptive task assignments handle the heterogeneity in the task difficulties by assigning more workers to difficult tasks, inferring such unknown difficulty of the tasks (as well as inferring unknown heterogeneity of the worker reliabilities) requires *inference*: estimating the latent parameters and the ground truth labels from crowdsourced responses thus far. Some workers are more reliable than the others, but we do not know their latent reliabilities. Some tasks are more difficult than the others, but we do not know their latent difficulty levels. We only get to observe the answers provided by those workers on their assigned tasks. Nevertheless, by comparing responses from multiple workers, we can estimate the true labels and the difficulties of the tasks, and use them in subsequent steps in our inference algorithm to learn the reliability of the workers. We perform such inferences at several points in time over the course of collecting all the labels we have budgeted for. The inference algorithm outputs the current estimates for the labels and difficulty levels of the tasks, which are used in subsequent time to assign tasks.\nModel and problem formulation\n-----------------------------\nWe assume that the requester has *m* binary classification tasks to be labelled by querying a crowdsourcing platform multiple times. For example, those might be image classification tasks, where the requester wants to classify *m* images as either suitable for children ( + 1) or not ( − 1). The requester has a budget Γ on how many responses she can collect on the crowdsourcing platform, assuming one unit of payment is made for each response collected. We use Γ interchangeably to refer to both a target budget and also the budget used by a particular task assignment scheme (as defined in ), and it should be clear from the context which one we mean. We want to find the true label by querying noisy workers who are arriving in an online fashion, one at a time.\nTypical crowdsourcing systems are modeled as a discrete time systems where at each time we have a new arriving worker. At time *j*, the requester chooses an action *T**j* ⊆ [*m*], which is a subset of tasks to be assigned to the *j*-th arriving worker. Then, the *j*-th arriving worker provides her answer *A**i**j* ∈ { + 1, − 1} for each task *i* ∈ *T**j*. We use the index *j* to denote both the *j*-th time step in this discrete time system as well as the *j*-th arriving worker. At this point (at the end of *j*-th time step), all previous responses are stored in a sparse matrix *A* ∈ {0, + 1, − 1}*m* × *j*, and this data matrix is increasing by one column at each time. We let *A**i**j* = 0 if task *i* is not assigned to worker *j*, i.e. $i\\notin T\\_j$, and otherwise we let *A**i**j* ∈ { + 1, − 1} be the previous worker *j*’s response on task *i*. At the next time *j* + 1, the next task assignment *T**j* + 1 is chosen, and this process is repeated. At time *j*, the action (or the task assignment) can depend on all previously collected responses up to the current time step stored in a sparse (growing) matrix *A* ∈ {0, + 1, − 1}*m* × (*j* − 1). This process is repeated until the task assignment scheme decides to stop, typically when the total number of collected responses (the number of nonzero entires in *A*) meet a certain budget constraint or when a certain target accuracy is estimated to be met.\nWe consider both a *non-adaptive scenario* and an *adaptive scenario*. In a non-adaptive scenario, a fixed number *n* of workers to be recruited are pre-determined (and hence the termination time is set to be *n*) and also fixed task assignments *T**j*’s for all *j* ∈ [*n*] are pre-determined, before any response is collected. In an *adaptive scenario*, the requester chooses *T**j*’s in an online fashion based on all the previous answers collected thus far. For both adaptive and non-adaptive scenarios, when we have determined that we have collected all the data we need, an inference algorithm is applied on the collected data *A* ∈ {0, + 1, − 1}*m* × *n* to output an estimate *t̂**i* ∈ { + 1, − 1} for the ground truth label *t**i* ∈ { + 1, − 1} for the *i*-th task for each *i* ∈ [*m*]. Note that we use *n* to denote the total number of workers recruited, which is a random variable under the adaptive scenario. Also, note that the estimated labels for all the tasks do not have to be simultaneously output in the end, and we can choose to output estimated labels on some of the tasks in the middle of the process before termination. The average accuracy of our estimates is measured by the average probability of error $P\\_{\\rm error} = (1/m) \\sum\\_{i=1}^m \\P[t\\_i\\neq \\hat{t}\\_i]$ under a probabilistic model to be defined later in this section in Eq. .\nThe total *budget* used in one instance of such a process is measured by the total number of responses collected, which is equal to the number of non-zero entries in *A*. This inherently assumes that there is a prefixed fee of one unit for each response that is agreed upon, and the requester pays this constant fee for every label that is collected. The expected budget used by a particular task assignment scheme will be denoted by\n \n$$\\begin{aligned}\n \\Gamma \\;\\;\\equiv\\;\\; \\E \\Big[ \\sum\\_{j=1}^n |T\\_j| \\Big] \\;, \n \\label{eq:defgamma}\\end{aligned}$$ \nwhere the expectation is over all the randomness in the model (the problem parameters representing the quality of the tasks and the quality of the workers, and the noisy responses from workers) and any randomness used in the task assignment. We are interested in designing task assignment schemes and inference algorithms that achieve the best accuracy within a target expected budget, under the following canonical model of how workers respond to tasks.\nWe assume that when a task is assigned to a worker, the response follows a probabilistic model introduced by, which is a recent generalization of the Dawid-Skene model originally introduced by. Precisely, each new arriving worker is parametrized by a latent worker quality parameter *p**j* ∈ [0, 1] (for the *j*-th arriving worker). Each task is parametrized by a latent task quality parameter *q**i* ∈ [0, 1] (for the *i*-th task). When a worker *j* is assigned a task *i*, the *generalized Dawid-Skene model* assumes that the response *A**i**j* ∈ { + 1, − 1} is a random variable distributed as\n \n$$\\begin{aligned}\n A\\_{ij} \\;\\; = \\;\\; \\left\\{ \n \\begin{array}{rl}\n +1, &\\;\\;\\;\\; \\text{w.p.} \\;\\;\\; q\\_ip\\_j + \\bar{q}\\_i \\bar{p}\\_j \\;, \\\\\n -1, &\\;\\;\\;\\; \\text{w.p.} \\;\\;\\; q\\_i \\bar{p}\\_j + \\bar{q}\\_i p\\_j \\;,\n \\end{array}\n \\right.\\;,\n \\label{eq:defA}\\end{aligned}$$ \nconditioned on the parameters *q**i* and *p**j*, where *q̄**i* = 1 − *q**i* and *p̄**j* = 1 − *p**j*. The task parameter *q**i* represents the probability that a task is perceived as a positive task to a worker, and the worker parameter *p**j* represents the probability the worker makes a mistake in labelling the task. Concretely, when a task *i* is presented to any worker, the task is perceived as a positive task with a probability *q**i* or a negative task otherwise, independent of any other events. Let *t̃**i**j* denote this perceived label of task *i* as seen by worker *j*. Conditioned on this perceived label of the task, a worker *j* with parameter *p**j* makes a mistake with probability 1 − *p**j*. She provides a ‘correct’ label *t̃**i**j* as she perceives it with probability *p**j*, or provides an ‘incorrect’ label − *t̃**i**j* with probability *p̄**j*. Hence, the response *A**i**j* follows the distribution in. The response *A**i**j* is, for example, a positive label if the task is perceived as a positive task and the worker does not make a mistake (which happens with a probability *q**i**p**j*), or if the task is perceived as a negative task and the worker does not make a mistake (which happens with a probability *q̄**i**p̄**j*). The strengths and weaknesses of this model are discussed in comparisons to related work in Section [sec:related].\nWe assume that worker parameters *p**j*’s are i.i.d. according to some prior distribution $\\cP$. For example, each arriving worker might be sampled with replacement from a pool of workers, and $\\cP$ denotes the discrete distribution of the quality parameters of the pool. The individual reliabilities *p**j*’s are hidden from us, and the prior distribution $\\cP$ is also unknown. namely\n \n$$\\begin{aligned}\n \\label{eq:defmu}\n \\mu \\; \\equiv\\; \\E\\_\\cP[2p\\_j-1] \\; \\text{, and } \\;\\;\\;\\;\\;\n \\sigma^2 \\;\\equiv \\; \\E\\_\\cP[(2p\\_j-1)^2] \\;, \\end{aligned}$$ \nwhere *p**j* is a random variable distributed as $\\cP$, and *μ* ∈ [ − 1, 1] is the (shifted and scaled) average reliability of the crowd and *σ*2 ∈ [0, 1] is the key quantity of $\\cP$ capturing the collective quality of the crowd as a whole. Intuitively, when all workers are truthful and have *p**j* close to a one, then the collective reliability *σ*2 will be close to its maximum value of one. On the other hand, if most of the workers are giving completely random answers with *p**j*’s close to a half, then *σ*2 will be close to its minimum value of a zero. The fundamental trade-off between the accuracy and the budget will primarily depend on the distribution of the crowd $\\cP$ via *σ*2. We do not impose any conditions on the distribution $\\cP$.\nWe assume that the task parameters *q**i*’s are drawn i.i.d. according to some prior distribution $\\cQ$. The individual difficulty of a task with a quality parameter *q**i* is naturally captured by\n \n$$\\begin{aligned}\n \\lambda\\_i \\;\\;\\equiv\\;\\; (2q\\_i-1)^2 \\;,\n \\label{eq:deflambdai}\\end{aligned}$$ \nas tasks with *q**i* close to a half are confusing and ambiguous tasks and hence difficult to correctly label (*λ**i* close to zero), whereas tasks with *q**i* close to zero or one are unambiguous tasks and easy to correctly label (*λ**i* close to one). and the collective difficulty of tasks drawn from a prior distribution $\\cQ$ are captured by the quantities *ρ* ∈ [0, 1] and *λ* ∈ [0, 1], defined as\n \n$$\\begin{aligned}\n \\textcolor{black}{\\rho \\;\\;\\equiv\\;\\; \\E\\_\\cQ\\left[ {(2q\\_i-1)^2}\\right] }\\;,\\quad \\lambda \\;\\;\\equiv\\;\\; \\left(\\E\\_\\cQ\\left[ \\frac{1}{(2q\\_i-1)^2}\\right]\\right)^{-1} \\;,\n \\label{eq:deflambda}\\end{aligned}$$ \nwhere *q**i* is distributed as $\\cQ$. The fundamental budget-accuracy trade-off depends on $\\cQ$ primarily via this *λ*. Another quantities that will show up in our main results is the worst-case difficulty in the given set of *m* tasks (conditioned on all the *q**i*’s) defined as\n \n$$\\begin{aligned}\n \\lambda\\_{\\min} \\;\\; \\equiv\\;\\; \\min\\_{i \\in [m]} (2q\\_i-1)^2\\,, \\qquad \\text{and} \\qquad \\lambda\\_{\\max} \\;\\; \\equiv\\;\\; \\max\\_{i \\in [m]} (2q\\_i-1)^2 \\;. \n \\label{eq:deflambdamin}\\end{aligned}$$ \nWhen we refer to a similar quantities from the population distributed as $\\cQ$, we abuse the notation and denote $\\lambda\\_{\\rm min} = \\min\\_{q\\_i \\in {\\rm supp}(\\cQ)} (2q\\_i-1)^2$ and $\\lambda\\_{\\rm max} = \\max\\_{q\\_i \\in {\\rm supp}(\\cQ)} (2q\\_i-1)^2$. The individual task parameters *q**i*’s are hidden from us. We do not have access to the prior distribution $\\cQ$ on the task qualities *q**i*’s, but we assume we know the statistics *ρ*, *λ*, *λ*min, and *λ*max, and we assume we also know a quantized version of the prior distribution on the task difficulties *λ**i*’s, which we explain below.\nGiven a distribution $\\cQ$ on *q**i*’s, let $\\lt\\cQ$ be the induced distribution on *λ**i*’s. For example, if $\\cQ(q\\_i)= (1/10)\\mathbb{I}\\_{(q\\_i=0.9)} + (3/10)\\mathbb{I}\\_{(q\\_i=0.1)} + (1/10)\\mathbb{I}\\_{(q\\_i=0.8)} + (3/10)\\mathbb{I}\\_{(q\\_i=0.2)} + (2/10)\\mathbb{I}\\_{(q\\_i=0.6)} $, then the induced distribution on *λ**i* is $\\lt\\cQ(\\lambda\\_i) = (4/10)\\mathbb{I}\\_{(\\lambda\\_i=0.64)} + (4/10)\\mathbb{I}\\_{(\\lambda\\_i=0.36)} + (2/10)\\mathbb{I}\\_{(\\lambda\\_i=0.04)}$. Our approach requires only the knowledge of a quantized version of the distribution $\\lt\\cQ$, namely $\\lq\\cQ$. This quantized distribution has support at $\\T$ discrete values $\\{\\lambda\\_{\\rm max}, \\lambda\\_{\\rm max}/2, \\ldots, \\lambda\\_{\\rm max}/2^{(\\T-1)} \\}$, where\n \n$$\\begin{aligned}\n \\T \\;\\; \\equiv \\;\\; 1 + \\Big\\lceil \\log\\_2 \\Big(\\frac{\\lambda\\_{\\max}}{\\lambda\\_{\\min}}\\Big) \\Big\\rceil \\;,\\end{aligned}$$ \nsuch that $\\lambda\\_{\\max} 2^{-(\\T-1)} \\leq \\lambda\\_{\\min} \\leq \\lambda\\_{\\max} 2^{-(\\T-2)}$. We denote these values by $\\{\\tlambda\\_a\\}\\_{a \\in [\\T]}$ such that $\\tlambda\\_{a} = \\lambda\\_{\\rm max}2^{-(a-1)}$ for each $a \\in [\\T]$. Then the quantized distribution is $\\sum\\_{a=1}^\\T \\tdelta\\_a {\\mathbb I}\\_{(\\lambda\\_i=\\tlambda\\_a)}$, where the probability mass $\\tdelta\\_a$ for the *a*-th partition is \n$$\\tdelta\\_a = \\lt\\cQ( \\, ( {\\lambda\\_{\\rm max}/2^{a}},{\\lambda\\_{\\rm max}/2^{(a-1)}}] \\,) \\,, \\qquad \\text{for}\\;\\; a \\in [\\T]\\,,$$ \n which is the fraction of tasks whose difficulty *λ**i* is in $(\\tlambda\\_{a+1},\\tlambda\\_a] $. We use the closed interval $[(1/2)\\tlambda\\_\\T,\\tlambda\\_\\T]$ for the last partition. In the above example, we have $\\T=5$, $\\{\\tlambda\\_a\\}\\_{a\\in\\T}=\\{0.64, 0.32, 0.16, 0.08, 0.04\\}$, and $\\{\\tdelta\\_a\\}\\_{a\\in\\T} = \\{0.8,0,0,0,0.2\\}$. For notational convenience, we eliminate those partitions with zero probability mass, and re-index the quantization $\\{\\tlambda\\_a, \\tdelta\\_a\\}\\_{a \\in [\\T]}$ to get {*λ**a*, *δ**a*}*a* ∈ [*T*], for $T \\leq \\T$, such that *δ**a* ≠ 0 for all *a* ∈ *T*. We define $\\lq\\cQ$ to be the re-indexed quantized distribution {*λ**a*, *δ**a*}*a* ∈ [*T*]. In the above example, we finally have $\\lq\\cQ(\\lambda\\_i)= 0.8{\\mathbb I}\\_{(\\lambda\\_i=0.64)} + 0.2{\\mathbb I}\\_{(\\lambda\\_i=0.04)}$.\nWe denote the maximum and minimum probability mass in $\\lq\\cQ$ as\n \n$$\\begin{aligned}\n \\delta\\_{\\max} \\;\\; \\equiv \\;\\; \\max\\_{a \\in [T]} \\delta\\_a\n \\;,\\;\\;\\text{ and }\\;\\;\\;\\;\\;\n \\delta\\_{\\min} \\;\\;\\equiv\\;\\; \\min\\_{a \\in [T]} \\delta\\_a\\;. \n \\label{eq:defdeltamin}\\end{aligned}$$ \nSimilar to the collective quality *λ* defined for the distribution $\\cQ$ in, we define $\\lq\\lambda$, collective quality for the quantized distribution $\\lq\\cQ$, which is used in our algorithm. $\\lq\\lambda \\equiv (\\sum\\_{a \\in [T]} (\\delta\\_a/\\lambda\\_a))^{-1}$.\nThe ground truth label of a task is also naturally defined as what the majority of the crowd would agree on if we ask all the workers to label that task, i.e. $t\\_i \\,\\equiv\\, {\\rm sign}(\\E[A\\_{ij}|q\\_i]) = {\\rm sign}(2q\\_i-1){\\rm sign}(\\mu)$, where the expectation is with respect to the prior distribution of $p\\_j\\sim \\cP$ and the randomness in the response as per the generalized Dawid-Skene model in. Without loss of generality, we assume that the average reliability of the worker is positive, i.e. ${\\rm sign}(\\mu)=+1$ and take ${\\rm sign}(2q\\_i-1)$ as the ground truth label *t**i* of task *i* conditioned on its difficulty parameter *q**i*:\n \n$$\\begin{aligned}\n t\\_i \\;\\; =\\;\\; {\\rm sign}(2q\\_i-1) \\,.\n \\label{eq:deft}\\end{aligned}$$ \nThe latent parameters {*q**i*}*i* ∈ [*m*], {*p**j*}*j* ∈ [*n*], and {*t**i*}*i* ∈ [*m*] are unknown, and we want to infer the true labels *t**i*’s from only *A**i**j*’s.\nThe accuracy of the final estimate is measured by the average probability of error:\n \n$$\\begin{aligned}\n \\label{eq:deferror}\n P\\_{\\rm error} \\;\\;= \\;\\; \\frac1m\\sum\\_{i=1}^m \\P[t\\_i\\neq \\hat{t}\\_i] \\;.\\end{aligned}$$ \nWe investigate the fundamental trade-off between budget and error rate by identifying the sufficient and necessary conditions on the expected budget Γ for achieving a desired level of accuracy $P\\_{\\rm error}\\leq \\varepsilon$. Note that we are interested in achieving the best trade-off, which in turn can give the best approach for both scenarios: when we have a fixed budget constraint and want to minimize the error rate, and when we have a target error rate and want to minimize the cost.\nRelated work\n------------\nThe generalized Dawid-Skene model studied in this paper allows the tasks to be heterogeneous (having different difficulties) and the workers to be heterogeneous (having different reliabilities). The original Dawid-Skene (DS) model introduced in and analyzed in is a special case, when only workers are allowed to be heterogeneous. All tasks have the same difficulty with *λ**i* = 0 for all *i* ∈ [*m*] and *q**i* can be either zero or one depending on the true label. Most of existing work on the DS model assumes that tasks are randomly assigned and focuses only on the inference problem of finding the true labels. Several inference algorithms have been proposed.\nA most relevant work is by. It is shown that in order to achieve a probability of error less than a small positive constant *ɛ* > 0, it is necessary to have an expected budget scaling as Γ = *O*((*m*/*σ*2)log(1/*ɛ*)), even for the best possible inference algorithm together with the best possible task assignment scheme, including all possible adaptive task assignment schemes. Further, a simple randomized non-adaptive task assignment is proven to achieve this optimal trade-off with a novel spectral inference algorithm. Namely, an efficient task assignment and an inference algorithm are proposed that together guarantees to achieve $p\\_{\\rm error}\\leq \\varepsilon$ with budget scaling as Γ = *O*((*m*/*σ*2)log(1/*ɛ*)). It is expected that this necessary and sufficient budget constraint scales linearly in *m*, the number of tasks to be labelled. The technical innovation of is in (*i*) designing a new spectral algorithm that achieves a logarithmic dependence in the target error rate *ɛ*; and (*i**i*) identifying *σ*2 defined in as the fundamental statistics of $\\cP$ that captures the collective quality of the crowd. The budget-accuracy trade-off mainly depends on the prior distribution of the crowd $\\cP$ via a single parameter *σ*2. When we have a reliable crowd with many workers having *p**j*’s close to one, the collective quality *σ*2 is close to one and the required budget Γ is small. When we have an unreliable crowd with many workers having *p**j*’s close to a half, then the collective quality is close to zero and the required budget is large. However, perhaps one of the most surprising result of is that the optimal trade-off is matched by a non-adaptive task assignment scheme. In other words, there is only a marginal gain in using adaptive task assignment schemes.\nThis negative result relies crucially on the fact that, under the standard DS model, all tasks are inherently equally difficult. As all tasks have *q**i*’s either zero or one, the individual difficulty of a task is *λ**i* ≡ (2*q**i* − 1)2 = 1, and a worker’s probability of making an error on one task is the same as any other tasks. Hence, adaptively assigning more workers to relatively more ambiguous tasks has only a marginal gain. However, simple adaptive schemes are widely used in practice, where significant gains are achieved. In real-world systems, tasks are widely heterogeneous. Some images are much more difficult to classify (and find the true label) compared to other images. To capture such varying difficulties in the tasks, generalizations of the DS model were proposed in and significant improvements have been reported on real datasets.\nThe generalized DS model serves as the missing piece in bridging the gap between practical gains of adaptivity and theoretical limitations of adaptivity (under the standard DS model). We investigate the fundamental question of “do adaptive task assignments improve accuracy?” under this generalized Dawid-Skene model of Eq..\nOn the theoretical understanding of the original DS model, the dense regime has been studied first, where all workers are assigned all tasks. A spectral method for finding the true labels was first analyzed in and an EM approach followed by spectral initial step is analyzed in to achieve a near-optimal performance. The minimax error rate of this problem was identified in by analyzing the MAP estimator, which is computationally intractable.\nIn this paper, we are interested in a more challenging setting where each task is assigned only a small number of workers of *O*(log*m*). For a non-adaptive task assignment, a novel spectral algorithm based on the non-backtracking operator of the matrix *A* has been analyzed under the original DS model by, which showed that the proposed spectral approach is near-optimal. Further, showed that any non-adaptive task assignment scheme will have only marginal improvement in the error rate under the original DS model. Hence, there is no significant gain in adaptivity.\nOne of the main weaknesses of the DS model is that it does not capture how some tasks are more difficult than the others. To capture such heterogeneity in the tasks, several practical models have been proposed recently. Although such models with more parameters can potentially better describe real-world datasets, there is no analysis on their performance under adaptive or non-adaptive task assignments. We do not have the analytical tools to understand the fundamental trade-offs involved in those models yet. In this work, we close this gap by providing a theoretical analysis of one of the generalizations of the DS model, namely the one proposed in. It captures the heterogeneous difficulties in the tasks, while remaining simple enough for theoretical analyses.\nContributions\n-------------\nTo investigate the gain of adaptivity, we first characterize the fundamental lower bound on the budget required to achieve a target accuracy. To match this fundamental limit, we introduce a novel *adaptive* task assignment scheme. The proposed adaptive task assignment is simple to apply in practice, and numerical simulations confirm the superiority compared to state-of-the-art non-adaptive schemes. Under certain assumptions on the choice of parameters in the algorithm, which requires a moderate access to an oracle, we can prove that the performance of the proposed adaptive scheme matches that of the fundamental limit up to a constant factor. Finally, we quantify the gain of adaptivity by proving a strictly larger lower bound on the budget required for any non-adaptive schemes to achieve a desired error rate of *ɛ* for some small positive *ɛ*.\nPrecisely, we show that the minimax rate on the budget required to achieve a target average error rate of *ɛ* scales as Θ((*m*/*λ**σ*2)log(1/*ɛ*)). The dependence on the prior $\\cP$ and $\\cQ$ are solely captured in *σ*2 (the quality of the crowd as a whole) and *λ* (the quality of the tasks as a whole). We show that the fundamental trade-off for *non-adaptive* schemes is $\\Theta((m/ \\lambda\\_{\\rm min} \\sigma^2 ) \\log(1/\\varepsilon))$, requiring a factor of $\\lambda/\\lambda\\_{\\rm min}$ larger budget for non-adaptive schemes. This factor of $\\lambda/\\lambda\\_{\\rm min}$ is always at least one and quantifies precisely how much we gain by adaptivity.\nOutline and notations\n---------------------\nWe present a list of notations and their definitions in Table [tbl:notation]. In Section [sec:main], we present the fundamental lower bound on the necessary budget to achieve a target average error rate of *ɛ*. We present a novel adaptive approach which achieves the fundamental lower bound up to a constant. In comparison, we provide the fundamental lower bound on the necessary budget for non-adaptive approaches in Section [sec:nonadub], and we present a non-adaptive approach that achieves this fundamental limit. In Section [sec:spectral], we give a spectral interpretation of our approach justifying the proposed inference algorithm, leading to a parameter estimation algorithm that serves as a building block in the main approach of Algorithm [algo:adapt]. As our proposed sub-routine using Algorithm [algo:mp] suffers when the budget is critically limited (known as spectral barrier in Section [sec:spectral]), we present another algorithm that can substitute Algorithm [algo:mp] in Section [sec:algo] and compare their performances. The proofs of the main results are provided in Section [sec:proof]. We present a conclusion with future research directions in Section [sec:discussion].\n[h!]\n | c | c | c| notation& data type & definition \n*m* & Z+ & the number of tasks \n*n* & Z+ & total number of workers recruited \n*A* = [*A**i**j*] & {0, + 1, − 1}*m* × *n* & labels collected from the workers \nΓ &$\\reals\\_+$ & budget used in collecting *A* is the number of nonzero entries in *A* \nℓ &Z+ & average budget per task : Γ/*m* \nΓ*ɛ* & $\\reals\\_+$ & the budget required to achieve error at most *ɛ* \nTΓ & & set of task assignment schemes using at most Γ queries in expectation \n*i* & [m]& index for tasks \n*j* & [n] & index for workers \n*W**i* & subset of [*n*] & a set of workers assigned to task *i* \n*T**j* & subset of [*m*] & a set of tasks assigned to worker *j* \n*q**i* & [0, 1] & quality parameter of task *i* \n*t**i* & { − 1, + 1} & ground truths label of task *i* \n*t̂**i* & { − 1, + 1} & estimated label of task *i* \n*p**j* & [0, 1] & quality parameter of worker *j* \n$\\cP$ & $[0,1] \\to \\reals$ & prior distribution of *p**j* \n$\\cQ$ & $[0,1]\\to\\reals$ & prior distribution of *q**i* \n$\\lt\\cQ$ & $[0,1]\\to\\reals$ & \n$\\lq\\cQ$ & $[0,1]\\to\\reals$ & \n*μ* & [ − 1, 1]& average reliability of the crowd as per $\\cP$: $\\E\\_{\\cP}[2p\\_j-1]$ \n*σ*2 & [0, 1]& collective reliability of the crowd as per $\\cP$: $\\E\\_{\\cP}[(2p\\_j-1)^2]$ \n*λ**i* & [0, 1]& individual difficulty level of task *i*: (2*q**i* − 1)2 \n$\\lambda\\_{\\rm min}$ & [0, 1]& worst-case difficulty as per $\\cQ$: $\\min\\_{q\\_i \\in {\\rm supp}(\\cQ)} (2q\\_i-1)^2$ \n$\\lambda\\_{\\rm max}$ & [0, 1]& \n*λ* & [0, 1]& collective difficulty level of the tasks as per $\\cQ$: $\\E\\_{\\cQ}[(2q\\_i-1)^{-2}]^{-1}$ \n$\\lq\\lambda$ & [0, 1]& collective difficulty level of the tasks as per $\\lq\\cQ$: (∑*a* ∈ [*T*]*δ**a*/*λ**a*)− 1 \n*ρ*2 & [0, 1] & average difficulty of tasks as per $\\cQ:$ $\\E\\_\\cQ[(2q\\_i-1)^2]$ \n*a* & [*T*] & \n*λ**a* & [0, 1] & difficulty level of *a*-th support point of $\\lq\\cQ$ \n*δ**a* & [0, 1] & \n$\\delta\\_{\\rm min}$ & [0, 1] & minimum probability mass in $\\lq\\cQ$: min*a* ∈ [*T*]*δ**a* \n$\\delta\\_{\\rm max}$ & [0, 1] & maximum probability mass in $\\lq\\cQ$: max*a* ∈ [*T*]*δ**a* \n*T* & Z+& number of rounds in Algorithm [algo:adapt] \n*t* & Z+ & index for a round in Algorithm [algo:adapt] \n*s**t* &Z+ & number of sub-rounds in round *t* of Algorithm [algo:adapt] \n*u*&Z+ & index for a sub-round of Algorithm [algo:adapt] \n[tbl:notation]\nMain Results under the Adaptive Scenario\n========================================\nFundamental limit under the adaptive scenario\n---------------------------------------------\nWith a slight abuse of notations, we let *t̂*(*A*) be a mapping from *A* ∈ {0, + 1, − 1}*m* × *n* to *t̂*(*A*) ∈ { + 1, − 1}*m* representing an inference algorithm outputting the estimates of the true labels. We drop *A* and write only *t̂* whenever it is clear from the context. We let $\\F\\_{\\sp}$ be the set of all the prior distributions on *p**j* such that the collective worker quality is $\\sp$, i.e.\n \n$$\\begin{aligned}\n\\F\\_{\\sp} \\equiv \\left\\{\\F \\, | \\, \\E\\_{\\F}[(2p\\_j-1)^2] = \\sp\\right\\}\\,.\\end{aligned}$$ \nWe let $\\cQ\\_{\\lambda}$ be the set of all the prior distributions on *q**i* such that the collective task difficulty is *λ*, i.e.\n \n$$\\begin{aligned}\n\\cQ\\_{\\lambda} \\equiv \\left\\{\\cQ \\, \\Big| \\, \\Big( \\E\\_{\\cQ}\\left[\\frac{1}{(2q\\_i-1)^2}\\right]\\Big)^{-1} = \\lambda \\right\\}\\,.\\end{aligned}$$ \nWe consider all task assignment schemes in TΓ, the set of all task assignment schemes that make at most Γ queries to the crowd in expectation. We prove a lower bound on the standard minimax error rate: the error that is achieved by the best inference algorithm $\\t$ using the best adaptive task assignment scheme *τ* ∈ TΓ under a worst-case worker parameter distribution $\\F \\in \\F\\_{\\sigma^2}$ and the worst-case task parameter distribution $\\cQ \\in \\cQ\\_{\\lambda}$. A proof of this theorem is provided in Section [sec:adaptlb].\n[thm:lbadapt] For *σ*2 < 1, there exists a positive constant *C*ʹ such that the average probability of error is lower bounded by\n \n$$\\begin{aligned}\n\\min\\_{\\tau \\in {\\mathscr{T}}\\_\\Gamma, \\t}\\;\\; \\;\\;\\max\\_{\\cQ \\in \\cQ\\_{\\lambda}, \\F \\in \\F\\_{\\sp} } \\;\\;\\;\\;\\frac{1}{m} \\sum\\_{i=1}^m \\P[t\\_i \\neq \\t\\_i] \n\\;\\; \\;\\;& \\geq & \\;\\; \\;\\; \\frac{1}{4} \\,e^{-C' \\frac{\\Gamma \\lambda \\sp}{m} } \\;,\n\\label{eq:lb\\_adapt}\\end{aligned}$$ \nwhere *m* is the number of tasks, Γ is the expected budget allowed in TΓ, *λ* is the collective difficulty of the tasks from a prior distribution $\\cQ$ defined in, and *σ*2 is the collective reliability of the crowd from a prior distribution $\\cP$ defined in.\nIn the proof, we provide a proof of a slightly stronger statement in Lemma [lem:lbadaptpre], where a similar lower bound holds for not only the worst-case $\\cQ$ but for all $\\cQ\\in\\cQ\\_{\\lambda}$. One caveat is that there is now an extra additive term in the error exponent in the RHS of the lower bound that depends on $\\cQ$, which is subsumed in the constant term (1/4) for the worst-case $\\cQ$ in the RHS of. We are assigning Γ/*m* queries per task on average, and it is intuitive that the error decays exponentially in Γ/*m*. The novelty in the above analysis is that it characterizes how the error exponent depends on the $\\cP$, which determines the quality of the crowd you have in your crowdsourcing platform, and $\\cQ$, which determines the quality of the tasks you have in your hand. If we have easier tasks and reliable workers, the error rate should be smaller. Eq. shows that this is captured by the error exponent scaling linearly in *λ**σ*2. This gives a lower bound (i.e. a necessary condition) on the budget required to achieve error at most *ɛ*; there exists a constant *C*ʺ such that if the total budget is\n \n$$\\begin{aligned}\n \\Gamma\\_\\varepsilon \\;\\; \\leq \\;\\; C'' \\frac{m}{\\lambda \\sigma^2 } \\log\\left(\\frac{1}{\\varepsilon}\\right) \\;, \n \\label{eq:budget\\_lb}\\end{aligned}$$ \nthen no task assignment scheme (adaptive or not) with any inference algorithm can achieve error less than *ε*. This recovers the known fundamental limit for standard DS model where all tasks have *λ**i* = 1 and hence *λ* = 1 in. For this standard DS model, it is known that there exists a constant *C*‴ such that if the total budget is less than \n$$\\Gamma\\_\\varepsilon \\;\\; \\leq \\;\\; C''' \\frac{m}{ \\sigma^2} \\log\\left(\\frac{1}{\\epsilon}\\right)\\;,$$ \n then no task assignment with any inference algorithm can achieve error rate less than *ɛ*. For example, consider two types of prior distributions where in one we have the original DS tasks with $\\cQ(q\\_i=0)= \\cQ(q\\_i=1) = 1/2$ and in the other we have $\\cQ'(q\\_i=0)=\\cQ'(q\\_i=1)=\\cQ'(q\\_i=3/4)=\\cQ'(q\\_i=1/4)=1/4$. We have *λ* = 1 under $\\cQ$ and *λ*ʹ = 2/5 under $\\cQ'$. Our analysis, together with the matching upper bound in the following section, shows that one needs 5/2 times more budget to achieve the same accuracy under the tasks from $\\cQ'$.\nUpper bound on the achievable error rate\n----------------------------------------\nWe present an adaptive task assignment scheme and an iterative inference algorithm that asymptotically achieve an error rate of *C*1*e*− (*C**δ*/4)(Γ/*m*)*λ**σ*2, when the number of tasks *m* grows large and the expected budget is increasing as Γ = Θ(*m*log*m*) where *C*1 = log2(2*δ*max/*δ*min)log2(2*λ*max/*λ*min) and *C**δ* is a constant that only depends on {*δ**a*}*a* ∈ [*T*]. This matches the lower bound in when *C*1 and *C**δ* are *O*(1). Comparing it to a fundamental lower bound in Theorem [thm:lbadapt] establishes the near-optimality of our approach, and the sufficient condition to achieve average error *ɛ* is for the average total budget to be larger than,\n \n$$\\begin{aligned}\n \\Gamma\\_\\varepsilon \\;\\; \\geq \\;\\; \\frac{4}{C\\_\\delta} \\frac{m}{\\lambda \\sigma^2 }\\log \\Big(\\frac{C\\_1}{\\varepsilon} \\Big)\\;. \n \\label{eq:budget\\_ad\\_ub}\n \\end{aligned}$$ \nOur proposed adaptive approach in Algorithm [algo:adapt] takes as input the number of tasks *m*, a target budget Γ, hyper parameter *C**δ* to be determined by our theoretical analyses in Theorem [thm:ubadapt], the quantized prior distribution $\\lq\\cQ$, the statistics *μ* and *σ*2 on the worker prior $\\cP$. The proposed scheme makes at most Γ queries in expectation to the crowd and outputs the estimated labels *t̂**i*’s for all the tasks *i* ∈ [*m*].\n[ht] [algo:adapt]\n[1] number of tasks *m*, allowed budget Γ, hyper parameter *C**δ*, quantized prior distribution {*λ**a*, *δ**a*}*a* ∈ [*T*], collective quality of the workers *σ*2, average reliability *μ* Estimated labels $\\{\\t\\_i\\}\\_{i \\in [m]}$ *M* ← {1, 2, ⋯, *m*} $\\lq\\lambda \\leftarrow \\Big(\\sum\\_{a \\in [T]} (\\delta\\_a/\\lambda\\_a)\\Big)^{-1}$ $\\ell\\_t \\leftarrow (C\\_{\\delta} \\lq\\lambda\\, \\,\\Gamma) / (m\\,\\lambda\\_t)$, *r**t* ← ℓ*t* [linegamma] $s\\_t \\leftarrow \\max\\Big\\{0, {\\left \\lceil{\\log\\Big(\\frac{2\\delta\\_t}{\\delta\\_{t+1}}\\Big)} \\right \\rceil}\\Big\\}\\I\\{t < T\\}+ 1\\, \\I\\{t = T\\}$[a0] *n* ← ∣*M*∣, $k \\leftarrow \\sqrt{\\log|M|}$ Draw *E* ∈ {0, 1}∣*M*∣ × *n* ∼ (ℓ*t*, *r**t*)-regular random graph [a1] Collect answers {*A**i*, *j* ∈ {1, − 1}}(*i*, *j*) ∈ *E* [a2] {*x**i*}*i* ∈ *M* ← Algorithm [algo:mp] $\\big[E, \\{A\\_{i,j}\\}\\_{(i,j) \\in E}, \\textcolor{black}{k} \\big]$ [a3] $\\rho^2\\_{t,u} \\leftarrow\\text{Algorithm \\ref{algo:est} }[E,\\{A\\_{i,j}\\}\\_{(i,j) \\in E},\\ell\\_t,r\\_t]$ [a4] $\\mathcal{X}\\_{t,u} \\leftarrow \\sqrt{\\lambda\\_t} \\mu \\ell\\_t \\big((\\ell\\_t-1)(r\\_t-1)\\textcolor{black}{\\rho^2\\_{t,u}\\sigma^2 }\\big)^{k-1}\\I\\{t < T\\}+ 0\\, \\I\\{t = T\\}$ [a5] *t̂**i* ← + 1 *t̂**i* ← − 1 *M* ← {*i* ∈ *M* : ∣*x**i*∣ ≤ X*t*, *u*} [a6]\n### The proposed adaptive approach: overview.\nAt a high level, our approach works in *T* *rounds* indexed by *t* ∈ [*T*], the support size of the quantized distribution $\\lq\\cQ$, and *s**t* *sub-rounds* at each round *t*, where *s**t* is chosen by the algorithm in line [a0]. In each sub-round, we perform both task assignment and inference, sequentially. *Inference* is done in line [a3] to get a confidence score *x**i*’s on the tasks *i* ∈ *M*, where *M* ⊆ [*m*] is the set of tasks that are remaining to be labelled at the current sub-round. The *adaptive task assignment* of our approach is entirely managed by the choice of this set *M* in line [a6], as only those tasks in *M* will be assigned new workers in the next sub-round in lines [a1] and [a2].\nAt each round, we choose how many responses to collect for each task present in that round as prescribed by our theoretical analysis. Given this choice of ℓ*t*, the number of responses collected for each task at round t, we repeat the key inner-loop in line [a1]-[a6] of Algorithm [algo:adapt]. In round *t* the sub-round is repeated *s**t* times to ensure that sufficient number of ‘easy’ tasks are classified. Given a set *M* of remaining tasks to be labelled, the sub-round collects ℓ*t* response per task on those tasks in *M* and runs an inference algorithm (Algorithm [algo:mp]) to give confidence scores *x**i*’s to all *i* ∈ *M*. Our theoretical analysis prescribes a choice of a threshold X*t*, *u* to be used in round *t* ∈ [*T*] sub-round *u* ∈ [*s**t*]. All tasks in *M* with confidence score larger than X*t*, *u* are permanently labelled as positive tasks, and those with confidence score less than − X*t*, *u* are permanently labelled as negative tasks. Those permanently labelled tasks are referred to as ‘classified’ and removed from the set *M*. The remaining tasks with confidence scores between X*t*, *u* and − X*t*, *u* are carried over to the next sub-round. The confidence scores are designed such that the sign of *x**i* provides the estimated true label, and we are more confident about this estimated label if the absolute value of the score *x**i* is larger. The art is in choosing the appropriate number of responses to be collected for each task ℓ*t* and the threshold X*t*, *u*, and our theoretical analyses, together with the provided statistics of the prior distribution $\\cP$, and the prior quantized distribution $\\lq\\cQ$ allow us to choose the ones that achieve a near optimal performance.\nNote that we are mixing inference steps and task assignment steps. Within each sub-round, we are performing both task assignment and inference. Further, the inner-loop within itself uses a non-adaptive task assignment, and hence our approach is a series of non-adaptive task assignments with inference in each sub-round. However, Algorithm [algo:adapt] is an adaptive scheme, where the adaptivity is fully controlled by the set of remaining unclassified tasks *M*. We are adaptively choosing which tasks to carry over in the set *M* based on all the responses we have collected thus far, and we are assigning more workers to only those tasks in *M*.\nSince difficulty levels are varying across the tasks, it is intuitive to assign fewer workers to easy tasks and more workers to hard tasks. Supposing that we know the difficulty levels *λ**i*’s, we could choose to assign the ideal number of workers to each task according to *λ**i*’s. However, the difficulty levels are not known. The proposed approach starts with a smaller budget in the first round classifying easier tasks, and carries over the more difficult tasks to the later rounds where more budget per task will be assigned.\n### The proposed adaptive approach: precise.\nMore precisely, given a budget Γ and the statistics of $\\cP$, and the known quantized distribution $\\lq\\cQ$ we know what target probability of error to aim for, say *ɛ*, from Theorem [thm:ubadapt]. The main idea behind our approach is to allocate the given budget Γ over multiple rounds appropriately, and at each round get an estimate of the labels of the remaining tasks in *M* and also the confidence scores, such that with an appropriate choice of the threshold X*t*, *u* those tasks we choose to classify in the current round achieve the desired target error rate of $\\P[t\\_i\\neq \\hat{t}\\_i \\,\\big|\\, |x\\_i|>\\mathcal{X}\\_{t,u}]\\leq\\varepsilon$. As long as this guarantee holds at each round for all classified tasks, then the average error rate will also be bounded by $(1/m)\\sum\\_{i=1}^m \\P[t\\_i\\neq \\hat{t}\\_i ]\\leq\\varepsilon$ when the process terminates eventually. The only remaining issue is how many queries are made in total when this process terminates. We guarantee that in expectation at most Γ queries are made under our proposed choices of ℓ*t*’s and X*t*, *u*’s in the algorithm.\nAt round zero, we initially put all the tasks in *M* = [*m*]. A fraction of tasks are permanently labelled in each round and the un-labelled ones are taken to the next round. At round *t* ∈ {1, …, *T*}, our goal is to classify a sufficient fraction of those tasks in the *t*-th difficulty group {*i* ∈ *M* ∣ *λ**i* ∈ [(1/2)*λ**t*, *λ**t*]} with the desired level of accuracy. The art is in choosing the right number of responses to be collected per task ℓ*t* for that round and also the right threshold X*t*, *u* on the confidence score, to be used in the inner-loop in line [a1]-[a6] of Algorithm [algo:adapt]. If ℓ*t* is too low and/or threshold X*t*, *u* too small, then misclassification rate will be too large. If ℓ*t* is too large and/or X*t*, *u* is too large, we are wasting our budget and achieving unnecessarily high accuracy on those tasks classified in the current round, and not enough tasks will be classified in that round. We choose ℓ*t* and X*t*, *u* appropriately to ensure that the misclassification probability is at most *C*1*e*− (*C**δ*/4)(Γ/*m*)*λ**σ*2 based on our analysis (see ) of the inner-loop. We run the identical sub-rounds *s**t* = max{0, ⌈log2(2*δ**t*/*δ**t* + 1)⌉} times to ensure that enough fraction of tasks with difficulty *λ**i* ∈ [(1/2)*λ**t*, *λ**t*] are classified. Precisely, the choice of *s**t* insures that the expected number of tasks with difficulty *λ**i* ∈ [(1/2)*λ**t*, *λ**t*] remaining unclassified after *t*-th round is at most equal to the number of tasks in the next group, i.e., difficulty level *λ**i* ∈ [(1/2)*λ**t* + 1, *λ**t* + 1].\nNote that statistically, the fraction of the *t*-th group (i.e. tasks with difficulty [*λ**t* + 1, *λ**t*]) that get classified before the *t*-th round is very small as the threshold set in these rounds is more than their absolute mean message. Most tasks in the *t*-th group will get classified in round *t*. Further, the proposed pre-processing step of binning the tasks ensures that ℓ*t* + 1 ≥ 2ℓ*t*. This ensures that the total extraneous budget spent on the *t*-th group of tasks is not more than a constant times the allocated budget on those tasks.\nThe main algorithmic component is the inner-loop in line [a1]-[a6] of Algorithm [algo:adapt]. For a choice of the (per task) budget ℓ*t*, we collect responses according to a (ℓ*t*, *r**t* = ℓ*t*)-regular random graph on ∣*M*∣ tasks and ∣*M*∣ workers. The leading eigen-vector of the non-backtracking operator on this bipartite graph, weighted by the ± 1 responses reveals a noisy observation of the true class and the difficulty levels of the tasks. Let $x \\in\\reals^{|M|}$ denote this top left eigenvector, computed as per the message-passing algorithm of Algorithm [algo:mp]. Then the *i*-th entry *x**i* asymptotically converges in the large number of tasks *m* limit to a Gaussian random variable with mean proportional to the difficulty level (2*q**i* − 1), with mean and variance specified in Lemma [lem:gaussian]. This non-backtracking operator approach to crowdsourcing was first introduced in for the standard DS model. We generalize their analysis to this generalized DS model in Theorem [thm:mainthm] for finite sample regime, and further give a sharper characterization based on central limit theorem in the asymptotic regime (Lemma [lem:gaussian]). For a detailed explanation of Algorithm [algo:mp] and its analyses, we refer to Section [sec:nonadub].\n### Justification of the choice of ℓ*t* and X*t*, *u*.\nThe main idea behind our approach is to allocate a target budget to each *i*-th task according to its quantized difficulty *λ**t* where *t* is such that *λ**i* ∈ [(1/2)*λ**t*, *λ**t*]. Given a total budget Γ and the quantized distribution $\\lq\\cQ$ which gives the collective difficulty of tasks *λ* (line 2, Algorithm [algo:adapt]), we target to assign $(\\lq\\lambda/\\lambda\\_t)(\\Gamma/m)$ workers to a task of quantized difficulty *λ**t*. This choice of the target budget is motivated from the proof of the lower bound Theorem [thm:lbadapt]. If we had identified the tasks with respect to their difficulty then the near-optimal choice of the budget that achieves the lower bound is given in. Our target budget is a simplified form of the near-optimal choice and ignores the constant part that does not depend upon the total budget. This choice of the budget would give the equal probability of misclassification for the tasks of varying difficulties. We refer to this error rate as the desired probability of misclassification. As we do not know which tasks belong to which quantized difficulty group *λ**t*, a factor of 1/*C**δ* is needed to compensate for the extra budget needed to infer those difficulty levels. This justifies our choice of budget in line 4 of the Algorithm [algo:adapt].\nFrom our theoretical analysis of the inner loop, we know the probability of misclassification for a task that belongs to difficulty group *λ**t* as a function of the classification threshold X and the budget that is assigned to it. Therefore, in each round we set the classification threshold X*t*, *u* such that even the possibly most difficult task achieves the desired probability of misclassification. This choice of X*t*, *u* is provided in line 13 of Algorithm [algo:adapt].\n### Numerical experiments.\nIn Figure [fig:fig1], we compare the performance of our algorithm with majority voting and also a non-adaptive version of our Algorithm [algo:adapt], where we assign to each task ℓ = Γ/*m* number of workers in one round and set classification threshold X1, 1 = 0 so as to classify all the tasks (choosing *T* = 1 and *s*1 = 1). Since this performs the non-adaptive inner-loop *once*, this is a non-adaptive algorithm, and has been introduced for the standard DS model in.\n (-145,114) (-141,-9 ) (-145,114) (-141,-9 )\n[fig:fig1]\nFor numerical experiments, we make a slight modification to our proposed Algorithm [algo:adapt]. In the final round, when the classification threshold is set to zero, we include all the responses collected thus far when running the message passing Algorithm [algo:mp], and not just the fresh samples collected in that round. This creates dependencies between rounds, which makes the analysis challenging. However, in practice we see improved performance and it allows us to use the given fixed budget efficiently.\nWe run synthetic experiments with *m* = 1800 and fix *n* = 1800 for the non-adaptive version. The crowds are generated from the spammer-hammer model where a worker is a hammer (*p**j* = 1) with probability 0.3 and a spammer (*p**j* = 1/2) otherwise. In the left panel, we take difficulty level *λ**a* to be uniformly distributed over {1, 1/4, 1/16}, that gives *λ* = 1/7. In the right panel, we take *λ**a* = 1 with probability 3/4, otherwise we take it to be 1/4 or 1/16 with equal probability, that gives *λ* = 4/13. Our adaptive algorithm improves significantly over its non-adaptive version, and our main results in Theorems [thm:ubadapt] and [thm:mainthm] predicts such gain of adaptivity. In particular, for the left panel, the non-adaptive algorithm’s error scaling depends on smallest $\\lambda\\_{\\rm min}$ that is 1/16 while for the adaptive algorithm it scales with *λ* = 1/7. In the left figure, it can be seen that the adaptive algorithm requires approximately a factor of $\\lambda\\_{\\rm min}/\\lambda=7/16$ more queries to achieve the same error as achieved by the non-adaptive scheme. For example, non-adaptive version of Algorithm [algo:adapt] requires Γ/*m* = 360 to achieve error rate 0.002, whereas the adaptive approach only requires 180 ≃ 360 × 7/16 = 157.5. Quantifying such a gap is one of our main results in Theorems [thm:ubadapt] and [thm:mainthm]. This gap widens in the right panel to approximately $\\lambda\\_{\\rm min}/\\lambda=13/64$ as predicted. For a fair comparison with the non-adaptive version, we fix the total budget to be Γ and assign workers in each round until the budget is exhausted, such that we are strictly using budget at most Γ deterministically.\n### Performance Guarantee\nAlgorithm [algo:adapt] is designed in such a way that we are not wasting any budget on any of the tasks; we are not getting unnecessarily high accuracy on easier tasks, which is the root cause of inefficiency for non-adaptive schemes. In order to achieve this goal, the internal parameter *ρ**t*, *u*2 computed in line [a4] of Algorithm [algo:adapt] has to satisfy *ρ**t*, *u*2 = (1/∣*M*∣)∑*i* ∈ [*M*]*λ**i*, which is the average difficulty of the remaining tasks. Such a choice is important in choosing the right threshold ${\\mathcal X}\\_{t,u}$.\nAs the set *M* of remaining tasks is changing over the course of the algorithm, we need to estimate this value in each sub-routine. We provide an estimator of *ρ**t*, *u*2 in Algorithm [algo:est] that only uses the sampled responses that are already collected. All numerical results are based on this estimator. However, analyzing the sensitivity of the performance with respect to the estimation error in *ρ**t*, *u*2 is quite challenging, and for a theoretical analysis, we assume we have access to an oracle that provides the exact value of *ρ**t*, *u*2 = (1/∣*M*∣)∑*i* ∈ [*M*]*λ**i*, replacing Algorithm [algo:est].\n[thm:ubadapt] Suppose Algorithm [algo:est] returns the exact value of *ρ**t*, *u*2 = (1/∣*M*∣)∑*i* ∈ *M**λ**i*. With the choice of *C**δ* = (4 + ⌈log(2*δ*max/*δ*min)⌉)− 1, for any given quantized prior distribution of task difficulty {*λ**a*, *δ**a*}*a* ∈ [*T*] such that $\\delta\\_{\\rm max}/\\delta\\_{\\rm min}=O(1)$ and *λ*max/*λ*min = *O*(1), and the budget Γ = Θ(*m*log*m*), the expected number of queries made by Algorithm [algo:adapt] is asymptotically bounded by\n \n$$\\begin{aligned}\n \\lim\\_{m\\to\\infty} \\sum\\_{t\\in[{T}],u\\in [s\\_t]} \\textcolor{black}{\\ell\\_t\\;\\E[m\\_{t,u}]} \\;\\; \\leq\\;\\; \\Gamma\\;,\\end{aligned}$$ \nFurther, Algorithm [algo:adapt] returns estimates $\\{\\t\\_i\\}\\_{i \\in [m]}$ that asymptotically achieve, if (Γ/*m*)*λ**σ*2 = Θ(1), where *C*1 = log2(2*δ*max/*δ*min)log2(2*λ*max/*λ*min), and\n \n$$\\begin{aligned}\n \\lim\\_{m \\rightarrow \\infty} \\frac{1}{m} \\sum\\_{i=1}^{m} \\P[t\\_i \\neq \\t\\_i] \n \\;\\; = \\;\\; 0 \\,,\\end{aligned}$$ \nif (Γ/*m*)*λ**σ*2 = *ω*(1).\nA proof of this theorem is provided in Section [sec:adaptub]. In this theoretical analysis, we are considering a family of problem parameters $(m,\\cQ,\\cP,\\Gamma)$ in an increasing number of tasks *m*. All the problem parameters $\\cQ$, $\\cP$, and Γ can vary as functions of *m*. For example, consider a family of $\\cQ(q\\_i)= (1/2)\\mathbb{I}(q\\_i=0) + (1/2) \\mathbb{I}(q\\_i=1)$ independent of *m* and $\\cP(p\\_j) = (1-1/\\sqrt{m})\\mathbb{I}(p\\_j=0.5)+ (1/\\sqrt{m})\\mathbb{I}(p\\_j=1)$. As *m* grows, most of the workers are spammers giving completely random answers. In this setting, we can ask how should the budget grow with *m*, in order to achieve a target accuracy of, say, *e*− 5? We have *λ* = 1 and $\\sigma^2=1/\\sqrt{m}$, indicating that the collective difficulty is constant but collective quality of the workers are decreasing in *m*. It is a simple calculation to show that *C*1 = 1 and *C**δ* = 1/5 in this case, and the above theorem proves that Γ = 100*m*3/2 is sufficient to achieve the desired error rate. Further such dependence of the budget in *m* is also necessary, as follows from our lower bound in Theorem [thm:lbadapt].\nConsider now a scenario where we have tasks with increasing difficulties in *m*. For example, $\\cQ(q\\_i) = \n(1/4) \\mathbb{I}(q\\_i=1/2+1/\\log m) + (1/4) \\mathbb{I}(q\\_i=1/2-1/\\log m) +\n(1/4) \\mathbb{I}(q\\_i=1/2+2/\\log m) + (1/4) \\mathbb{I}(q\\_i=1/2-2/\\log m) \n$ and $\\cP(p\\_j) = \\mathbb{I}(p\\_j=3/4)$. We have *λ* = 32/(5(log*m*)2) and *σ*2 = 1/4. It follows from simple calculations that *C*1 = 2 and *C**δ* = 1/5. It follows that it is sufficient and necessary to have budget scaling in this case as Γ = Θ(*m*(log*m*)2).\nFor families of problem parameters for increasing *m*, we give asymptotic performance guarantees. Finite regime of *m* is challenging as our analysis relies on a version of central limit theorem and the resulting asymptotic distribution of the score value *x**i*’s. However, the numerical simulations in Figure [fig:fig1] suggests that the improvement of the proposed adaptive approach is significant for moderate values of *m* as well.\nOur main result in Eq. gives the sufficient condition of our approach in. Compared to the fundamental lower bound in Theorem [thm:lbadapt], this proves the near-optimality of our adaptive approach. Under the regime considered in Theorem [thm:ubadapt], it is necessary and sufficient to have budget scaling as Γ = Θ((*m*/(*λ**σ*2))log(1/*ɛ*)).\nAnalysis of the inner-loop and the minimax error rate under the non-adaptive scenario\n=====================================================================================\nIn this section, we provide the analysis of the non-adaptive task assignment and inference algorithm in the sub-routine in line [a1]-[a6] of Algorithm [algo:adapt]. To simplify the notations, we consider the very first instance of the sub-round where we have a set *M* = [*m*] of tasks to be labelled, and all the subsequent subroutines will follow similarly up to a change of notations. Perhaps surprisingly, we show that this inner-loop itself achieves near optimal performance for *non-adaptive schemes*. We show that $\\Gamma = O((m/(\\lambda\\_{\\rm min}\\sigma^2))\\log(1/\\varepsilon))$ is sufficient to achieve a target probability of error *ɛ* > 0 in Theorem [thm:mainthm]. We show this is close to optimal by comparing it to a necessary condition that scales in the same way in Theorem [thm:thmlb]. First, here is the detailed explanation of the inner-loop.\nSuppose we are given a budget of Γ = *m*ℓ, so that each task can be assigned to ℓ workers on average. Further assume that each worker is assigned *r* tasks. We are analyzing a slightly more general setting than Algorithm [algo:adapt] where *r* = ℓ for all instances. We follow the recipe of and use a random regular graph for a non-adaptive task assignment. Namely, we know that we need to recruit *n* = *m*ℓ/*r* workers in total. Before any responses are collected, we make all the task assignments for all *n* workers in advance and store it in a bipartite graph *G*([*m*], [*n*], *E*) where [*m*] are the task nodes, [*n*] are the worker nodes, and *E* ⊆ [*m*] × [*n*] is the collection of edges indicating that task *i* is assigned to worker *j* if (*i*, *j*) ∈ *E*. This graph *E* is drawn from a random regular graph with task degree ℓ and worker degree *r*. Such random graphs can be drawn efficiently, for example, using the configuration model.\nUnder the *original Dawid-Skene model*, showed that this non-adaptive task assignment achieves the minimax optimal error rate when labels are estimated using Algorithm [algo:mp]. This was surprising, as adaptive task assignments were shown to have no gain over this non-adaptive scheme. Under the *generalized Dawid-Skene model*, we are significantly improving upon this simple non-adaptive scheme by applying this to multiple rounds with adaptive choices in each round on which tasks to carry over to the next round. Our adaptive scheme uses this non-adaptive task assignments in the inner-loop repeatedly, making adaptive choices on which tasks are carried over to the next rounds.\nThe message passing algorithm of Algorithm [algo:mp], is a state-of-the-art spectral method based on non-backtracking operators, first introduced for inference in. A similar approach has been later applied to other inference problems, e.g.. This is a message passing algorithm that operates on two sets of messages: the task messages {*x**i* → *j*}(*i*, *j*) ∈ *E* capturing how likely the task is to be a positive task and the worker messages {*y**j* → *i*}(*i*, *j*) ∈ *E* capturing how reliable the worker is. Consider a data collected on *m* tasks and *n* workers such that *A* ∈ {0, + 1, − 1}*m* × *n* under the non-adaptive scenario with task assigned according to a random regular graph *E* of task degree ℓ and worker degree *r*. In each round, all messages are updated as\n \n$$\\begin{aligned}\n \\label{eq:update1}\n x\\_{i\\to j} &=& \\sum\\_{j'\\in W\\_i \\setminus j} A\\_{ij'} y\\_{j'\\to i}\\;, \\text{ and } \\\\\n y\\_{j\\to i} &=& \\sum\\_{i'\\in T\\_j \\setminus i} A\\_{i'j} x\\_{i'\\to j}\\;, \n \\label{eq:update2}\\end{aligned}$$ \nwhere *W**i* ⊆ [*n*] is the set of workers assigned to task *i*, and *T**j* ⊆ [*m*] is the set of workers assigned to worker *j*. The first is taking the weighted majority according to how reliable each worker is, and the second is updating the reliability according to how many times the worker agreed with what we believe. After a prefixed $k\\_{\\rm max}$ iterations, we provide a confidence score by aggregating the messages at each task node *i* ∈ [*m*]:\n \n$$\\begin{aligned}\n \\label{eq:update3}\n x\\_{i} &=& \\sum\\_{j'\\in W\\_i } A\\_{ij'} y\\_{j'\\to i}\\;. \\end{aligned}$$ \nThe precise description is given in Algorithm [algo:mp]. Perhaps surprisingly, this algorithm together with the random regular task assignment achieve the minimax optimal error rate among all non-adaptive schemes. This will be made precise in the upper bound in Theorem [thm:mainthm] and a fundamental lower bound in Theorem [thm:thmlb]. An intuitive explanation of why this algorithm works is provided in Section [sec:spectral] via spectral interpretation of this approach.\n[ht] [algo:mp]\n[1] *E* ∈ {0, 1}∣*M*∣ × *n*, {*A**i**j* ∈ {1, − 1}}(*i*, *j*) ∈ *E*, *k*max $\\{x\\_i \\in \\reals\\}\\_{i \\in [|M|]}$ Initialize *y**j* → *i*(0) with a Gaussian random variable $Z\\_{j \\rightarrow i } \\sim \\N(1,1)$ *x**i* → *j*(*k*) ← ∑*j*ʹ ∈ *W**i* \\ *j**A**i**j*ʹ*y**j*ʹ → *i*(*k* − 1) *y**j* → *i*(*k*) ← ∑*i*ʹ ∈ *T**j* \\ *i**A**i*ʹ*j**x**i*ʹ → *j*(*k*) $x\\_i^{(k\\_{\\rm max})} \\leftarrow \\sum\\_{j \\in W\\_i} A\\_{ij} y\\_{j \\rightarrow i}^{(k\\_{\\rm max}-1)}$\nPerformance guarantee\n---------------------\nFor this non-adaptive scenario, we provide a sharper upper bound on the achieved error, that holds for all (non-asymptotic) regimes of *m*. Define *σ**k*2 as\n \n$$\\begin{aligned}\n \\sigma\\_k^2 &\\equiv& \\frac{2\\sp }{\\mu^2 \\big(\\hat{\\ell}\\hat{r}(\\sq\\sp)^2\\big)^{k-1}} + 3\\bigg( 1 + \\frac{1}{\\hat{r}\\sq\\sp} \\bigg) \\frac{1 - 1/\\big(\\hat{\\ell}\\hat{r}(\\sq\\sp)^2\\big)^{k-1}}{1 - 1/ \\big(\\hat{\\ell}\\hat{r}(\\sq\\sp)^2\\big)} \\;,\\end{aligned}$$ \nwhere $\\hat{\\ell}=\\ell-1$, *r̂* = *r* − 1, $\\mu=\\E\\_\\cP[2p\\_j-1]$, $\\sigma^2 = \\E\\_\\cP[(2p\\_j-1)^2]$, and $\\rho^2 = \\E\\_\\cQ[(2q\\_i-1)^2]$. This captures the effective variance in the sub-Gaussian tail of the messages *x**i*’s after *k* iterations of Algorithm [algo:mp], as shown in the proof of the following theorem in Section [sec:nonadaptub].\n[thm:mainthm] For any ℓ > 1 and *r* > 1, suppose *m* tasks are assigned according to a random (ℓ, *r*)-regular graph drawn from the configuration model. If *μ* > 0, $\\l\\r \\rho^4 \\sigma^4 > 1$, and $\\r\\sq > 1$, then for any *t* ∈ { ± 1}*m*, the estimate $\\hat{t}\\_i^{(k)} = {\\rm sign}(x\\_i^{(k)})$ after *k* iterations of Algorithm [algo:mp] achieves\n \n$$\\begin{aligned}\n \\P\\big[t\\_{i} \\neq \\hat{t}\\_{i}^{(k)} \\big| \\lambda\\_{i}\\big] \\;\\;&\\leq&\\;\\; e^{-\\ell\\sp\\lambda\\_i/ (2\\sigma\\_k^2)} + \\frac{3\\ell r }{m}(\\hat{\\ell}\\hat{r})^{2k-2}.\n \\end{aligned}$$ \nTherefore, the average error rate is bounded by\n \n$$\\begin{aligned}\n \\frac{1}{m}\\sum\\_{i = 1}^m \\P[t\\_i \\neq \\hat{t}\\_i^{(k)}] \\;\\;& \\leq &\\;\\; \\E\\_{\\cQ}\\bigg[e^{\\frac{-\\ell\\sp\\lambda\\_i}{2\\sigma\\_k^2}}\\bigg] \n+ \\frac{3\\ell r }{m}(\\hat{\\ell}\\hat{r})^{2k-2}. \\label{eq:average}\\end{aligned}$$ \nThe second term, which is the probability that the resulting (ℓ, *r*)-regular random graph is not locally tree-like, can be made small for large *m* as long as $k=O(\\sqrt{\\log m})$ (which is the choice we make in Algorithm [algo:adapt]). Hence, the dominant term in the error bound is the first term. Further, when we run our algorithm for large enough numbers of iterations, *σ**k*2 converges linearly to a finite limit *σ*∞2 ≡ lim*k* → ∞*σ**k*2 such that\n \n$$\\begin{aligned}\n\\sigma\\_{\\infty}^2 \\;\\; =\\;\\; 3\\Big(1 + \\frac{1}{\\r\\sq\\sp } \\Big) \\frac{(\\l\\r\\sq\\sp)^2} {(\\l\\r\\sq\\sp)^2 - 1} \\;, \\end{aligned}$$ \nwhich is upper bounded by a constant for large enough *r̂**ρ*2*σ*2 and $\\hat{\\ell} $, for example *r̂**ρ*2*σ*2 ≥ 1 and $\\hat{\\ell} \\geq 2$. Hence, for a wide range of parameters, the average error in is dominated by $\\E\\_{\\cQ}\\big[e^{{-\\ell\\sp\\lambda\\_i}/{2\\sigma\\_k^2}}\\big]$. When the fraction of tasks with worst-case difficulty *λ*min is strictly positive, the error is dominated by them as illustrated in Figure [fig:fig0]. Hence, it is sufficient to have budget\n \n$$\\begin{aligned}\n \\Gamma\\_\\varepsilon \\;\\; \\geq \\;\\; \\frac{C'' m}{\\lambda\\_{\\rm min} \\sigma^2} \\log(1/\\varepsilon)\\;, \\end{aligned}$$ \nto achieve an average error of *ɛ* > 0. Such a scaling is also necessary as we show in the next section.\n (-145,120) (-141,-7 ) (-145,120) (-111,-7 ) [fig:fig0]\nThis is further illustrated in Figure [fig:fig0]. The error decays exponentially in ℓ and $\\sp$ as predicted, but the rate of decay crucially hinges on the individual difficulty level of the task being estimated. We run synthetic experiments with *m* = *n* = 1000 and the crowds are generated from the spammer-hammer model where *p**j* = 1 with probability *σ*2 and *p**j* = 1/2 with probability 1 − *σ*2, where the choice of this probability is chosen to match the collective difficulty $\\sigma^2=\\E[(2p\\_j-1)^2]$. We fix *σ*2 = 0.3 and vary ℓ in the left figure and fix ℓ = 30 and vary *σ*2 in the right figure. We let *q**i*’s take values in {0.6, 0.8, 1} with equal probability such that *ρ*2 = 1.4/3. The error rate of each task grouped by their difficulty is plotted in the dashed lines, matching predicted $e^{-\\Omega(\\ell \\sp (2q\\_i-1)^2)}$. The average error rates in solid lines are dominated by those of the difficult tasks, which is a universal drawback for all non-adaptive schemes.\nFundamental limit under the non-adaptive scenario\n-------------------------------------------------\nTheorem [thm:mainthm] implies that it suffices to assign $\\ell \\geq (c/(\\sp\\lambda\\_i))\\log(1/\\varepsilon)$ workers to achieve an error smaller than *ɛ* for a task *i*. We show in the following theorem that this scaling is also necessary when we consider all *non-adaptive* schemes. Even the best non-adaptive task assignment with the best inference algorithm still required budget scaling in the same way. Hence, applying *one round* of Algorithm [algo:adapt] (which is a non-adaptive scheme) is near-optimal in the non-adaptive scenario compared to a minimax rate where the nature chooses the worst distribution of worker *p**j*’s among the set of distributions with the same $\\sp$. We provide a proof of the theorem in Section [sec:nonadaptlb].\n[thm:thmlb]\nFor formal comparisons with the upper bound, consider a case where the induced distribution on task difficulties *λ**i*’s, $\\lt\\cQ$, is same as its quantized version $\\lq\\cQ$ such that $\\lt\\cQ(\\lambda\\_i) = \\sum\\_{a=1}^T \\delta\\_a {\\mathbb I}\\_{(\\lambda\\_i=\\lambda\\_a)}$. Since in this non-adaptive scheme, task assignments are done a priori, there are on average ℓ workers assigned to any task, regardless of their difficulty. In particular, if the total budget is less than\n \n$$\\begin{aligned}\n \\Gamma\\_\\varepsilon &\\leq& C' \\frac{m}{\\lambda\\_{\\rm min} \\sigma^2} \\log \\frac{\\delta\\_{\\rm min}}{\\varepsilon},\n \\label{eq:budger\\_nonad\\_lb}\\end{aligned}$$ \nthen there will a a proportion of at least $\\delta\\_{\\rm min}$ tasks with error larger than $\\varepsilon/\\delta\\_{\\rm min}$, resulting in overall average error to be larger than *ɛ* even if the rest of the tasks are error-free. Compared to the adaptive case in (nearly achieved up to a constant factor in ), the gain of adaptivity is a factor of $\\lambda/\\lambda\\_{\\rm min}$. When $\\delta\\_{\\rm min} < \\varepsilon$, the above necessary condition is trivial as the RHS is negative. In such a case, the necessary condition can be tightened to *C*ʹ(*m*/*λ**a**σ*2)log(∑*b* = 1*a**δ**b*/*ɛ*) where *a* is the smallest integer such that ∑*b* = 1*a**δ**b* > *ɛ*.\nSpectral interpretation of Algorithm [algo:mp] and parameter estimation\n=======================================================================\nIn this section, we give a spectral analysis of Algorithm [algo:mp], which leads to a spectral algorithm for estimating *ρ*2 (Algorithm [algo:est]), to be used in the inner-loop of Algorithm [algo:adapt]. This spectral interpretation provides a natural explanation of how Algorithm [algo:mp] is extracting information and estimating the labels. Precisely, we are computing the top eigenvector of a matrix known as a weighted non-backtracking operator, via standard power method. Note that the above mapping is a *linear mapping* from the messages to the messages. This mapping, if formed into a 2∣*E*∣ × 2∣*E*∣ dimensional matrix *B* is known as the non-backtracking operator. Precisely, for (*i*, *j*), (*i*ʹ, *j*ʹ) ∈ *E*,\n \n$$\\begin{aligned}\n B\\_{(i\\to j),(j'\\to i')} = \\left\\{ \n \\begin{array}{rl}\n A\\_{i'j'} & \\text{if $j=j'$ and $i\\neq i'$ }\\;, \\\\\n A\\_{i'j'} & \\text{if $j\\neq j'$ and $i= i'$ }\\;, \\\\\n 0 & \\text{ otherwise }, \n \\end{array}\n \\right.\\end{aligned}$$ \nand the message update of Equations and are simply\n \n$$\\begin{aligned}\n \\begin{bmatrix}\n x\\\\\n y\n \\end{bmatrix}\n \\;\\; = \\;\\;B\\;\n \\begin{bmatrix}\n x\\\\\n y\n \\end{bmatrix}\\;,\\end{aligned}$$ \nwhere *x* and *y* denote vectorizations of *x**i* → *j*’s and *y**i* → *j*’s. This is exactly the standard power method to compute the singular vector of the matrix $B\\in\\reals^{2|E|\\times 2|E|}$.\nThe spectrum, which is the set of eigenvalues of this square but non-symmetric matrix *B* illustrates when and why spectral method might work. First consider decomposing the data matrix as\n \n$$\\begin{aligned}\n A \\;\\; = \\;\\; \\underbrace{\\E[A]}\\_{\\text{true signal}} + \\underbrace{(A-\\E[A])}\\_{\\text{random noise}} \\;.\\end{aligned}$$ \nSimple analysis shows that $\\E[A|q,p]$, where the expectation is taken with respect to the randomness in the graph and also in the responses, is a rank one matrix with spectral norm $\\|\\E[A|q,p]\\| = \\sqrt{\\ell r \\hat{\\rho}^2 \\hat{\\sigma}^2 }$, where\n \n$$\\begin{aligned}\n \\hat{\\rho}^2 \\; \\equiv \\; \\frac1m \\sum\\_{i=1}^m (2q\\_i-1)^2 \\;,\\;\\; \\text{ and } \\;\n \\hat{\\sigma}^2 \\; \\equiv \\; \\frac1n \\sum\\_{j=1}^n (2p\\_j-1)^2 \\;. \\end{aligned}$$ \nThis is easy to see as $\\E[A\\_{ij}|q,p] = (\\ell/n) (2q\\_i-1)(2p\\_j-1) $. It follows that the expected matrix is $\\E[A|q,p] = \\sqrt{\\ell\\,r/(mn)}\\, \\sqrt{\\hat{\\rho}^2 \\hat{\\sigma}^2 mn} \\, uv^T $, where *u* and *v* are norm-one vectors with $u\\_i=(1/\\sqrt{\\sum\\_{i'\\in[m]}(2q\\_{i'}-1)^2})(2q\\_i-1)$ and $v\\_j=(1/\\sqrt{\\sum\\_{j'\\in[n]}(2p\\_{j'}-1)^2})(2p\\_j-1)$.\nAlso, typical random matrix analyses, such as those in, show that the spectral norm (the largest singular value) of the noise matrix $(A-\\E[A|q,p])$ is bounded by *C*(ℓ*r*)1/4 with some constant *C*. Hence, when the spectral norm of the signal is larger then that of the noise, i.e. $\\|\\E[A|q,p]\\| > \\|(A-\\E[A|q,p])\\|$, the top eigenvector of this matrix *A* corresponds to the true underlying signal, and we can hope to estimate the true labels from this top eigenvector. On the other hand, if $\\|\\E[A|q,p]\\| < \\|(A-\\E[A|q,p])\\|$, one cannot hope to recover any signal from the top eigenvector of *A*. This phenomenon is known as the spectral barrier.\nThis phenomenon is more prominent in the non-backtracking operator matrix *B*. Note that *B* is not symmetric and hence the eigen values are complex valued. Similar spectral analysis can be applied to show that when we are above the spectral barrier, the top eigenvalue is real-valued and concentrated around the mean $\\Lambda\\_1(B) \\simeq \\sqrt{(\\ell-1) (r-1) \\hat{\\rho}^2 \\hat{\\sigma}^2}$ and the mode of the rest of the complex valued eigenvalues are bounded within a circle of radius: ∣Λ*i*(*B*)∣ ≤ ((ℓ − 1)(*r* − 1))1/4. Hence, the spectral barrier is exactly when Λ1(*B*) = ∣Λ*i*(*B*)∣ which happens at (ℓ − 1)(*r* − 1)*ρ̂*4*σ̂*4 = 1, and this plays a crucial role in the performance guarantee in Theorem [thm:mainthm]. Note that because of the bipartite nature of the graph we are considering, we always have a pair of dominant eigenvalue as $\\Lambda\\_1(B) = \\sqrt{(\\ell-1) (r-1) \\hat{\\rho}^2 \\hat{\\sigma}^2}$ and $\\Lambda\\_2(B) = - \\sqrt{(\\ell-1) (r-1) \\hat{\\rho}^2 \\hat{\\sigma}^2}$.\n(-104,-9)real part of Λ*i*(*B*) (-260,75)imaginary part of Λ*i*(*B*) \n(-65,-9)real part of Λ*i*(*B*)\n[fig:spectrum]\nFigure [fig:spectrum] illustrates two sides of the spectral barrier. The one on the left shows the scatter plot of the complex valued eigen values of *B*. Notice a pair of top eigen values at $\\sqrt{0.3 \\times (1.4/3) \\times 14\\times 14} \\simeq 5.24$ and − 5.24 as predicted by the analysis. They always appear in pairs, due to the bipartite nature of the graph involved. The rest of the spurious eigenvalues are constrained within a circle of radius (14 × 14)1/4 ≃ 3.74 as predicted. The figure on the right is when we are below the spectral barrier, since the eigenvalue corresponding to the signal is $\\sqrt{0.3 \\times (1.4/3) \\times 4\\times 4} \\simeq 1.5$ which is smaller than (4 × 4)1/4 = 2. The relevant eigenvalue is buried under other spurious eigenvalues and does not show.\nAmong other things, this spectral interpretation gives an estimator for the problem parameter *ρ*2, to be used in the inner-loop of Algorithm [algo:adapt]. Consider the data matrix $\\widetilde{A}$ defined below. Again, a simple analysis shows that $\\E[\\widetilde{A}|q,p]$ is a rank one matrix with $\\|\\E[\\widetilde{A}|q,p]\\| = \\sqrt{\\ell r \\hat{\\rho}^2 \\hat{\\sigma}^2}$. Since the spectral norm of the noise matrix $\\|\\widetilde{A} - \\E[\\widetilde{A}|q,p]\\|$ is upper bounded by *C*(ℓ*r*)1/4 for some constant *C*, we have $\\|\\widetilde{A}\\|/\\sqrt{ \\ell r \\sigma^2} = \\sqrt{\\hat{\\rho}^2 \\hat{\\sigma}^2/\\sigma^2} + O((\\ell r \\hat{\\rho}^4 \\hat{\\sigma}^4)^{-1/4})$. We know ℓ and *r*, and assuming we know *σ*, this provides a natural estimator for *ρ̂*2. Note that $\\hat{\\sigma}^2 = \\sigma^2 + O(\\log (n) /\\sqrt{n})$ with high probability. The performance of this estimator is empirically evaluated, as we use this in all our numerical simulations to implement our adaptive scheme in Algorithm [algo:adapt].\n[ht] [algo:est]\n[1] assignment graph adjacency matrix *E* ∈ {0, 1}∣*M*∣ × *n*, binary responses from the crowd {*A**i**j*}(*i*, *j*) ∈ *E*, task degree ℓ, worker degree *r*, worker collective quality parameter *σ*2 estimate *ρ*2 of (1/∣*M*∣)∑*i* ∈ *M**λ**i* Construct matrix $\\widetilde{A}\\in\\{0,\\pm1\\}^{|M|\\times n }$ such that i,j={\nrl Ai,j& \n0 &\n. for all *i* ∈ [∣*M*∣], *j* ∈ [*n*]. Set $\\sigma\\_1(\\widetilde{A})$ to be the top singular value of matrix $\\widetilde{A}$ $\\rho^2 \\leftarrow \\big(\\sigma\\_1(\\widetilde{A})/\\sqrt{\\ell r\\sigma^2}\\big)^2$\nAlternative inference algorithm for the generalized DS model\n============================================================\nOur main contribution is a general framework for adaptive crowdsourcing: starting with a small-budget, classify tasks with high-confidence, and then gradually increase the budget per round, classifying remaining tasks. If we have other inference algorithms with which we can get reliable confidence levels in the estimated task labels, we can replace Algorithm [algo:mp]. In this section, we propose such a potential candidate and discuss the computational challenges involved.\nUnder the original DS model, various standard methods such as Expectation Maximization (EM) and Belief Propagation (BP) provide efficient inference algorithms that also work well in practice. However, under the generalized DS model, both approaches fail to give computationally tractable inference algorithms. In this section, we propose an alternative inference algorithm based on alternating minimization. This approach enjoys the benefits of EM and BP, such as seamlessly extending to *k*-ary alphabet labels, while remaining computationally manageable. Figure [fig:comp] illustrates how this alternating minimization performs at least as well as the iterative algorithm (Algorithm [algo:mp]), and improves significantly when the budget is critically small, i.e. only a few workers are assigned to each task.\nWe propose to maximize the posterior distribution,\n \n$$\\begin{aligned}\n\\label{eq:eqgm1}\n\\P[\\q, \\p| \\A ] \\;\\; \\propto \\;\\; \\prod\\_{i \\in [m]} \\P\\_\\cQ[q\\_i] \\; \\prod\\_{j \\in [n]} \\bigg( \\P\\_\\cP[p\\_j ] \n\\prod\\_{i' \\in W\\_j} \\P[A\\_{i'j}| p\\_{i'},q\\_j]\\bigg)\\,.\\end{aligned}$$ \nAlthough this function is not concave, maximizing over $\\q$ (or $\\p$) fixing $\\p$ (or $\\q$) is simple due to the bipartite nature of the graph. Define a function *g* : { ± 1} × [0, 1] × [0, 1] → [ − ∞, 0] such that\n \n$$\\begin{aligned}\ng(A\\_{ij},q\\_i,p\\_j) = \\begin{cases}\n \\log(q\\_ip\\_j + \\bq\\_i\\bp\\_j)\\;\\;\\; \\text{if}\\; A\\_{ij}=1 \\\\\n \\log(\\bq\\_ip\\_j + q\\_j\\bp\\_i) \\;\\;\\; \\text{if}\\; A\\_{ij} = -1\n \\end{cases}\\end{aligned}$$ \nThe logarithm of the joint posterior distribution is\n \n$$\\begin{aligned}\n \\cL(\\q,\\p | \\A )\n\\;\\; = \\;\\;\\sum\\_{i \\in [m]}\\sum\\_{j \\in W\\_i} g(A\\_{ij},q\\_i,p\\_j) \n + \\sum\\_{i \\in [m]}\\log(\\P\\_\\cQ[q\\_i ]) + \\sum\\_{j \\in [n]}\\log(\\P\\_\\cP[p\\_j ])\\,. \\label{eq:eq\\_am1}\\end{aligned}$$ \nWith properly chosen prior distributions $\\cQ$ and $\\cP$, in particular Beta priors, it is easy to see that the log likelihood is a concave function of $\\p$ for fixed $\\q$. The same is true when fixing $\\p$ and considering a function over $\\q$. Further, each coordinate *p**j* (and *q**i*) can be maximized separately. We start with *q**i* = ∣*W**i*+∣/(∣*W**i*+∣ + ∣*W**i*−∣) and perform alternating minimization on with respect to $\\q$ and $\\p$ iteratively until convergence, where *W**i*+ = {*j* ∈ *W**i* : *A**i**j* = 1}, *W**i*− = {*j* ∈ *W**i* : *A**i**j* = − 1}, and *W**i* is the set of workers assigned to task *i*.\nIn Figure [fig:comp], we compare our algorithm with alternating minimization and majority voting on simulated data and real data. The first plot is generated under the same settings as the first plot of Figure [fig:fig0] except that here we use *n* = *m* = 300 and $\\sp = 0.2$. It shows that Algorithm [algo:mp] and alternating minimization performs almost the same after the spectral barrier, while the proposed Algorithm [algo:mp] fails below the spectral barrier as expected from the spectral analysis of Section [sec:spectral]. For the figure on the left, we choose $\\sp = 0.2, \\sq = 1.4/3$. From the analyses in Section [sec:spectral], we predict the spectral barrier to be at ℓ = 11. In the second plot, we compare all the three algorithms on real data collected from Amazon Mechanical Turk in. This dataset considers binary classification tasks for comparing closeness in human perception of colors; three colors are shown in each task and the worker is asked to indicate ``whether the first color is more similar to the second color or the third color.\" This is asked on 50 of such color comparison tasks and 28 workers are recruited to complete all the tasks. We take the ground truth according to which color is closer to the first color in pairwise distances in the *Lab color* space. The second plot shows probability of error of the three algorithms when number of queries per task ℓ = Γ/*m* is varied. We generated responses for different values of Γ/*m* by uniformly sub-sampling. The alternating minimization and iterative algorithm perform similarly. However, for very small Γ, alternating minimization outperforms the iterative algorithm.\n (-160,117) (-141,-7 )\n (-160,117) (-141,-7 )\n[fig:comp]\nProofs\n======\nIn this section, we provide the proofs of the main technical results.\nProof of Theorem [thm:lbadapt]\n------------------------------\nIn this section, we first prove a slightly stronger result in Lemma [lem:lbadaptpre] and prove Theorem [thm:lbadapt] as a corollary. Lemma [lem:lbadaptpre] is stronger as it is an *adaptive* lower bound that holds for *all* discrete prior distribution $\\cQ$. The lower bound in Equation is adaptive in the sense that it automatically adjusts for any given $\\cQ$ as shown in its explicit dependence in $\\delta\\_{\\rm min}$ and *λ*. On the other hand, Theorem [thm:lbadapt] only has to hold for *one* worst-case prior distribution $\\cQ$.\nLet $\\cQ\\_{\\lambda,\\delta\\_{\\rm min}}$ be the set of all discrete prior distributions on *q**i* such that the collective task difficulty is *λ*, and the minimum probability mass in it is *δ*min, i.e.\n \n$$\\begin{aligned}\n\\label{eq:Q\\_lambda\\_delta}\n\\cQ\\_{\\lambda,\\delta\\_{\\rm min}} \\equiv \\left\\{{\\rm discrete\\;} \\cQ \\, \\Big| \\, \\Big( \\E\\_{\\cQ}\\left[\\frac{1}{(2q\\_a-1)^2}\\right]\\Big)^{-1} = \\lambda \\;,\\; \\min\\_{\\lambda\\_a\\in {\\rm supp}(\\lt\\cQ)} \\lt\\cQ(\\lambda\\_a) =\\delta\\_{\\rm min}\\right\\}\\,,\\end{aligned}$$ \nwhere $\\lt\\cQ$ is the induced distribution on *λ**a*’s. We let TΓ be the set of all task assignment schemes that make at most Γ queries to the crowd in expectation. We prove a lower bound on the standard minimax error rate: the error that is achieved by the best inference algorithm $\\t$ using the best adaptive task assignment scheme *τ* ∈ TΓ under a worst-case worker parameter distribution $\\F \\in \\F\\_{\\sigma^2}$ and any task parameter distribution $\\cQ \\in \\cQ\\_{\\lambda,\\delta\\_{\\rm min}}$. Note that instead of maximizing over $\\cQ \\in \\cQ\\_{\\lambda,\\delta\\_{\\rm min}}$, our result holds for all discrete $\\cQ\\in \\cQ\\_{\\lambda,\\delta\\_{\\rm min}}$. A proof of this lemma is provided in the following section.\n[lem:lbadaptpre] For *σ*2 < 1, for any discrete $\\cQ \\in \\cQ\\_{\\lambda,\\delta\\_{\\rm min}}$, there exists a positive constant *C*ʹ such that the average probability of error is lower bounded by\n \n$$\\begin{aligned}\n\\min\\_{\\tau \\in {\\mathscr{T}}\\_\\Gamma, \\t}\\;\\; \\;\\;\\max\\_{\\F \\in \\F\\_{\\sp} } \\;\\;\\;\\;\\frac{1}{m} \\sum\\_{i=1}^m \\P[t\\_i \\neq \\t\\_i] \n\\;\\; \\;\\;& \\geq & \\;\\; \\;\\; \\frac{1}{2} \\,\\delta\\_{\\rm min} e^{-C' \\frac{\\Gamma \\lambda \\sp}{m} } \\;,\n\\label{eq:lb\\_lem\\_adapt}\\end{aligned}$$ \nwhere *m* is the number of tasks, Γ is the expected budget allowed in TΓ, *λ* is the collective difficulty of the tasks from a prior distribution $\\cQ \\in \\cQ\\_{\\lambda,\\delta\\_{\\rm min}}$ defined in, *σ*2 is the collective reliability of the crowd from a prior distribution $\\cP$ defined in, and $\\delta\\_{\\rm min}$ is defined in.\nTheorem [thm:lbadapt] follows immediately from Lemma [lem:lbadaptpre] as it considers the worst-case $\\cQ \\in \\cQ\\_{\\lambda}$ whereas the Lemma is proved for any discrete $\\cQ \\in \\cQ\\_{\\lambda,\\delta\\_{\\rm min}}$. For any given *λ*, there exists a discrete distribution $\\cQ \\in \\cQ\\_{\\lambda,\\delta\\_{\\rm min}}$, namely a distribution that is supported at two points $q = (1 \\pm \\sqrt{\\lambda})/2$ with equal probability mass of 1/2. Such a distribution has *δ*min = 1/2 and therefore the Theorem [thm:lbadapt] follows.\nProof of Lemma [lem:lbadaptpre]\n-------------------------------\nLet *W**i* ⊆ [*n*] denote the (random) set of workers assigned to task *i* in the end, when *n* (random) number of workers have provided their responses. For a task assignment scheme *τ*, we let\n \n$$\\begin{aligned}\n \\ell\\_{i,q\\_i}^{(\\tau)}(\\cQ,\\cP) \\; \\equiv\\; \\E[ \\, |W\\_i| | q\\_i \\, ] \\;, \\end{aligned}$$ \ndenote the conditional expectation of number of workers assigned to a task *i* conditioned on its quality *q**i*. Let\n \n$$\\begin{aligned}\n {\\mathscr{T}}\\_{\\ell\\_{i,q\\_i}} \\;\\equiv\\; \\left\\{ \\tau: \\ell\\_{i,q\\_i}^{(\\tau)}=\\ell\\_{i,q\\_i}\\right\\}\\;, \\end{aligned}$$ \ndenote the set of all task assignment schemes that in expectation assign ℓ*i*, *q**i* workers to the *i*-th task conditioned on its quality *q**i*. Further, let\n \n$$\\begin{aligned}\n {\\mathscr{T}}\\_{\\{{\\ell\\_{i,q\\_i}}\\}\\_{i= 1}^m} \\;\\equiv\\; \\left\\{ \\tau: \\left({\\{\\ell\\_{i,q\\_i}\\}}\\_{i=1}^m\\right)^{(\\tau)} ={\\{\\ell\\_{i,q\\_i}\\}}\\_{i=1}^m\\right\\}\\;,\\end{aligned}$$ \ndenote the set of all task assignment schemes that in expectation assign ℓ*i*, *q**i* workers to each task *i* ∈ [*m*] conditioned on its quality *q**i*. The fundamental lower bound crucially relies on the following technical lemma, whose proof is provided in the following section.\n[lem:lbadapt] For any *σ*2 < 1, there exists a positive constant *C*ʹ and a prior distribution $\\cP^\\* \\in \\cP\\_{\\sigma^2}$ such that for each task *i* ∈ [*m*], for all $\\cQ$,\n \n$$\\begin{aligned}\n \\min\\_{\\tau \\in {\\mathscr{T}}\\_{\\ell\\_{i,q\\_i}}, \\t}\\;\\; \\P[t\\_i \\neq \\t\\_i | q\\_i] & \\geq & \\frac{1}{2} e^{-C'\\lambda\\_i \\sp \\, \\ell\\_{i,q\\_i} }\\,,\\end{aligned}$$ \nwhere *λ**i* = (2*q**i* − 1)2.\nThis proves a lower bound on *per task* probability of error that decays exponentially with exponent scaling as $\\lambda\\_i \\sigma^2\\ell\\_{i,\\q\\_i}$. The easier the task (*λ**i* = (2*q**i* − 1)2 large), the more reliable the workers are (*σ*2 large), and the more workers assigned to that task ($\\ell\\_{i,\\q\\_i}$ large), the smaller the achievable error. To get a lower bound on the minimax *average* probability of error, where the error probability is over the randomness in the latent variables from $(\\cQ,\\cP)$ and the randomness in the task assignment scheme *τ* and the responses *A*, we have,\n \n$$\\begin{aligned}\n&&\\min\\_{\\tau \\in {\\mathscr{T}}\\_\\Gamma, \\t}\\;\\; \\max\\_{ \\F \\in \\F\\_{\\sp} } \\frac{1}{m} \\sum\\_{i=1}^m \\P\\_{(\\cP,\\cQ,\\tau)} [t\\_i \\neq \\t\\_i] \\nonumber\\ \\\\\n&=& \\min\\_{\\tau \\in {\\mathscr{T}}\\_\\Gamma, \\t}\\;\\; \\max\\_{ \\F \\in \\F\\_{\\sp} } \\frac{1}{m} \\sum\\_{i=1}^m \\E\\_{ q\\_i\\sim \\cQ } \\big[ \\, \\P\\_{(\\cP,\\cQ,\\tau)}[t\\_i \\neq \\t\\_i | q\\_i] \\,\\big] \\label{eq:opt\\_2}\\\\\n&= & \\min\\_{\\ell\\_{i,\\q\\_i}: \\sum\\_{i \\in[m]} \\E\\_\\cQ [\\ell\\_{i,\\q\\_i}] \\leq \\Gamma} \\;\\; \\;\\;\n \\min\\_{\\tau \\in {\\mathscr{T}}\\_{\\{\\ell\\_{i,\\q\\_i}\\}\\_{i=1}^m}, \\t}\\;\\; \\;\\;\\max\\_{\\F \\in \\F\\_{\\sp} } \\;\\; \\frac{1}{m} \\sum\\_{i=1}^m \\E\\_{q\\_i \\sim \\cQ } \\big[ \\, \\P\\_{(\\cP,\\cQ,\\tau)}[t\\_i \\neq \\t\\_i | q\\_i ] \\, \\big] \\label{eq:opt\\_3}\\\\\n&\\geq& \\min\\_{\\ell\\_{i,\\q\\_i}: \\sum\\_{i \\in[m]} \\E\\_\\cQ [\\ell\\_{i,\\q\\_i}]= \\Gamma} \\;\\; \\;\\; \n \\min\\_{\\tau \\in {\\mathscr{T}}\\_{\\{\\ell\\_{i,\\q\\_i}\\}\\_{i=1}^m}, \\t}\\;\\; \\;\\;\n \\frac{1}{m} \\sum\\_{i=1}^m \\E\\_{q\\_i \\sim \\cQ } \\big[ \\, \\P\\_{(\\cP^\\*,\\cQ,\\tau)}[t\\_i \\neq \\t\\_i | q\\_i ] \\, \\big] \\label{eq:opt\\_3\\_1}\\\\\n&\\geq& \\min\\_{\\ell\\_{i,\\q\\_i}: \\sum\\_{i \\in[m]} \\E\\_\\cQ [\\ell\\_{i,\\q\\_i}] = \\Gamma} \\;\\; \\;\\; \n \\frac{1}{m} \\sum\\_{i=1}^m \\left\\{ \\; \\E\\_{q\\_i \\sim \\cQ } \\left[ \\, \\min\\_{\\tau \\in {\\mathscr{T}}\\_{ \\ell\\_{i,\\q\\_i} }, \\t}\\;\\; \\;\\; \\P\\_{(\\cP^\\*,\\cQ,\\tau)}[t\\_i \\neq \\t\\_i | q\\_i ] \\, \\right] \\right\\} \\label{eq:opt\\_3\\_2}\\\\\n&\\geq& \\min\\_{\\ell\\_{i,\\q\\_i}: \\sum\\_{i \\in[m]} \\E\\_\\cQ [\\ell\\_{i,\\q\\_i}] = \\Gamma} \\;\\; \\;\\;\n \\frac{1}{m} \\sum\\_{i=1}^m \\E\\_{q\\_i \\sim \\cQ } \\left[ \\, \\frac{1}{2} e^{-C'\\lambda\\_i \\sp \\ell\\_{i,q\\_i}} \\,\\right] \\label{eq:opt\\_4}\\\\\n&=&\\min\\_{\\ell\\_a: \\sum\\_{a\\in[T]} \\delta\\_a \\ell\\_a = \\Gamma/m} \\;\\;\\; \\sum\\_{a = 1}^{T} \\frac{1}{2} \\delta\\_a e^{-C' \\lambda\\_a \\sp \\ell\\_a }\\, \\label{eq:lb\\_ad1}\\\\\n&= &\\frac{1}{2} e^{-C' \\frac{\\Gamma \\lambda \\sp }{m}} \\bigg(\\sum\\_{a = 1}^T \\delta\\_a e^{-\\lambda \\sum\\_{a' \\neq a} (\\delta\\_{a'}/\\lambda\\_{a'}) \\log(\\lambda\\_{a}/\\lambda\\_{a'})} \\bigg) \\label{eq:opt\\_soln}\\\\\n&\\geq& \\frac{1}{2} \\delta\\_{\\min} e^{-C' \\frac{\\Gamma \\lambda \\sp }{m}} \\,,\\end{aligned}$$ \nwhere *T* is the support size of the discrete distribution $\\lt\\cQ$. follows from the fact that fixing a prior $\\cP^\\*$ provides a lower bound. follows from the fact that exchanging min and sum (and also expectation which is essentially a weighted sum) provides a lower bound. uses Lemma [lem:lbadapt]. follows by change of notations in. follows by solving the optimization problem in where the optimal choice of ℓ*a* is,\n \n$$\\begin{aligned}\n\\label{eq:opt\\_ell}\n\\ell\\_a & \\;\\;=\\;\\; & \\frac{\\lambda}{\\lambda\\_a} \\frac{\\Gamma}{m} \\;+\\; \\frac{\\lambda}{\\lambda\\_a C' \\sp} \\bigg (\\sum\\_{a' \\neq a} \\frac{\\delta\\_{a'}}{\\lambda\\_{a'}} \\log\\Big(\\frac{\\lambda\\_{a}}{\\lambda\\_{a'}}\\Big) \\bigg)\\,.\\end{aligned}$$ \nThe summand in does not depend upon the budget Γ/*m*, and it is lower bounded by *δ*min > 0. This follows from the fact that in the summand the term corresponding to *a* such that *λ**a* = *λ*min is lower bounded one *δ*min.\nProof of Lemma [lem:lbadapt]\n----------------------------\nWe will show that there exists a family of worker reliability distributions $\\F^\\* \\in \\F\\_{\\sp}$ such that for any adaptive task assignment scheme that assigns $\\E[|W\\_i||q\\_i]$ workers in expectation to a task *i* conditioned on its difficulty *q**i*, the conditional probability of error of task *i* conditioned on *q**i* is lower bounded by $\\exp{(-C'\\lambda\\_i\\sigma^2 \\E[|W\\_i||q\\_i])}$. We define the following family of distributions according to the spammer-hammer model with imperfect hammers. We assume that *σ*2 < *a*2 and\n \n$$\\begin{aligned}\n p\\_j = \\left\\{ \n \\begin{array}{rl}\n 1/2, & \\text{w.p.} \\;\\;\\; 1-\\sigma^2/a^2, \\\\\n 1/2(1+a), & \\text{w.p.} \\;\\;\\; \\sigma^2/a^2.\n \\end{array}\n \\right.\\;,\\end{aligned}$$ \nsuch that *E*[(2*p**j* − 1)2] = *σ*2. Let $\\E[W\\_i|q\\_i]$ denote the expected number of workers conditioned on the task difficulty *q**i*, that the adaptive task assignment scheme assigns to the task *i*. We consider a labeling algorithm that has access to an oracle that knows reliability of every worker (all the *p**j*’s). Focusing on a single task *i*, since we know who the spammers are and spammers give no information about the task, we only need the responses from the reliable workers in order to make an optimal estimate. Let E*i* denote the conditional error probability of the optimal estimate conditioned on the realizations of the answers {*A**i**j*}*j* ∈ *W**i* and the worker reliability {*p**j*}*j* ∈ *W**i*. We have $\\E[\\mathscr{E}\\_i | q\\_i] \\equiv \\P[t\\_i \\neq \\t\\_i | q\\_i]$. The following lower bound on the error only depends on the number of reliable workers, which we denote by ℓ*i*.\nWithout loss of generality, let *t**i* = + 1. Then, if all the reliable workers agreed on “–” answers, the maximum likelihood estimation would be “–” for this task, and vice-versa. For a fixed number of ℓ*i* responses, the probability of error is minimum when all the workers agreed. Therefore, since probability that a worker gives “–” answer is *p**j*(1 − *q**i*) + *q**i*(1 − *p**j*) = (1 − *a*(2*q**i* − 1))/2 from, we have,\n \n$$\\begin{aligned}\n\\E[\\mathscr{E}\\_i |q\\_i, \\ell\\_i] \\geq \\E[\\mathscr{E}\\_i | \\text{all } \\ell\\_i \\text{ reliable workers agreed, } q\\_i, \\ell\\_i ] \\geq \\frac{1}{2}\\bigg(\\frac{1 - a(2q\\_i-1))}{2}\\bigg)^{\\ell\\_i}\\,,\\end{aligned}$$ \nfor any realizations of {*A**i**j*} and {*p**j*}. By convexity and Jensen’s inequality, it follows that\n \n$$\\begin{aligned}\n\\E[\\ell\\_i |q\\_i] \\geq \\frac{\\log(2\\E[\\mathscr{E}\\_i | q\\_i])}{\\log((1 - a(2q\\_i-1))/2)}\\,.\\label{eq:adapt\\_lb0}\\end{aligned}$$ \nWhen we recruit ∣*W**i*∣ workers, using Doob’s Optional-Stopping Theorem, conditional expectation of reliable number of workers is\n \n$$\\begin{aligned}\n\\E[\\ell\\_i|q\\_i] = (\\sigma^2/a^2)\\E[|W\\_i| \\,| q\\_i]\\,. \\label{eq:adapt\\_lb1}\\end{aligned}$$ \nTherefore, from and, we get\n \n$$\\begin{aligned}\n\\E[|W\\_i|\\,| q\\_i] \\geq \\frac{1}{\\sigma^2} \\frac{a^2}{\\log((1- a(2q\\_i-1))/2)} \\log (2\\E[\\mathscr{E}\\_i | q\\_i])\\,.\\end{aligned}$$ \nMaximizing over all choices of *a* ∈ (0, 1), we get,\n \n$$\\begin{aligned}\n\\label{eq:adapt\\_lb2}\n\\E[|W\\_i|\\,| q\\_i] \\geq \\frac{-0.27}{\\sigma^2(2q\\_i-1)^2} \\log (2\\E[\\mathscr{E}\\_i | q\\_i])\\,,\\end{aligned}$$ \nfor *a* = 0.8/(2*q**i* − 1) which as per our assumption of *σ*2 < *a*2 requires that *σ*2(2*q**i* − 1)2 < 0.64. By changing the constant in the bound, we can ensure that the bound holds for any value of *σ*2 and *q**i*. Theorem readily follows from Equation.\nTo verify Equation [eq:adaptlb1]: Define *X**i*, *k* for *k* ∈ [∣*W**i*∣] to be a Bernoulli random variable, for a fixed *i* ∈ [*m*] and fixed task difficulty *q**i*. Let *X**i*, *k* take value one when the *k*-th recruited worker for task *i* is reliable and zero otherwise. Observe that the number of reliable workers is ℓ*i* = ∑*k* = 1∣*W**i*∣*X**i*, *k*. From the spammer-hammer model that we have considered, $\\E[X\\_{k,i} - \\sigma^2/a^2] = 0$. Define *Z**i*, *k* ≡ ∑*k*ʹ = 1*k*(*X**i*, *k*ʹ − *σ*2/*a*2) for *k* ∈ [∣*W**i*∣]. Since {(*X**i*, *k* − *σ*2/*a*2)}*k* ∈ [∣*W**i*∣] are mean zero i.i.d. random variables, {*Z**i*, *k*}*k* ∈ [∣*W**i*∣] is a martingale with respect to the filtration F*i*, *k* = *σ*(*X**i*, 1, *X**i*, 2, ⋯, *X**i*, *k*). Further, it is easy to check that the random variable ∣*W**i*∣ for a fixed *q**i* is a stopping time with respect to the same filtration F*i*, *k* and is almost surely bounded assuming the budget is finite. Therefore using Doob’s Optional-Stopping Theorem, we have $\\E[Z\\_{i,|W\\_i|}] = \\E[Z\\_{i,1}] = 0$. That is we have, $\\E[X\\_{i,1}+X\\_{i,2}+\\cdots+ X\\_{i,|W\\_i|}] = (\\sigma^2/a^2)\\E[|W\\_i|]$. Since this is true for any fixed task difficulty *q**i*, we get Equation.\nProof of Theorem [thm:ubadapt]\n------------------------------\nFirst we show that the messages returned by Algorithm [algo:mp] are normally distributed and identify their conditional means and conditional variances in the following lemma. Assume in a sub-round (*t*, *u*), *t* ∈ [*T*], *u* ∈ [*s**t*], the number of tasks remaining unclassified are *m**t*, *u* and the task assignment is performed according to an (ℓ*t*, *r**t*)-regular random graph. To simplify the notation, let $\\hat{\\ell\\_t} \\equiv \\ell\\_t-1$, $\\hat{r\\_t} \\equiv r\\_t - 1$, and recall $\\mu = \\E[2p\\_j-1]$, $\\sigma^2 = \\E\\_\\cP[(2p\\_j-1)^2]$. Note that *μ*, *σ*2 remain same in each round. Let *ρ**t*, *u*2 = (1/∣*M*∣)∑*i* ∈ [*M*]*λ**i* be the exact value of average task difficulty of the tasks present in the (*t*, *u*) sub-round. When ℓ*t* and *r**t* are increasing with the problem size, the messages converge to a Gaussian distribution due to the central limit theorem. We provide a proof of this lemma in Section [sec:gaussian].\n[lem:gaussian] Suppose for tasks are assigned according to (ℓ*t*, *r**t*)-regular random graphs. In the limit *m**t*, *u* → ∞, if *μ* > 0, then after number of iterations in Algorithm [algo:mp], the conditional mean *μ**q*(*k*) and the conditional variance $\\big( \\rho^{(k)}\\_q \\big)^2$ conditioned on the task difficulty *q* of the message *x**i* corresponding to the task *i* returned by the Algorithm [algo:mp] are\n \n$$\\begin{aligned}\n \\label{eq:abcd}\n &&\\mu^{(k)}\\_q = (2q-1) \\mu \\ell\\_t (\\l\\_t\\r\\_t\\rho\\_{t,u}^2\\sp)^{(k-1)} \\;, \\nonumber\\\\\n &&\\big( \\rho^{(k)}\\_q \\big)^2 \\nonumber\\\\\n &&= \\mu^2 \\ell\\_t ( \\l\\_t\\r\\_t\\rho\\_{t,u}^2\\sp)^{2(k-1)} \n\\bigg( \\rho\\_{t,u}^2-(2q-1)^2 + \\frac{\\rho\\_{t,u}^2\\l\\_t(1-\\rho\\_{t,u}^2\\sp)(1+\\r\\_t\\rho\\_{t,u}^2\\sp)\\big(1-(\\l\\_t\\r\\_t\\rho\\_{t,u}^4\\sigma^4)^{-(k-1)}\\big)}{\\l\\_t\\r\\_t\\rho\\_{t,u}^4\\sigma^4-1}\\bigg) \\nonumber \\\\\n&&+ \\ell\\_t (2-\\mu^2\\rho\\_{t,u}^2)(\\l\\_t\\r\\_t)^{k-1}\\,. \n \\end{aligned}$$ \nWe will show in that the probability of misclassification for any task in sub-round (*t*, *u*) in Algorithm [algo:adapt] is upper bounded by *e*− (*C**δ*/4)(Γ/*m*)*λ**σ*2. Since, there are at most rounds, using union bound we get the desired probability of error. In, we show that the expected total number of worker assignments across all rounds is at most Γ.\nLet’s consider any task *i* ∈ [*m*] having difficulty *λ**i*. Without loss of generality assume that *t**i* = 1 that is *q**i* > 1/2. Let us assume that the task *i* gets classified in the (*t*, *u*) sub-round, *t* ∈ [*T*], *u* ∈ [*s**t*]. That is the number of workers assigned to the task *i* when it gets classified is $\\ell\\_t = C\\_{\\delta} (\\Gamma/m) (\\widehat\\lambda/\\lambda\\_t)$ and the threshold X*t*, *u* set in that round for classification is $\\mathcal{X}\\_{t,u} = \\sqrt{\\lambda\\_t} \\mu \\ell\\_t \\big((\\ell\\_t-1)(r\\_t-1)\\rho^2\\_{t,u}\\sigma^2 \\big)^{k\\_t-1}$. From Lemma [lem:gaussian] the message *x**i* returned by Algorithm [algo:mp] is Gaussian with conditional mean and conditional variance as given in. Therefore in the limit of *m*, the probability of error in task *i* is\n \n$$\\begin{aligned}\n \\lim\\_{m \\rightarrow \\infty} \\P\\big[\\t\\_i \\neq t\\_i | q\\_i\\big] &=& \\lim\\_{m \\rightarrow \\infty} \\P\\big[x\\_i < -\\mathcal{X}\\_{t,u} | q\\_i\\big] \\nonumber\\\\\n &=& \\lim\\_{m \\rightarrow \\infty} Q\\Big(\\frac{\\mu^{(k)}\\_{q\\_i} + \\mathcal{X}\\_{t,u}}{\\rho^{(k)}\\_{q\\_i}}\\Big) \\label{eq:adapt1}\\\\\n &\\leq & \\lim\\_{m \\rightarrow \\infty} \\exp\\Big(\\frac{-(\\mu^{(k)}\\_{q\\_i} + \\mathcal{X}\\_{t,u})^2}{2(\\rho^{(k)}\\_{q\\_i})^2}\\Big) \\label{eq:adapt2}\\\\\n&=& \\exp\\Big(\\frac{-((2q\\_i-1)+ \\sqrt{\\lambda\\_t})^2\\ell\\_t\\sp}{2(1 - (2q\\_i-1)^2\\sp)}\\Big) \\label{eq:adapt3} \\\\\n&\\leq & \\exp\\Big(\\frac{-\\lambda\\_t\\ell\\_t\\sp}{2}\\Big) \\nonumber\\\\\n&=& \\exp\\Big(\\frac{-C\\_{\\delta} (\\Gamma/m) \\widehat\\lambda \\sp}{2}\\Big) \\label{eq:adapt5}\\\\\n&\\leq & \\exp\\Big(\\frac{-C\\_{\\delta} (\\Gamma/m) \\lambda \\sp}{4}\\Big)\\,, \\label{eq:adapt6}\\end{aligned}$$ \nwhere *Q*( ⋅ ) in is the tail probability of a standard Gaussian distribution, and uses the Chernoff bound. follows from substituting conditional mean and conditional variance from Equation, and using ℓ*t* = Θ(log*m**t*, *u*), $k = \\Theta(\\sqrt{\\log m\\_{t,u}})$ where *m* grows to infinity. uses $\\ell\\_t = C\\_{\\delta} (\\Gamma/m) (\\widehat\\lambda/\\lambda\\_t)$, our choice of ℓ*t* in Algorithm [algo:adapt] line 4. uses the fact that for the quantized distribution {*λ**a*, *δ**a*}*a* ∈ [*T*], $\\widehat \\lambda = \\big(\\sum\\_{a \\in [T]} (\\delta\\_a/\\lambda\\_a)\\big)^{-1} \\geq \\lambda/2$. We have established that our approach guarantees the desired level of accuracy. We are left to show that we use at most Γ assignments in expectation.\nWe upper bound the expected total number of workers used for tasks of quantized difficulty level *λ**a*’s for each 1 ≤ *a* ≤ *T*. Recall that our adaptive algorithm runs in *T* rounds indexed by *t*, where each round *t* further runs *s**t* sub-rounds. The total expected number of workers assigned to *δ**a* fraction of tasks of quantized difficulty *λ**a* in *t* = 1 to *t* = *a* − 1 rounds is upper bounded by *m**δ**a*∑*t* = 1*a* − 1*s**t*ℓ*t*. The upper bound assumes the worst-case (in terms of the budget) that these tasks do not get classified in any of these rounds as the threshold X set in these rounds is more than absolute value of the conditional mean message *x* of these tasks.\nNext, in *s**t* = *a* sub-rounds the threshold X is set less than or equal to the absolute value of the conditional mean message *x* of these tasks, i.e. X ≤ ∣*μ**q**a*(*k*)∣ for (2*q**a* − 1)2 = *λ**a*. Therefore, in each of these *s**a* sub-rounds, probability of classification of these tasks is at least 1/2. That is the expected total number of workers assigned to these tasks in *s**a* sub-rounds is upper bounded by 2*m**δ**a*ℓ*a*. Further, *s**a* is chosen such that the fraction of these tasks remaining un-classified at the end of *s**a* sub-rounds is at most same as the fraction of the tasks having difficulty *λ**a* + 1. That is to get the upper bound, we can assume that the fraction of *λ**a* + 1 difficulty tasks at the start of *s**a* + 1 sub-rounds is 2*δ**a* + 1, and the fraction of *λ**a* difficulty tasks at the start of *s**a* + 1 sub-rounds is zero. Further, recall that we have set *s**T* = 1 as in this round our threshold X is equal to zero. Therefore, we have the following upper bound on the expected total number of worker assignments.\n \n$$\\begin{aligned}\n\\sum\\_{i = 1}^m \\E[|W\\_i|] &\\leq & 2m\\delta\\_1\\ell\\_1 + \\sum\\_{a=2}^{T-1} 4 m\\delta\\_a \\ell\\_a + 2m\\delta\\_{T}\\ell\\_{T} + \\sum\\_{a =2}^{T}\\Big(m\\delta\\_a \\sum\\_{b=1}^{a-1} s\\_b\\ell\\_b \\Big) \\nonumber \\\\\n& \\leq & \\sum\\_{a=1}^{T} 4 m\\delta\\_a \\ell\\_a + s\\_{\\max} \\sum\\_{a=1}^{T} m\\delta\\_a \\ell\\_a \\label{eq:adapt7} \\\\\n & \\leq & (4 + {\\left \\lceil{\\log(2\\delta\\_{\\max}/\\delta\\_{\\min})} \\right \\rceil}) \\sum\\_{a=1}^{T}m \\delta\\_a \\ell\\_a \\label{eq:adapt8} \\\\\n& \\leq & (4 + {\\left \\lceil{\\log(2\\delta\\_{\\max}/\\delta\\_{\\min})} \\right \\rceil}) \\Gamma C\\_{\\delta} \\label{eq:adapt9} \\\\\n&=& \\Gamma\\,,\\label{eq:adapt10}\\end{aligned}$$ \nEquation uses the fact that $\\ell\\_t = ( C\\_{\\delta} (\\Gamma/m)(\\widehat\\lambda/\\lambda\\_t)$ where *λ**t*’s are separated apart by at least a ratio of 2 (recall the quantized distribution), therefore ∑*t* = 1*a* − 1ℓ*t* ≤ ℓ*a*. Equation follows from the choice of *s**t*’s in the algorithm. Equation follows from using $\\ell\\_t = ( C\\_{\\delta} (\\Gamma/m)(\\widehat\\lambda/\\lambda\\_t)$ and *λ* = (∑*a* ∈ [*T*](*δ**a*/*λ**a*))− 1, and Equation uses *C**δ* = (4 + ⌈log(2*δ*max/*δ*min)⌉)− 1.\nProof of Lemma [lem:gaussian]\n-----------------------------\nWe omit subscripts *t* and (*t*, *u*) from all the quantities for simplicity of notations. Also, we use notation ℓ, average budget per task, ℓ = Γ/*m*. We will prove it for a randomly chosen task $\\i$, and all the analyses naturally holds for a specific *i*, when conditioned on *q**i*. Let *n* be the number of workers, that is *n* = (*m**r*)/ℓ. In our algorithm, we perform task assignment on a random bipartite graph $\\G([m]\\cup[n],E)$ constructed according to the configuration model. Let $\\G\\_{i,k}$ denote a subgraph of $\\G([m]\\cup[n],E)$ that includes all the nodes that are within *k* distance from the the ``root\" *i*. If we run our inference algorithm for one run to estimate $\\t\\_i$, we only use the responses provided by the workers who were assigned to task *i*. That is we are running inference algorithm only on the local neighborhood graph $\\G\\_{i,1}$. Similarly, when we run our algorithm for *k* iterations to estimate $\\t\\_i$, we perform inference only on the local subgraph $\\G\\_{i,2k-1}$. Since we update both task and worker messages at each iteration, the local subgraph grows by distance two at each iteration. We use a result from to show that the local neighborhood of a randomly chosen task node $\\i$ is a tree with high probability. Therefore, assuming that the graph is locally tree like with high probability, we can apply a technique known as *density evolution* to estimate the conditional mean and conditional variance. The next lemma shows that the local subgraph converges to a tree in probability, in the limit *m* → ∞ for the specified choice of ℓ, *r* and *k*.\n[Lemma 5 from ] [lem:lemma1] For a random (ℓ, *r*)-regular bipartite graph generated according to the configuration model,\n \n$$\\begin{aligned}\n\\label{eq:eq2}\n\\P\\big[\\G\\_{\\i,2k-1} \\;{\\text{\\rm{is not a tree}}}\\big] \\leq \\big((\\ell-1)(r-1)\\big)^{2k-2} \\frac{3\\ell r}{m}.\\end{aligned}$$ \n**Density Evolution.** Let {*x**i* → *j*(*k*)}(*i*, *j*) ∈ *E* and {*y**j* → *i*(*k*)}(*i*, *j*) ∈ *E* denote the messages at the *k*-th iteration of the algorithm. For an edge (*i*, *j*) chosen uniformly at random, let $\\x^{(k)}\\_q$ denote the random variable corresponding to the message *x**i* → *j*(*k*) conditioned on the *i*-th task’s difficulty being *q*. Similarly, let $\\y\\_p^{(k)}$ denote the random variable corresponding to the message *y**j* → *i*(*k*) conditioned on the *j*-th worker’s quality being *p*.\nAt the first iteration, the task messages are updated according to $x\\_{i \\rightarrow j}^{(1)} = \\sum\\_{j' \\in \\partial i \\setminus j } \\A\\_{ij'}y\\_{j' \\rightarrow i }^{(0)}$. Since we initialize the worker messages {*y**j* → *i*(0)}(*i*, *j*) ∈ *E* with independent Gaussian random variables with mean and variance both one, if we know the distribution of $\\A\\_{ij'}$’s, then we have the distribution of *x**i* → *j*(1). Since, we are assuming that the local subgraph is tree-like, all *x**i* → *j*(1) for $i \\in \\G\\_{\\i,2k-1}$ for any randomly chosen node $\\i$ are independent. Further, because of the symmetry in the construction of the random graph $\\G$ all messages *x**i* → *j*(1)’s are identically distributed. Precisely, *x**i* → *j*(1) are distributed according to $\\x\\_q^{(1)}$ defined in Equation."},"data_id":{"kind":"string","value":"arxiv_0182382"}}},{"rowIdx":45593,"cells":{"content":{"kind":"string","value":"**Supplementary Materials: Resource requirements for efficient quantum communication using all-photonic graph states generated from a few matter qubits**\n=========================================================================================================================================================\nQuantum communication technologies show great promise for applications ranging from the secure transmission of secret messages to distributed quantum computing. Due to fiber losses, long-distance quantum communication requires the use of quantum repeaters, for which there exist quantum memory-based schemes and all-photonic schemes. While all-photonic approaches based on graph states generated from linear optics avoid coherence time issues associated with memories, they outperform repeater-less protocols only at the expense of a prohibitively large overhead in resources. Here, we consider using matter qubits to produce the photonic graph states and analyze in detail the trade-off between resources and performance, as characterized by the achievable secret key rate per matter qubit. We show that fast two-qubit entangling gates between matter qubits and high photon collection and detection efficiencies are the main ingredients needed for the all-photonic protocol to outperform both repeater-less and memory-based schemes.\nThe ability to share entangled states over long distances is a major milestone for the realization of a fully-functional quantum internet . Beyond secure communications via quantum key distribution (QKD) , the implementation of such a quantum internet would also have various applications ranging from distributed quantum computing , secure access to a remote quantum computer , accurate clock synchronization , and improved telescope observations .\nHowever, enabling world-wide quantum communication requires addressing the major problem of photonic losses, which significantly reduces the range of quantum information transfer. Even though direct amplification of a quantum state is made impossible due to the no-cloning theorem , this exponential photon loss can still be overcome through the realization of quantum repeaters (QR) .\n [figboth]\nSeveral QR approaches have been proposed and can be divided into two main categories depending on the method used to propagate quantum information between two adjacent repeater nodes . The first approach proposed, referred to here as a “memory-based” approach, relies on heralded-entanglement generation between two quantum memories (QM) situated at adjacent repeater nodes. Examples of memory-based schemes are depicted schematically in Figs. [figboth](a),(b). The heralded entanglement generally succeeds when a detector situated at an intermediate node measures photons emitted by the QMs . A classical signal carrying the outcome of an entanglement generation attempt must be sent back to the QMs, thus limiting the repetition rate of these protocols. When successful, entanglement swapping transfers the entanglement through these repeater nodes.\nThe second category of repeater relies on logically-encoded multi-photon states , resistant both to photonic losses and errors, to transfer quantum information across a network. One particularly promising approach of this type was put forward in Ref. , which proposed an all-optical QR protocol based on repeater graph states (RGS). An example of an RGS with logical encoding is shown in Fig. [figboth](c), while the corresponding communication protocol is summarized in Fig. [figboth](d) and will be detailed later. Ref. also proposed a method to construct these states probabilistically using single-photon sources, linear optics, and detectors . However, this generation procedure requires about 106 single-photon sources per repeater node to just barely outperform direct fiber transmission . These findings suggest that the RGS protocol may be well beyond the reach of current and near-future technological capabilities.\nHowever, it remains unclear whether the RGS protocol can become more feasible if alternative methods to create the states are used. Ref. proposed a deterministic generation procedure to produce a linear graph state using one quantum emitter which emits spin-entangled photons ; this procedure was later demonstrated experimentally . Ref. showed theoretically that emitters that undergo entangling gates can also be used to generate two-dimensional cluster states. Refs. built on this protocol and demonstrated that entanglement between emitters can be harnessed for the generation of more complex photonic graph states. These ideas have been extended further to general prescriptions and to protocols tailored to specific physical systems . Ref. introduced a protocol for producing an arbitrary-sized RGS using only a few matter qubits, thus significantly decreasing the resource overhead required for RGS generation, making it deterministic in principle. Using this generation technique, the RGS protocol only necessitates a few-qubit processor at each repeater node, which would ease its practical implementation compared to other error-correction-based proposals that generally require several hundreds of qubits per repeater node (with the notable exception of Ref. which uses techniques introduced in Ref. and requires deterministic spin-photon Bell measurements). Although the RGS protocol with deterministic state generation seems promising, a systematic and detailed evaluation of its performance and resource requirements has not been carried out.\nIn this paper, we compare the resource-efficiency—characterized by the achievable secret key rate per matter qubit—of this protocol to direct fiber transmission and to QR schemes based on memories and heralded entanglement generation. We first show that the rate per matter qubit has a fundamental upper bound in the case of memory-based QRs. We then review the RGS protocol and how RGSs can be generated using a few matter qubits. We evaluate the performance of this scheme, show that its rate per matter qubit does not have a theoretical upper-bound, and find the conditions under which it outperforms both the repeater-less and the memory-based QR approaches. These conditions depend sensitively on the speed with which two-qubit gates between the matter qubits can be executed and on the collection and detection efficiencies of the photons emitted by these matter qubits.\nUpper bound on rate for memory-based repeater schemes\n=====================================================\nIn this section, we show that there is a theoretical upper bound, *R*max(*Q**M*), on the rate per matter qubit for protocols based on quantum memories and heralded entanglement generation. In such protocols, the total distance *L* between Alice and Bob is divided into smaller distances *L*0 by *N**Q**R* = *L*/*L*0 − 1 repeater nodes. Quantum memories at adjacent repeater nodes are entangled via a heralded entanglement procedure (see Fig. [figboth](a)). When a repeater node shares entanglement connections with its two adjacent nodes, entanglement swapping is performed on the two memory qubits within that node to create a direct entanglement connection between memories on the adjacent nodes. This procedure can be repeated until Alice and Bob share an entangled qubit pair.\nIt is clear that creating an entangled pair between Alice and Bob requires generating an entanglement connection between each adjacent pair of repeaters. This means that the overall protocol rate *R*(*Q**M*) is limited by the entanglement generation rate ⟨*T*ent⟩− 1 between two adjacent repeaters. Here, we use this fact to derive an upper bound on the rate per matter qubit for QR protocols that use quantum memories and heralded entanglement generation.\nTo determine an upper bound on ⟨*T*ent⟩− 1, we focus for concreteness on the protocol presented in Fig. [figboth](a), which uses two quantum memories per node. A quantum memory should emit a photon that is maximally entangled with one of its degrees of freedom. Two photons generated at adjacent repeater nodes arrive at the same measurement node situated halfway between the two repeaters, where they are measured in a Bell state basis. Because a photon Bell state measurement using only linear optics succeeds with probability at best 1/2 (without ancillary qubits or QND measurements ), the overall success probability of the distant heralded entanglement generation is *P*ent ≤ 1/2. It is worth mentioning that a method for achieving heralded entanglement generation with higher success probability has been proposed , but its efficacy is restricted to qubits separated by a short distance, so we exclude this from our analysis (see Supplementary Materials for more details). A classical signal must then inform the repeater nodes of the success or failure of the Bell state measurement. Because the distance from the measurement node to the repeater nodes is *L*0/2, this means that the overall heralded entanglement generation attempt takes total time *T*trial ≥ *L*0/*c*. This includes the time *L*0/(2*c*) for the single-photon transfer from the repeater to the measurement node and also the time *L*0/(2*c*) for the classical signaling in the opposite direction. Here, we are neglecting the time it takes to prepare and pump the quantum memories. Throughout this protocol, a QM can be maximally entangled with at most one other qubit (either a photon or another QM). Therefore, it cannot emit another spin-entangled photon before receiving the classical signal carrying the information about the success or failure of the Bell measurement, hence limiting the repetition rate of the protocol. To generate an entanglement connection between the two adjacent nodes, the procedure must be repeated on average *P*ent− 1 times. The entanglement generation rate is therefore ⟨*T*ent⟩− 1 = *P*ent/*T*trial < *c*/(2*L*0). We derived this result for a specific heralded entanglement generation protocol but it also holds for all known protocols (see Supplementary Materials).\n |p1.6cmp7.0cm| Notation & Definition \n*L* & Total distance between Alice and Bob. \n*L*0 & Distance between adjacent nodes. \n*N**Q**R* & Number of repeater nodes. \n2*m* & Number of arms of the RGS. \n*b⃗* & Branching vector of an error-correction tree (*b⃗* = (*b*0, *b*1, …, *b**n* − 1)). \n*T**C**Z* & *C**Z* gate time. \n*η**t*(*l*) & Transmission of a fiber of length *l* (*η**t*(*l*) = exp( − *l*/*L*att)). \n*η**c*, *η**d* & In-fiber collection and detection efficiencies of photons. \n*L*att, *c* & Attenuation distance of the fiber and speed of light in a fiber. ($L\\_{\\mathrm{att}} \\approx 20 \\kilo\\meter$) \n*t*att & Average time of flight of photons in the fiber (*t*att = *L*att/*c*). \n*T**R**G**S* & Generation time of an RGS. \n*R*, *R**m* & Rate and rate per matter qubit of the protocol. \n*ε* & Single-photon error rate. \n[table3]\nFrom these results, we can show that the rate per matter qubit (where the number of matter qubits is *N**m* = 2(*N**Q**R* + 1) = 2*L*/*L*0) has a theoretical upper bound, *R*max(*Q**M*):\n \n$$\\frac{R^{(QM)}}{N\\_m} \\leq \\frac{\\langle T\\_{\\mathrm{ent}} \\rangle^{-1}}{N\\_m} \\leq \\frac{c}{4L} =R\\_{\\mathrm{max}}^{(QM)}.$$ \nThis theoretical upper bound also holds if there are more than two QMs at each repeater node (see Fig. [figboth](b)) as the rate would linearly increase with the number of matter qubits. Therefore, we have derived a general theoretical upper bound for memory-based protocols based on heralded entanglement generation. It is worth noting that the fundamental reason for this upper bound comes from the need for classical signaling in these protocols. Such classical signaling is not required for RGS protocols, enabling them to surpass this limit, as we show below. In the Supplementary Materials, we also show that a tighter bound, $R\\_{\\rm max}^{(QM)}=c/7L$, can be obtained for memory-based schemes in which there are two quantum memories per repeater node, and heralded entanglement swapping is used.\nWe emphasize that the upper bound derived in this section holds for QR protocols that are based on quantum memories and distant heralded entanglement generation. This corresponds to the first and second generations of QRs, as categorized in Ref. . Consequently, in the following, the RGS-based protocol will be compared only to these categories of QRs, for which the performance is limited by classical signaling.\nRate of the RGS protocol with deterministic graph state generation\n==================================================================\nIn this section, we review the RGS protocol as introduced in Ref. and the deterministic generation of RGSs using a few matter qubits as proposed in Ref. . We show how the rate of the RGS protocol depends on various parameters in the case where deterministic state generation methods are used.\nRGS protocol and rate\n---------------------\nAn RGS is a quantum state $\\ket{G}$ that can conveniently be represented in the form of a graph *G* = (*V*, *E*) with *V* vertices and *E* edges. Each vertex corresponds to a photonic qubit prepared in the $\\ket{+}$ state, and each edge corresponds to the application of a *C**Z* gate between the two qubits it connects:\n \n$$\\ket{G} = \\prod\\_{(i,j) \\in E} CZ\\_{ij} \\ket{+}^{\\otimes V}.$$ \nAn example of the graph representing an RGS is shown in Fig. [figboth](c). These states include 2*m* inner photonic qubits that are referred to as the first-leaf qubits. All the first-leaf qubits are fully connected to each other and each of them is also connected to one additional qubit, referred to as a second-leaf qubit. The first-leaf qubits are logically-encoded using tree graph states; further details on this are given below.\nIn an RGS protocol (see Fig. [figboth](d)), the distance *L* separating Alice and Bob is also divided into smaller steps *L*0 by *N**Q**R* = *L*/*L*0 − 1 source nodes where the RGSs are created. The RGS is divided into two equal parts, each containing *m* arms, and one part is sent to the left adjacent measurement node and the other to the right. Thus, half of one RGS meets half of another RGS at each measurement node, where each second-leaf qubit from one of the half-RGSs undergoes a Bell measurement with its counterpart from the other half-RGS. Further details about this entanglement swapping procedure at the measurement node are given later, but it is important to note that the RGS protocol does not use quantum memories at all and thus cannot store the information. This implies that an entanglement connection between Alice and Bob should be realized in only one trial with a probability *P**A**B* much higher than direct fiber transmission: *P**A**B* ≫ *η**t*(*L*). The generation of an entanglement connection between Alice and Bob requires the realization of successful entanglement connections between all the adjacent RGSs. It is important to note that the measurements performed at each measurement node do not require information from other measurement nodes so that, in contrast to memory-based approaches, the RGS protocol does not require any classical signaling while entanglement is being extended through the network. Classical signaling is needed only once at the end of the protocol to recover the Pauli frame of the Bell pair shared by Alice and Bob, i.e. to determine which local Pauli rotations Alice and Bob’s qubits should undergo. This means that in the RGS protocol, it is not necessary to wait for any classical signaling before proceeding to generate the next batch of RGSs needed to create the next Bell pair shared between Alice and Bob (see Supplementary Materials for more details). Consequently, unlike the memory-based scheme, the rate of the RGS protocol does not depend on the time it takes for the photon to get from one node to the next. It is limited only by the generation time *T**R**G**S* of an RGS:\n \n$$R^{(RGS)} = \\frac{{P\\_{RGS \\leftrightarrow RGS}}^{L/L\\_0}}{T\\_{RGS}},\n \\label{eq\\_rate}$$ \nwith *P**R**G**S* ↔ *R**G**S* the probability to generate an entanglement connection between two RGSs. The main notations used in this work are defined in Table [table3].\n[!ht] [figgeneration]\nA successful entanglement link can be generated if at least one of the Bell measurements at a measurement node succeeds. In that case, the two first-leaf qubits attached to the second-leaf qubits that underwent the successful Bell measurement are measured in the *X* basis, while the remaining 2*m* − 2 first-leaf qubits are measured in the *Z* basis. The *X* measurements transfer the entanglement connection to the next two adjacent measurement nodes, while the *Z* measurements disentangle all the excess qubits associated with failed Bell measurements. All these first-leaf qubit measurements must be successful in order to reliably create an entanglement link. Therefore, the probability to successfully create an entanglement link between two RGSs is given by:\n \n$$\\begin{aligned}\n P\\_{RGS \\leftrightarrow RGS} & = \\left(1 - \\left(1 - P\\_{\\mathrm{Bell}}\\right)^m\\right) \\\\\n & \\times \\mathrm{Pr}(M\\_{X,\\ell})^2 \\mathrm{Pr}(M\\_{Z,\\ell})^{2m-2},\\label{eq:PRGStoRGS}\n \\end{aligned}$$ \nwhere *P*Bell = *P*ph2/2 is the probability of a successful Bell measurement. This depends on *P*ph = *η**c**η**d**η**t*(*L*0/2), which is the probability that a single photon is emitted and collected into the fiber (*η**c*), is transmitted to the measurement node (*η**t*(*L*0/2)) and detected (*η**d*). Pr(*M**X*, ℓ) and Pr(*M**Z*, ℓ) are the probabilities that the logical *X* and *Z* measurements on the first-leaf qubits succeed. Note that if the first-leaf qubits were not logically encoded, we would have Pr(*M**X*)2Pr(*M**Z*)2*m* − 2 = *P*ph2*m* ≤ *η**t*(*L*0), and it would be impossible to have an advantage over direct fiber transmission. Therefore, the loss-tolerance of logically-encoded qubits is crucial for this protocol. Next, we review this encoding, which was introduced in Refs. and .\nLoss-tolerance with tree graph states\n-------------------------------------\nWe review how the probabilities Pr(*M**X*, ℓ) and Pr(*M**Z*, ℓ) depend on the single-photon transfer probability *P*ph and on the shape of the tree graph state used for the logical encoding. Ref. demonstrated that this encoding remains loss-tolerant as long as *P*ph is above 50%.\nWe consider the calculation of the probabilities of successful measurements of the logically encoded qubits in the presence of loss errors on a tree graph state. A tree is characterized by its branching vector *b⃗* = (*b*0, *b*1, ..., *b**n* − 1) (see Fig. [figboth](c)), which describes the connectivity between the different levels of the tree. To perform a *Z* measurement *M**Z*, *k* on a qubit at level *k*, it is possible to either perform a direct measurement on this qubit (with success probability *P*ph) or, if it fails (with probability 1 − *P*ph), perform an indirect measurement (with probability *r**k*). Thus, the overall success probability of a *Z* measurement at level *k* is:\n \nPr(*M**Z*, *k*) = *P*ph + (1 − *P*ph)*r**k*. \nTo perform an indirect measurement on a qubit (call it A) at level *k*, one can use the stabilizing property of a graph state . It is possible to deduce the outcome of the *Z* measurement on A by performing an *X* measurement on another qubit (B) at level *k* + 1 and a *Z* measurement on all the qubits, *C**i*, that are in the neighborhood of B at level *k* + 2 (see Fig. [figgeneration](a)). This works because of the invariance of graph states when they are acted upon by their stabilizers:\n \n$$\\begin{aligned}\n \\ket{G}& = X\\_B {\n \\mathbin{\\mathop{\\otimes}\\limits\\_{j \\in N(B)}}\n } Z\\_j \\ket{G} \\\\\n & = X\\_B \\otimes Z\\_A {\n \\mathbin{\\mathop{\\otimes}\\limits\\_{i\\in \\{1, b\\_{k+1}\\}}}\n } Z\\_{C\\_i} \\ket{G},\n \\end{aligned}$$ \nso we have:\n \n$$Z\\_A \\ket{G} =X\\_B {\n \\mathbin{\\mathop{\\otimes}\\limits\\_{i\\in \\{1, b\\_{k+1}\\}}}\n } Z\\_{C\\_i} \\ket{G}.$$ \nA single indirect measurement has a success probability *s**k*. Note, however, that the tree structure allows *b**k* indirect measurement attempts, and only one needs to succeed to indirectly measure a qubit at level *k*. So the probability that at least one indirect measurement succeeds is\n \n*r**k* = 1 − (1 − *s**k*)*b**k*, \nwith\n \n*s**k* = *P*phPr(*M**Z*, *k* + 2)*b**k* + 1. \nIt is thus possible to derive the success probability of a measurement recursively, given that the qubits at the lowest level can only be measured directly: Pr(*M**Z*, *n*) = *P*ph. Logical measurements in the *X* or *Z* basis are given by :\n \n$$\\begin{aligned}\n \\mathrm{Pr}(M\\_{X,\\ell}) & = r\\_0, \\\\\n \\mathrm{Pr}(M\\_{Z,\\ell}) & = \\left(P\\_{\\mathrm{ph}} + (1 - P\\_{\\mathrm{ph}}) r\\_1 \\right)^{b\\_0} \\\\\n & = \\mathrm{Pr}(M\\_{Z,1})^{b\\_0}.\n \\end{aligned}$$ \nIt is interesting to note that logical encoding with tree graph states can also correct single-qubit errors, as shown in Ref. and described in the Supplementary Materials. This will be used later when we evaluate the sensitivity of the RGS protocol to errors.\nGeneration of an RGS\n--------------------\nThe achievable rate between Alice and Bob also depends on the repetition rate of the protocol, which is given by the generation time of the RGS. Because it is impossible to realize deterministic two-qubit gates on photons with linear optics, such a graph state can either be generated probabilistically by the recursive fusion of smaller graphs using linear optics and Bell state measurements as shown in Ref. and , or deterministically using a few matter qubits as shown in Ref. . We now review the latter.\nAn arbitrary-sized RGS can be generated deterministically by following a given sequence based on four operations on matter qubits: the emission of a photon maximally entangled with the matter qubit *E*ph, the Hadamard gate *H*, measurements in the Pauli bases *M**X*, *M**Y*, *M**Z* and the *C**Z* gate. The generation of an RGS with 2*m* arms and a tree graph encoding with branching vector *b⃗* = (*b*0, *b*1, ..., *b**n* − 1) requires *n* + 1 matter qubits *Q*1,..., *Q**n* + 1, and is given by the sequence (see also Fig. [figgeneration](b))\n \n$$\\begin{aligned}\n & M\\_{Y,{Q\\_1}}(M\\_{X,Q\\_3} M\\_{X,Q\\_2} CZ\\_{Q\\_1,Q\\_3} CZ\\_{Q\\_2,Q\\_3} E\\_{\\mathrm{ph},Q\\_3} {G\\_3}^{b\\_0})^{2m} \\\\\n& \\mathrm{with} \\; G\\_{k} = M\\_{Z,Q\\_{k}} H\\_{Q\\_{k}} E\\_{\\mathrm{ph},Q\\_{k}} CZ\\_{Q\\_{k-1}, Q\\_{k}} {G\\_{k+1}}^{b\\_{k-2}} \\\\\n& \\mathrm{and} \\; G\\_{n+ 2} = E\\_{\\mathrm{ph},Q\\_{n+ 1}},\n \\end{aligned}$$ \nwhere, for simplicity, we have omitted the single photonic qubit rotations.\nThe overall generation time of an RGS using this procedure is therefore\n \n$$\\begin{aligned}\n T\\_{RGS} & = 2m \\left(1 + f(\\vec{b}, n-1)\\right) T\\_{E\\_{\\mathrm{ph}}} +T\\_M\\\\\n & + 2m \\left(2 + f(\\vec{b}, n-2)\\right) (T\\_M + T\\_{CZ}) \\\\\n & + 2m f(\\vec{b}, n-2) T\\_{H}, \\\\\n \\end{aligned}$$ \nwith *f*(*b⃗*, *k*) = ∑*i* = 0*k*∏*j* = 0*i**b**j* and with *T**E*ph, *T**M*, *T**H* and *T**C**Z* the times for photon emission, matter qubit measurement, Hadamard and *C**Z* gates, respectively.\nIn the following, we make the realistic assumption that the *C**Z* gate time *T**C**Z* is much longer than the durations of the other operations, and so we set *T**H* = *T**M* = *T**E*ph = 0 for simplicity. With this assumption, the generation time *T**R**G**S* only depends on the number of *C**Z* gates and their duration:\n \n*T**R**G**S* = 2*m*(2 + ∑*k* = 0*n* − 2∏*j* = 0*k**b**j*)*T**C**Z*. \nThis is for an RGS with 2*m* arms and a logical tree encoding with branching vector *b⃗* = (*b*0, ..., *b**n* − 1). In the following, we will assume a depth-two tree graph state (*b⃗* = (*b*0, *b*1)), so that the full RGS can be generated with only three matter qubits (*n* = 2). The total number of matter qubits in the network is therefore *N**m* = (*L*/*L*0 − 1)(*n* + 1) = 3(*L*/*L*0 − 1).\nPerformance comparisons\n=======================\nWe now evaluate the performance of the RGS protocol when it is generated deterministically by a few matter qubits. We compare our results to the direct fiber transmission limit derived by Refs. and to the memory-based upper bound *R*max(*Q**M*) = *c*/4*L* found in Sec. [secQM].\nOptimizing the RGS protocol\n---------------------------\n [figoptimizationtreeRGS]\n |p1.2cm|p1.2cm|p0.6cm|p0.6cm|p0.6cm|p1.8cm| *L*/*L*att & *L*0/*L*att & *m* & *b*0 & *b*1 & $R\\_{\\rm max}^{(RGS)} T\\_{CZ}$ \n10 & 0.23 & 11 & 8 & 4 & 2.4 × 10− 5 \n25 & 0.21 & 13 & 10 & 5 & 6.8 × 10− 6 \n50 & 0.19 & 14 & 10 & 5 & 2.8 × 10− 6 \n100 & 0.17 & 15 & 10 & 5 & 1.1 × 10− 6 \n150 & 0.15 & 16 & 10 & 5 & 6.8 × 10− 7 \n[table]\nIn the following, we show how the RGS protocol parameters can be optimized to maximize the overall rate per matter qubit, *R**m*(*R**G**S*), or in the presence of errors, the secret key rate per matter qubit. For the moment, we assume perfect photon collection and detection efficiencies (*η**c**η**d* = 1) and no single-photon errors (*ε* = 0); we take these effects into account later on. From Eqs. ,, and , the achievable rate per matter qubit *R**m*(*R**G**S*) of the RGS protocol for a total distance *L* is inversely proportional to the *C**Z* gate time *T**C**Z*, but it depends non-linearly on the separation distance, *L*0, between two RGS nodes and the RGS shape (number of arms 2*m* and the tree branching vector *b⃗* = (*b*0, *b*1)). Therefore, for each choice of the total distance *L*, there is a certain node separation and RGS shape that maximize the achievable rate.\nThe optimization of the rate per matter qubit for a total distance *L* = 50*L*att ($\\approx 1000\\; \\kilo\\meter$) is shown in Fig. [figoptimizationtreeRGS]. The position of each point corresponds to a specific RGS shape, the color of the point indicates the node separation *L*0 that optimizes the rate for that shape, and the size of the point represents the maximal rate per matter qubit for these parameters. This optimization converges, allowing us to extract the optimal RGS shape and distance *L*0 for this particular choice of the total distance *L*. The optimization can be repeated for various choices of *L*, and the extracted optimal parameters are recorded in Table [table].\nWe compare the RGS protocol to direct fiber transmission in Fig. [figoptimizationL](a). The maximum achievable rate per matter qubit for the RGS protocol, *R*max(*R**G**S*), is shown as a function of the total distance $L/L\\_{\\rm att}$. In the case of direct transmission, we show seven different curves corresponding to the achievable rate for seven different values of the single-photon source repetition rate. We see that the RGS protocol outperforms direct transmission with the highest repetition rate for $L\\gtrsim30L\\_{\\rm att}$. In the same figure, we also show how well the protocol works if we keep the optimized parameters fixed and change the total distance *L*. To demonstrate this, we fix the RGS parameters and node separation *L*0 to the values that optimize the rate for a total distance *L*tar. We then adjust *L* away from *L*tar without changing the RGS parameters or *L*0, and we calculate the new rate for each value of *L*. In Fig. [figoptimizationL](a), we show the resulting rates as a function of *L*/*L*0 for five different choices of *L*tar. We see that the RGS protocol continues to work well over a broad range of total distance *L* when we use parameters that are optimized for a large total distance *L*tar.\n [figoptimizationL]\nFor long distances *L* in the range 50*L*att − 200*L*att, *R*max(*R**G**S*) scales approximately as *L*− 1.27, and thus the scaling with distance is slightly worse than the upper bound scaling for memory-based schemes, *R*max(*Q**M*) ∼ *L*− 1, obtained in Sec. [secQM]. However, contrary to the memory-based schemes, there are no fundamental upper limits imposed by classical signaling on the maximal achievable rate *R*max(*R**G**S*), as the latter is inversely proportional to the *C**Z* gate time, *T**C**Z*. If *T**C**Z* is sufficiently small, then the RGS protocol should surpass the upper bound of memory-based schemes. This is illustrated in Fig. [figoptimizationL](b), where the red region of the plot indicates the regime where the RGS protocol outperforms memory-based schemes.\nFrom our calculations, it seems that the maximum achievable rate per matter qubit, *R**m**a**x*(*R**G**S*), is in principle unbounded since it is inversely proportional to the *C**Z* gate time. We should recall however that these results are based on the assumption that the *C**Z* gate takes much longer than the other operations *O* made on the matter qubits: *T**C**Z* ≫ *T**O*. Decreasing *T**C**Z* initially increases the rate, but if *T**C**Z* becomes small enough, neglecting the durations of other operations eventually becomes invalid. Consequently, to increase the rate further, not only the *C**Z* gate time but all the operation times should be reduced simultaneously. If the durations of all these operations could be made arbitrarily small, then the rate per matter qubit would increase to infinity. In practice however, the operation times could also have an intrinsic lower bound that limits the performance of the RGS protocol, but these lower bounds would depend on the specific system on which the protocol is implemented, while the limit imposed by classical signaling is much more stringent and general.\nFor distances below $200L\\_{\\mathrm{att}} \\approx 4000\\;\\kilo\\meter$, a gate time *T**C**Z* below $6\\times10^{-4} t\\_{\\mathrm{att}} \\approx 60\\;\\nano\\second$ is sufficiently short to outperform any memory-based scheme. For this range of total distance, the optimal value of the node separation *L*0 ranges from $0.15-0.19 L\\_{\\mathrm{att}} \\approx 3-3.8\\;\\kilo\\meter$. As an illustration of the RGS performance, for $L = 1000 {\\rm \\; km}$ and $T\\_{CZ} = 10 {\\rm \\; ns}$, the total rate is $R = 220 {\\rm \\; kHz} $ for *N**m* = 786 matter qubits used, leading to $R\\_{\\rm max}^{(RGS)}=276 {\\rm \\; Hz}$ per matter qubit, while $R\\_{\\rm max}^{(QM)} = 50 {\\rm \\; Hz}$ per matter qubit for this distance.\nSensitivity to errors\n---------------------\nSo far, we have considered the generation of a perfect RGS, which is a pure entangled state of many photons. Generating such a perfect state is not feasible experimentally, and so we need to evaluate the sensitivity of the RGS protocol to errors. To do so, we now consider single-photon loss and single-photon errors in our optimization process.\nThe single-photon loss (other than the fiber losses) depends on the probability that an emitted photon is neither collected into the fiber nor detected; this corresponds to the case where *η**c**η**d* ≠ 1. Fig. [figoptimizationL](c) shows the optimized rate per matter qubit for different values of *η**c**η**d*. Similarly, Fig. [figoptimizationL](d) compares the performance of the RGS protocol with the upper bound for memory-based protocols, *R*max(*Q**M*), as a function of *T**C**Z* and *η**c**η**d* for a total distance *L* = 50*L*att. These results show that the photon losses must be below 15% (*η**c**η**d* > 85%) in order for the RGS protocol to outperform both the direct fiber transmission and memory-based protocols. This rather stringent requirement is mainly due to the loss-tolerance of the tree graph state, which only works when the total photon loss (including fiber losses) is below 50%.\nApart from the photon losses, the RGS protocol also depends on other kinds of errors that reduce the fidelity *F**A**B* of the final entangled photon pair shared by Alice and Bob. These errors include, but are not limited to, single-photon measurement errors, photon Bell-measurement errors, and depolarization errors. In the present case where the RGS is generated using matter qubits, the limited coherence time of the matter qubits should also limit the fidelity of the entangled photons. For QKD applications, the final secret key rate depends on the total rate *R* and the fidelity *F**A**B* :\n \n*R**s**k**r* = *R*(1 − 2*h*(*F**A**B*)), \nwhere *h* is the binary entropy function:\n \n*h*(*F*) = − *F*log2(*F*) − (1 − *F*)log2(1 − *F*). \nIn the remainder of this paper, we now use *R**m*(*R**G**S*) to refer to the secret key rate per matter qubit, which coincides with the rate per matter qubit in the absence of errors. We model these imperfections by using a single-qubit error probability *ε* that affects all the photonic qubits. This model works well for qubit measurement errors as well as depolarization errors. In Ref. , it was also shown that decoherence errors on quantum emitters induce only local errors in the emitted cluster state. The detailed derivation of *F**A**B* is given in the Supplementary Materials. Fig. [figoptimizationL](e) compares the optimized secret key rate of the RGS protocol against that of direct fiber transmission as a function of the single-qubit error probability and the total distance *L*. Fig. [figoptimizationL](f) compares the performance of the RGS protocol to the upper bound for memory-based protocols for *L* = 50*L*att. We find that the single-qubit error should be below *ε* < 10− 4 to outperform both the direct fiber transmission and the memory-based protocol upper bound, $R^{(QM)}\\_{\\rm max}$.\nOne might expect a better tolerance to single-qubit errors since they are also corrected by the logical encoding of the 1st-leaf qubits. This is not the case because the 2nd-leaf qubits are not logically encoded and thus are very sensitive to errors. The dependence on single-qubit errors is actually similar to the protocols using heralded entanglement generation. It is also possible to use entanglement purification to improve the fidelity *F**A**B* and maximize the secret key rate.\nFrom the maximum single-qubit error rate, we can derive an order-of-magnitude estimate for the coherence time requirements for the matter qubits. To do so, we use two strong assumptions. We first assume that the fidelity *F**R**G**S* of an RGS depends on the ratio between the generation time *T**R**G**S* of the RGS and the coherence time *T*2 of the matter qubits:\n \n$$F\\_{RGS} \\approx e^{- \\frac{3 T\\_{RGS}}{T\\_2}}$$ \nThe coefficient 3 corresponds to the number of matter qubits that degrade the RGS fidelity. Here, we have made the simplifying assumption that the main source of infidelity is the matter qubit decoherence, and we have assumed that the effect of this decoherence is to reduce the overall fidelity of the graph state by *e*− *T**R**G**S*/*T*2, which decays exponentially with the time during which the qubit is attached to the RGS (*T**R**G**S*) divided by its coherence time. This assumption may or may not be justified depending on the system used to implement the protocol, but we use it simply to get an order of magnitude estimate of the typical coherence time that is required for this protocol. We also assume that the errors are homogeneously distributed over the *N*ph photons of an RGS, such that *F**R**G**S* = (1 − *ε*)*N*ph. This assumption is consistent with Ref. , which shows that errors generated by a quantum emitter and imprinted onto the generated graph state are localized. With these assumptions, the coherence time should satisfy *T*2 ≥ 2500*T**C**Z*.\nDiscussion\n==========\n |p1.8cm|p1.2cm|p1.0cm|p1.2cm|p1.1cm| **Matter qubit** & $\\bm{T\\_{CZ}}$ & & $\\bm{T\\_2}$ & $\\bm{\\beta}$ \nCoupled QDs & & & & \ndefects (diamond) & & & & \nTrapped ions & & & & \n[table2]\nIn the previous section, we derived a set of requirements that need to be fulfilled in order for the RGS protocol to have a clear advantage over both memory-based QRs and repeater-less protocols:\n1. $T\\_{CZ}\\leq T\\_{lim}(L) \\approx 60 \\nano\\second$,\n2. *T*2 ≥ 2500*T**C**Z*,\n3. *η**c**η**d* > 85%.\nIt is important to note that these bounds were obtained using RGSs with a depth-2 logical tree encoding. We may expect an increase in the rate per matter qubit if deeper trees were used but the improvement would certainly not be large as *T**R**G**S* grows quickly with tree depth. Checking this directly is challenging because the process of optimizing the RGS protocol becomes significantly more complex. We leave this intensive numerical analysis to future work.\nTable [table2] shows the current state-of-the-art for three candidate systems that can be used to generate RGSs. So far, the characteristics of the current systems are not good enough yet to exceed the memory-based resource-efficient upper bound *R*max(*Q**M*). The main limitation comes from long two-qubit gate times, since fast and high-fidelity gates are critical for the performance of the RGS protocol. Currently, most of the two-qubit gates that have been experimentally demonstrated are slow because they rely on local interactions, such as the hyperfine coupling, which are generally weak. Efficient light-matter interfaces using cavities can also be used to realize deterministic, photon-mediated *C**Z* gates between two matter qubits , using spin-photon gates . The crucial advantage of this technique is that it only requires optical interactions, which are intrinsically much faster than hyperfine interactions. Progress in the experimental realization of strong spin-dependent phase shifts has been made recently . The second condition (ii) will also be easier to achieve if the *C**Z* gate time is reduced.\nThe third condition (iii) requires the efficient collection of single photons emitted by a single quantum emitter. The collection of single photons needs to be facilitated by increasing the emission of photons into a single optical mode. This is generally realized using waveguides or cavities with single-mode emission probabilities *β* that can reach 99.7% . This single mode should then be efficiently coupled to a single-mode fiber. Finally, superconducting photon detectors can be used to achieve high detection efficiencies .\nIn conclusion, we have identified the main requirements that must be met in order for all-photonic repeater protocols based on deterministically generated graph states to outperform memory-based protocols. We did this by finding the graph state structures that maximize the secret key rate and by comparing this to an upper bound on the rate for memory-based schemes that we derived. We found that two-qubit gate times and the efficient collection and detection of emitted photons are the most important factors in determining whether the all-photonic approach is superior to memory-based ones. An interesting future direction would be to improve the protocol for the emitter-based generation of photonic graph states by incorporating atom-photon *C**Z* gates, which would both simplify the procedure and decrease gate times, and thus enhance the performance of deterministic, all-photonic repeaters.\nAcknowledgements\n================\nWe thank Yuan Zhan and Shuo Sun for their contribution in solving issues in the code. This research was supported by the NSF (Grant No. 1741656) and by the EU Horizon 2020 programme (GA 862035 QLUSTER).\nMethods\n=======\nThe numerical model used for the results in this article is available here:\n<https://github.com/Paulhilaire/performance_repeater_graph_state>.\nBounds for memory-based repeater protocols\n==========================================\nAs discussed in the main text, the rate of a memory-based quantum repeater based on heralded entanglement generation is limited by the rate ⟨*T**Q**M*⟩− 1 at which a given memory qubit can be re-used. ⟨*T**Q**M*⟩ is given by the sum of the average time required for generating entanglement ⟨*T*ent⟩ and the time during which the entanglement is stored ⟨*T*store⟩. ⟨*T*ent⟩ is given by *T*trial/*P*ent, where *T*trial is the time required for a heralded entanglement generation attempt, and *P*ent is the probability that an attempt is successful. Let’s consider three heralded entanglement procedures :\n* In Ref. , *T*trial = *L*0/*c*, and the probability is *P*ent = *e*− *L*0/*L*att/2 (assuming perfect photon emission and detection probability).\n* In Ref. , *T*trial = 2*L*0/*c*, and the probability is *P*ent = *e*− *L*0/*L*att.\n* In Ref. , *T*trial = *L*0/*c*, and the probability is *P*ent ∝ *e*− *L*0/(2*L*att). However, the protocol needs to be implemented in a regime where the probability to emit a photon is very small to avoid two-photon emission, which degrades the entanglement fidelity. Consequently, *P*ent ≪ 1/2.\nThus in all these cases, if we consider small values of *L*0 ≪ *L*att, we have:\n \n$$\\langle T\\_{\\mathrm{ent}} \\rangle \\geq \\frac{2L\\_0}{c} e^{L\\_0/L\\_{\\mathrm{att}}} \\geq \\frac{2L\\_0}{c}.$$ \nThis limitation can be understood from the fact that the photonic qubit and the classical signal need to travel back and forth either to a measurement node (twice the distance *L*0/2) or to the next QR (twice the distance *L*0). If the photon travels to a quantum repeater, it is possible to deterministically entangle the QM and the photon, so the entanglement probability is only bounded by the photonic losses. If the photon travels to an intermediate node, the entanglement generation procedure uses linear optics and has a success probability that cannot reach more than 1/2 (unless ancillary photonic qubits are used, but this requires another quantum emitter to produce them ).\nFrom this limitation, we find the upper bound *R*max(*Q**M*) ≥ max(*R*/*N**m*) in the case of QR protocols with two QMs. In this case, the rate *R* is limited by ⟨*T**Q**M*⟩− 1 ≤ ⟨*T*ent⟩− 1 = *c*/(2*L*0). The number of matter qubits is given by twice the number of repeaters plus Alice and Bob’s matter qubits, thus *N**m* = 2(*N**Q**R* + 1) = 2*L*/*L*0, therefore:\n \n*R*max(*Q**M*) = *c*/4*L*. \nIn the case of multiplexed repeaters, by multiplying the number of memories per repeater by *N*, the entangling rate between two repeaters is bounded by *N**c*/(2*L*0), but the number of matter qubits is now *N**m* = 2*N**L*/*L*0, so that the total rate is still bounded by *R*max(*Q**M*).\nRef. also proposes a method to generate heralded entanglement deterministically by detecting photons. The idea is to excite the two quantum emitters (each consisting of two ground states and one excited state that couples to only one of the ground states) such that they both emit a photon whose which-path information is erased using linear optics. When the first photon is detected a signal is sent to each quantum emitter to perform a rotation between the ground states. This rotation needs to be performed before the second photon is emitted. Therefore, the latency between the photon detection and the rotation should be much smaller than the lifetime of the quantum emitter’s excited state. A direct implication of this is that the photon detectors should be placed very close to the quantum emitters, which is impractical. For example, for quantum emitters with a lifetime under $10 {\\rm \\; ns}$ (which is typical for QDs and defects in diamond), the position of the detectors should be much smaller than $1 {\\rm \\; m}$. In addition, to obtain high fidelities with this scheme, the two quantum emitters are excited for a typical time *T*1 which should be much larger than the classical signaling time *L*0/*c*, and thus the previous analysis should still hold in that case.\nIn the case of memory-based schemes using heralded entanglement and 2 QMs per QR, it is possible to find a stricter upper bound. To derive this bound, consider two repeater nodes labeled 1 and 2. Node 1 contains two memory qubits QM1 and QM1ʹ, while node 2 contains QM2 and QM2ʹ. Suppose that we succeed in entangling QM1 and QM2 via a Bell measurement at an intermediate measurement node. How long do we have to maintain this entanglement before further entanglement swappings at neighboring nodes free up QM1 and QM2, allowing them to be re-used for creating the next Bell pair to be shared between Alice and Bob? The answer to this question is what we called $\\langle T\\_{\\rm store}\\rangle$ above, the average amount of time we have to store the entanglement. The upper bound on the rate is then $\\langle T\\_{\\rm QM}\\rangle^{-1}$, where $\\langle T\\_{\\rm QM}\\rangle=\\langle T\\_{\\rm ent}\\rangle+\\langle T\\_{\\rm store}\\rangle$. We obtained a bound for $\\langle T\\_{\\rm ent}\\rangle$ above and used this to derive the bound on the rate given in Eq. . There, we neglected the role of $\\langle T\\_{\\rm store}\\rangle$. Here, we reinstate this quantity to obtain a stricter bound.\nTo determine $\\langle T\\_{\\rm store}\\rangle$, we first note that the entanglement between QM1 and QM2 needs to be maintained until QM1ʹ and QM2ʹ become entangled with memory qubits on other nodes. Once this happens, the entanglement can then be swapped, leaving QM1 and QM2 completely disentangled and ready for re-use. Of course, QM1ʹ and QM2ʹ could already be entangled with other nodes by the time QM1 and QM2 become entangled. Thus, there are four equally probable cases to consider:\n* If QM1ʹ and QM2ʹ are already connected to other nodes, we immediately swap the entanglement and thus do not wait any longer. In this case $\\langle T\\_{\\mathrm{store}}^{\\rm case 1}\\rangle = 0$.\n* If one QM (either QM1ʹ or QM2ʹ) is already connected to other nodes, we need to wait an average time of ⟨*T*ent⟩ for the second QM to become entangled as well. In this case, $\\langle T\\_{\\rm store}^{\\rm case 2}\\rangle=\\langle T\\_{\\rm store}^{\\rm case 3}\\rangle= \\langle T\\_{\\mathrm{ent}} \\rangle$.\n* If neither QM1ʹ or QM2ʹ are connected, we need to wait for both to become connected. In this case, $\\langle T\\_{\\mathrm{store}}^{\\rm case 4}\\rangle = \\frac{3-2P\\_{\\mathrm{ent}}}{2- P\\_{\\mathrm{ent}}}\\times \\langle T\\_{\\mathrm{ent}} \\rangle $.\nThe expression for $\\langle T\\_{\\mathrm{store}}^{\\rm case 4}\\rangle$ above requires some explanation. It comes from the fact that both QM1ʹ and QM2ʹ become successfully entangled with other nodes after *n* attempts with probability\n \n$$\\mathrm{Pr}(n, P\\_{\\mathrm{ent}}) = [1-(1-P\\_{\\rm ent})^n]^2.$$ \nThe average storage time in this case then depends on the average number of attempts needed before entanglement is successfully created and on the time $T\\_{\\rm trial}$ each attempt takes:\n \n$$\\langle T\\_{\\rm store}^{\\rm case 4}\\rangle = T\\_{\\rm trial}\\sum\\_{n=0}^{\\infty} n [\\mathrm{Pr}(n,P\\_{\\rm ent})- \\mathrm{Pr}(n-1,P\\_{\\rm ent})] = \\frac{3-2P\\_{\\mathrm{ent}}}{2- P\\_{\\mathrm{ent}}}\\langle T\\_{\\rm ent}\\rangle.$$ \nThis implies that the overall average storage time is\n \n$$\\langle T\\_{\\rm store}\\rangle=\\frac{\\langle T\\_{\\rm store}^{\\rm case 1}\\rangle+\\langle T\\_{\\rm store}^{\\rm case 2}\\rangle+\\langle T\\_{\\rm store}^{\\rm case 3}\\rangle+\\langle T\\_{\\rm store}^{\\rm case 4}\\rangle}{4}=\\frac{7/4-P\\_{\\rm ent}}{2-P\\_{\\rm ent}}\\langle T\\_{\\rm ent}\\rangle.$$ \nThe upper bound on the rate per matter qubit is therefore\n \n$$R\\_{\\mathrm{max}}^{(2QM)}=\\frac{1}{(\\langle T\\_{\\rm ent}\\rangle+\\langle T\\_{\\rm store}\\rangle)N\\_m}=\\frac{2-P\\_{\\rm ent}}{15-8P\\_{\\rm ent}}\\frac{c}{L},$$ \nwhere we set $\\langle T\\_{\\rm ent}\\rangle\\to2L\\_0/c$ and *N**m* = 2*L*/*L*0 as appropriate for the memory-based scheme. If we set $P\\_{\\rm ent} \\rightarrow 1$ , this becomes\n \n$$R\\_{\\mathrm{max}}^{(2QM)}=\\frac{c}{7L}\\quad\\hbox{for}\\quad P\\_{\\rm ent}=1.$$ \nClassical signaling does not limit the repetition rate of the RGS protocol\n==========================================================================\nIn this section, we show that classical signaling does not limit the repetition rate of the RGS protocol. This is important to consider because some of the *X* and *Z* measurements that are performed on the RGS will, depending on the measurement outcomes, leave the remaining photons in a state that is only locally equivalent to a graph state. In this case, additional single-qubit gates must be applied to bring the state to a proper graph state before the next measurements are performed. Otherwise, the bases for these subsequent measurements will be effectively rotated, and the measurements will not modify the state in the intended way. In the following, we show that the extra single-qubit gates only need to be applied on qubits located at the same measurement node where the measurements on which they depend occur. Thus, classical communication between nodes is not needed. The only exception to this occurs at the very end of the protocol, when we are left with a linear cluster state in which the two end qubits are those of Alice and Bob. The final step is to turn this state into a Bell pair shared by Alice and Bob by performing *X* measurements on the inner qubits of the linear cluster state. Afterward, Alice and Bob’s qubits form a Bell pair whose Pauli frame depends on the *X* measurement outcomes. This information therefore needs to be transferred to them using classical signaling. However, the protocol can be repeated before this classical signal is received and thus, the classical signaling does not limit the repetition rate. The latter is only limited by the generation time of RGSs.\nRecursive construction of a repeater graph state.\n-------------------------------------------------\n [figclassicalsignaling]\nWe denote the RGS with *k* arms by $\\ket{RGS\\_k}$. For the purposes of this argument, we allow *k* to be either odd or even. We want to define this quantum state recursively. As illustrated in Fig. [figclassicalsignaling](a), we can make this state by starting with an RGS with *k* − 1 arms and then adding one new 1st-leaf qubit and one new 2nd-leaf qubit by applying *C**Z* gates between these two new qubits and between the new 1st-leaf qubit and all the existing 1st-leaf qubits in $\\ket{RGS\\_{k-1}}$. In other words, we can start from a 2-qubit cluster state and $\\ket{RGS\\_{k-1}}$ in a product state: $(\\ket{0\\_{k\\_1},+\\_{k\\_2}} + \\ket{1\\_{k\\_1},-\\_{k\\_2}}) \\otimes \\ket{RGS\\_{k-1}}$. Here, *k*1 and *k*2 denote the 1st-leaf qubit and the 2nd-leaf qubit from the ktharm. We ignore the normalization factors troughout this section. The remaining step is to apply *C**Z* gates between *k*1 and all the 1st-leaf qubits in $\\ket{RGS\\_{k-1}}$, which yields\n \n$$\\ket{RGS\\_k} = \\ket{0\\_{k\\_1},+\\_{k\\_2}} \\otimes \\ket{RGS\\_{k-1}} + \\ket{1\\_{k\\_1},-\\_{k\\_2}} \\otimes \\ket{\\overline{RGS}\\_{k-1}},\\label{eq:RGSdecomp}$$ \nwhere we have introduced the notation\n \n$$\\ket{\\overline{RGS}\\_{k}} \\equiv \\prod\\_{i=1}^{k} Z\\_{i\\_1} \\ket{RGS\\_{k}}.$$ \nHere, *Z**i**j* is the Pauli *Z* operator applied to the jth-leaf qubit of the itharm of the RGS.\nNote that if the RGS has a logical tree encoding, then instead of attaching a 2-qubit cluster state to $\\ket{RGS\\_{k-1}}$, we need to attach a star graph with *b*0 arms (in the case of a depth-two tree), where *b*0 is the first branching parameter of the tree. To create $\\ket{RGS\\_{k}}$, each arm of the star graph needs to connect to every 1st-leaf qubit of $\\ket{RGS\\_{k-1}}$; this requires a total of *b*0(*k* − 1) *C**Z* gates. The resulting RGS can still be decomposed as in Eq. , but we need to replace $\\ket{0\\_{k\\_1}}$ by a sum over all the even-parity bit strings on *b*0 qubits, and $\\ket{1\\_{k\\_1}}$ needs to be replaced by all the odd-parity bit strings on *b*0 qubits. In what follows, we will continue to use Eq. with the understanding that the *k*1 qubit states need to be replaced in this manner if a logical tree encoding is included. This does not otherwise alter the analysis below.\nBell state measurement\n----------------------\nLet’s now consider the effect of a Bell measurement on 2nd-leaf qubits from two different RGSs. We suppose that one of these qubits resides in arm *k* of one RGS, and the other in arm *k*ʹ of the other RGS. To analyze this measurement, we expand the two-RGS product state using the recursive definition of an RGS shown above:\n \n$$\\begin{aligned}\n \\ket{RGS\\_k} \\otimes \\ket{RGS'\\_k} & = \\ket{RGS\\_{k-1}} \\otimes \\ket{0\\_{k\\_1}, +\\_{k\\_2}, +\\_{{k'}\\_2}, 0\\_{{k'}\\_1}} \\otimes \\ket{RGS'\\_{k-1}} \\\\\n & + \\ket{RGS\\_{k-1}} \\otimes \\ket{0\\_{k\\_1}, +\\_{k\\_2}, -\\_{{k'}\\_2}, 1\\_{{k'}\\_1}} \\otimes \\ket{\\overline{RGS'}\\_{k-1}} \\\\\n & + \\ket{\\overline{RGS}\\_{k-1}} \\otimes \\ket{1\\_{k\\_1}, -\\_{k\\_2}, +\\_{{k'}\\_2}, 0\\_{{k'}\\_1}} \\otimes \\ket{RGS'\\_{k-1}} \\\\\n & + \\ket{\\overline{RGS}\\_{k-1}} \\otimes \\ket{1\\_{k\\_1}, -\\_{k\\_2}, -\\_{{k'}\\_2}, 1\\_{{k'}\\_1}} \\otimes \\ket{\\overline{RGS'}\\_{k-1}}.\n \\end{aligned}$$ \nWe want to perform a Bell state measurement on qubits *k*2 and *k*ʹ2. After the Bell measurement, we want to obtain the state:\n \n$$\\begin{aligned}\n \\ket{RGS\\_{k,\\mathrm{bridge}}} & = \\ket{RGS\\_{k-1}} \\otimes \\ket{0\\_{k\\_1}, 0\\_{{k'}\\_1}} \\otimes \\ket{RGS'\\_{k-1}} \\\\\n & + \\ket{RGS\\_{k-1}} \\otimes \\ket{0\\_{k\\_1}, 1\\_{{k'}\\_1}} \\otimes \\ket{\\overline{RGS'}\\_{k-1}} \\\\\n & + \\ket{\\overline{RGS}\\_{k-1}} \\otimes \\ket{1\\_{k\\_1}, 0\\_{{k'}\\_1}} \\otimes \\ket{RGS'\\_{k-1}} \\\\\n & - \\ket{\\overline{RGS}\\_{k-1}} \\otimes \\ket{1\\_{k\\_1}, 1\\_{{k'}\\_1}} \\otimes \\ket{\\overline{RGS'}\\_{k-1}},\n \\end{aligned}$$ \nwhich corresponds to the state represented in Fig. [figclassicalsignaling](b) where an edge replaces a pair of 2nd-leaf qubits shared by two RGSs.\nThe Bell state measurements are realized in the basis:\n \n$$\\begin{aligned}\n \\ket{\\phi^{\\pm}} = \\ket{0,+}\\pm \\ket{1,-}, \\\\\n \\ket{\\psi^{\\pm}} = \\ket{1,+} \\pm \\ket{0,-}.\n \\end{aligned}$$ \nThese are not the standard Bell states but are related to them by a Hadamard gate on the second qubit. By applying such a Hadamard gate, a standard Bell state analyzer can be used. Using this basis for the 2nd-leaf qubits *k*2 and *k*ʹ2, we can rewrite the state before the Bell measurement as\n \n$$\\begin{aligned}\n \\ket{RGS\\_k} \\otimes \\ket{RGS'\\_k} & = \\ket{RGS\\_{k-1}} \\otimes \\ket{0}\\_{k\\_1} \\otimes \\left(\\ket{\\phi^+}+\\ket{\\phi^-}+\\ket{\\psi^+}+\\ket{\\psi^-}\\right)\\_{k\\_2, {k'}\\_2} \\otimes \\ket{0}\\_{{k'\\_1}} \\otimes \\ket{RGS'\\_{k-1}} \\\\\n & + \\ket{RGS\\_{k-1}} \\otimes \\ket{0}\\_{k\\_1} \\otimes \\left(\\ket{\\phi^+}-\\ket{\\phi^-}+\\ket{\\psi^+}-\\ket{\\psi^-}\\right)\\_{k\\_2, {k'}\\_2} \\otimes \\ket{1}\\_{{k'\\_1}} \\otimes \\ket{\\overline{RGS'}\\_{k-1}} \\\\\n & + \\ket{\\overline{RGS}\\_{k-1}} \\otimes \\ket{1}\\_{k\\_1} \\otimes \\left(\\ket{\\phi^+}+\\ket{\\phi^-}-\\ket{\\psi^+}-\\ket{\\psi^-}\\right)\\_{k\\_2, {k'}\\_2} \\otimes \\ket{0}\\_{{k'\\_1}} \\otimes \\ket{RGS'\\_{k-1}} \\\\\n & + \\ket{\\overline{RGS}\\_{k-1}} \\otimes \\ket{1}\\_{k\\_1} \\otimes \\left(-\\ket{\\phi^+}+\\ket{\\phi^-}+\\ket{\\psi^+}-\\ket{\\psi^-}\\right)\\_{k\\_2, {k'}\\_2} \\otimes \\ket{1}\\_{{k'\\_1}} \\otimes \\ket{\\overline{RGS'}\\_{k-1}}.\n \\end{aligned}$$ \nSuppose that we can resolve only the two states $\\ket{\\psi^{\\pm}}$ with a linear optics Bell state analyzer. Depending on the measurement outcomes, the state becomes:\n \n$$\\begin{aligned}\n \\ket{\\psi^-} &\\rightarrow Z\\_{k\\_1} Z\\_{{k'}\\_1}\\ket{RGS\\_{k,\\mathrm{bridge}}}, \\\\\n \\ket{\\psi^+} &\\rightarrow Z\\_{k\\_1} \\ket{RGS\\_{k,\\mathrm{bridge}}}.\n \\end{aligned}$$ \nThus, we find that if the outcome is $\\ket{\\psi^+}$ (respectively $\\ket{\\psi^-}$) we need to apply a *Z* rotation on qubit *k*1 (on qubits *k*1 and *k*ʹ1) to recover the desired state. Note that this qubit is at the measurement node where we perform the Bell measurement.\n*Z* measurements\n----------------\nIf the measurement fails or if one Bell measurement has already succeeded, we need to perform *Z* measurements on 1st-leaf qubits to disconnect them and their associated 2nd-leaf qubits from the graph. Depending on the outcomes of these measurements, the resulting state is one of four possible states:\n \n$$\\begin{aligned}\n \\ket{RGS\\_{k-1}}\\otimes \\ket{RGS'\\_{k-1}}, \\\\\n \\ket{\\overline{RGS}\\_{k-1}}\\otimes \\ket{RGS'\\_{k-1}}, \\\\\n \\ket{RGS\\_{k-1}}\\otimes \\ket{\\overline{RGS'}\\_{k-1}}, \\\\\n \\ket{\\overline{RGS}\\_{k-1}}\\otimes \\ket{\\overline{RGS'}\\_{k-1}}.\n \\end{aligned}$$ \nNote that if a Bell measurement has already succeeded, then each of the above outcomes needs to be acted upon by a *C**Z* gate on the two 1st-leaf qubits associated with the successful measurement. We leave this possible *C**Z* gate implicit in the following. Ideally, we want to recover $\\ket{RGS\\_{k-1}}\\otimes \\ket{RGS'\\_{k-1}}$. Therefore, we need to be able to convert a state $\\ket{\\overline{RGS}\\_{k-1}}$ into $\\ket{RGS\\_{k-1}}$ by performing measurements only on qubits present in the measurement node. Let’s consider any other arm *l* of the RGS that is present at the same measurement node and rewrite the state as\n \n$$\\ket{RGS\\_{k-1}} = \\ket{0\\_{l\\_1}, +\\_{l\\_2}} \\otimes \\ket{RGS\\_{k-2}} + \\ket{1\\_{l\\_1}, -\\_{l\\_2}} \\otimes \\ket{\\overline{RGS}\\_{k-2}}.$$ \nBy applying *Z* gates on all the 1st-leaf qubits, we also have:\n \n$$\\ket{\\overline{RGS}\\_{k-1}} = \\ket{0\\_{l\\_1}, +\\_{l\\_2}} \\otimes \\ket{\\overline{RGS}\\_{k-2}} - \\ket{1\\_{l\\_1}, -\\_{l\\_2}}) \\otimes \\ket{RGS\\_{k-2}}.$$ \nA comparison of these two results leads to the identity:\n \n$$\\begin{aligned}\n Y\\_{l\\_1}Z\\_{l\\_2}\\ket{\\overline{RGS}\\_{k-1}} = i \\ket{RGS\\_{k-1}}.\n \\end{aligned}$$ \nBy applying local rotations of qubits situated at the same measurement node, it is therefore possible to recover the desired state. The last two *Z* measurements on the two qubits denoted *k*1 and *k*ʹ1 requires special care as the only two other qubits, *l*1 and *l*ʹ1, positioned at this measurement node, are the one which connect the two RGSs (if a Bell measurement has succeeded) and should undergo an *X* measurement. In that case, it is still possible to use *l*1 and *l*ʹ1 to recover the desired graph state by performing single-qubit rotations that depends on the measurement outcomes of *k*1 and *k*ʹ1:\n \n$$\\begin{aligned}\n \\ket{0\\_{k\\_1},0\\_{k'\\_1}} & \\rightarrow I\\_{l\\_1} \\otimes I\\_{l'\\_1}, \\\\\n \\ket{0\\_{k\\_1},1\\_{k'\\_1}} & \\rightarrow - i Z\\_{l\\_1} \\otimes Y\\_{l'\\_1}, \\\\\n \\ket{1\\_{k\\_1},0\\_{k'\\_1}} & \\rightarrow - i Y\\_{l\\_1} \\otimes Z\\_{l'\\_1}, \\\\\n \\ket{1\\_{k\\_1},1\\_{k'\\_1}} & \\rightarrow - X\\_{l\\_1} \\otimes X\\_{l'\\_1}, \\\\\n \\end{aligned}$$ \nwhere *I**i* denotes the identity on qubit *i*.\n*X* measurements and classical signaling\n----------------------------------------\nIf at least one Bell measurement succeeds at each measurement node and if all the required *Z* measurements succeed, we are left with a linear cluster state with two 1st-leaf qubits at each measurement node (see Fig. [figclassicalsignaling](c)). The next step is to perform *X* measurements on these qubits. Performing *X* measurements on two connected qubits of a graph state has the effect of connecting each qubit in the neighborhood of the first qubit to all the qubits in the neighborhood of the second one . Therefore, after the *X* measurements of all the qubits at each measurement node, the two remaining qubits shared by Alice and Bob are in a maximally entangled state. More precisely, this state is equivalent to a Bell pair up to local qubit rotations. It is important to note that all the *X* measurements can be performed independently of each other. Although the state can deviate from a proper graph state depending on the measurement outcomes, it is straightforward to show that the state only differs by at most local *Z* gates. This can effectively convert some of the *X* measurements to *Y* measurements, but both types of measurements have a similar effect on the state, and a Bell pair is still obtained once all the measurements are completed. However, the precise state depends on all the measurement outcomes of each *X* measurement. This information needs to be sent via a classical channel to Alice and Bob to recover the desired state.\nThis classical signal does not limit the repetition rate of the protocol, which can either be used for quantum teleportation of quantum states or for QKD. Indeed, for quantum teleportation, two quantum memories situated at Alice’s and Bob’s nodes can be used to store the Bell pair while waiting for the classical signal to arrive. In that case, the protocol can still be repeated at a high rate as long as there are enough quantum memories at Alice’s and Bob’s nodes to store the states that are already transferred. It is still the case that the protocol does not require quantum memories at the nodes inside the network, which is a clear resource improvement compared to the memory-based approach. For QKD, Alice’s and Bob’s qubits can be measured in a random basis upon arrival at their nodes, and they don’t need to be stored in quantum memories until the classical signaling is received. The classical signaling is still required to recover the Pauli frame of the measured qubits and to check if Alice’s and Bob’s qubits were measured in compatible basis for QKD.\nError correction using tree graph states\n========================================\nIn the main text, we have shown that logical encoding with tree graph states allows measuring a qubit with a probability Pr(*M**Z*, ℓ) or Pr(*M**X*, ℓ) that is higher than the single-photon measurement probability $P\\_{\\rm ph}$. We now show that the logical encoding based on tree graphs also increases the tolerance of the RGS protocol to single-qubit errors. Indeed, if *m**k* indirect measurements are performed on a qubit at level *k* in the tree, it is possible to use a majority vote of the measurements to reduce the effect of errors. Suppose that all *b**k* of the possible indirect measurements are performed, where *b**k* is the *k*thbranching parameter of the tree. The probability that exactly *m**k* of these measurements succeed (perhaps with errors) is\n \n$$p\\_k(m\\_k) = \\binom{b\\_k}{m\\_k} {s\\_k}^{m\\_k} {(1 - s\\_k)}^{b\\_k - m\\_k},$$ \nwhere *s**k* is the probability that a single indirect measurement succeeds.\nThe average error after *b**k* indirect measurement attempts at level k is:\n \n$$e\\_{I\\_k} = \\frac{1}{r\\_k} \\sum\\_{m\\_k=1}^{b\\_k} p\\_k(m\\_k) e\\_{I\\_k|m\\_k},$$ \nwhere *e**I**k*∣*m**k* is the average error in the case of *m**k* successful measurements, and *r**k* is the probability that at least one indirect measurement succeeds. For *m**k* indirect measurements, an error still occurs in the majority vote if more than half of the indirect measurements (*m**k*/2) are faulty. Thus, if *m**k* is odd:\n \n$$\\begin{aligned}\n e\\_{I\\_k|m\\_k} &=\\mathrm{Pr}(N\\_{\\mathrm{errors}} > m\\_k/2) \\\\\n &= \\sum\\_{j=\\lceil m\\_k/2 \\rceil}^{m\\_k} \\binom{m\\_k}{j} {(e\\_{I\\_k|1})}^j {(1 - e\\_{I\\_k|1})}^{m\\_k - j},\n \\end{aligned}$$ \nwhere *e**I**k*∣1 is the average error of a single indirect measurement. If *m**k* is even, we have:\n \n$$e\\_{I\\_k|m\\_k} = \\sum\\_{j=\\lceil m\\_k/2 \\rceil}^{m\\_k - 1} \\binom{m\\_k - 1}{j} {(e\\_{I\\_k|1})}^j {(1 - e\\_{I\\_k|1})}^{m\\_k -1 - j}.$$ \nThe sum goes only up to *m**k* − 1 because we can randomly remove one result to return to the odd-number case and re-use the above formula for the probability.\nThe single indirect *Z* measurement error *e**I**k*∣1 of a qubit A at level *k* depends on the error of the *X* measurement of qubit B at level *k* + 1 (which we take to be a constant error probability *ε*) and on the errors of the *b**k* + 1 measurements on the qubits at level *k* + 2, denoted *C**i* in Fig. [figgeneration](a). These can either be direct (with error *ε*) or indirect (with error *e**I**k* + 2). Let *n**k* be the number of qubits that are only measured directly (with error *ε*) while *b**k* + 1 − *n**k* qubits are measured indirectly (with error *e**I**k* + 2). *e**I**k*∣1 is thus given by:\n \n$$e\\_{I\\_k|1} = \\sum\\_{n\\_k = 0}^{b\\_{k+1}} \\binom{b\\_{k+1}}{n\\_k} \\mathrm{Pr}(I\\_{Z,k+2}|M\\_{Z,k+2})^{b\\_{k+1}-n\\_k} (1 - \\mathrm{Pr}(I\\_{Z,k+2}|M\\_{Z,k+2}))^{n\\_k} e\\_{n\\_k},$$ \nwhere *e**n**k* is the error in the case of *n**k* direct qubit measurements, and\n \n$$\\mathrm{Pr}(I\\_{Z, k}|M\\_{Z,k}) = \\frac{\\mathrm{Pr}(I\\_{Z,k}) \\mathrm{Pr}(M\\_{Z,k}|I\\_{Z,k})}{\\mathrm{Pr}(M\\_{Z,k})} = \\frac{r\\_k}{\\mathrm{Pr}(M\\_{Z,k})}.$$ \nHere, *I**Z*, *k* denotes a successful indirect *Z* measurement of a photon at level *k* and thus Pr(*I**Z*, *k*) is what is called *r**k* in the main text. Therefore, Pr(*I**Z*, *k*∣*M**Z*, *k*) is the probability that a photon at level *k* was successfully measured indirectly (*I**Z*, *k*) given that it has been successfully measured (*M**Z*, *k*). Note that a photon can be both successfully directly and indirectly measured. In that case, we keep the indirect measurement outcome which should have a smaller error *e**I**k* compared to the intrinsic single qubit error *ε*, thanks to the error correction.\nThe only remaining quantity to determine is the error *e**n**k*. The *X* measurement on qubit B and the *n**k* direct *Z* measurements on the qubits *C**i* are performed with error *ε*, while the remaining *b**k* + 1 − *n**k* indirect *Z* measurements are performed with error *e**I**k* + 2. Note that the indirect measurement outcome is given by the net parity of all the *X* and *Z* measurement outcomes. Thus, the indirect measurement can still yield the correct outcome even if there are errors in the *X* and *Z* measurements, provided there are an even number of errors so that the parity remains unchanged. Only an odd number of errors causes the indirect measurement to fail. Thus:\n \n$$e\\_{n\\_k} = \\sum\\_{i = 0}^{n\\_k+1}{\\left( \\binom{n\\_k+1}{i}{\\epsilon}^i (1- \\epsilon)^{n\\_k+1 - i} \\sum\\_{\\substack{j = 0,\\\\ i+j=1[2]}}^{b\\_{k+1} - n\\_k} \\binom{b\\_{k+1} - n\\_k}{j} {e\\_{I\\_{k+2}}}^j (1- e\\_{I\\_{k+2}})^{b\\_{k+1} -n\\_k - j}\\right) }$$ \nIn this equation, we take into account the fact that only odd numbers of measurement errors contribute to *e**n**k* by restricting the second sum to only values of *i* + *j* = 1[2]. This is different than what was obtained in Ref. , perhaps due to additional approximations in that reference.\n[!ht] [figlogicalmeasurement]\nAs shown in Fig. [figlogicalmeasurement](a), the logical *Z* measurement *M**Z*, ℓ succeeds if all the 1st-level qubits are measured directly or indirectly in the *Z* basis. The logical *X* measurement *M**X*, ℓ succeeds if any of the 1stlevel qubits is measured in the *X* basis and all its corresponding 2ndlevel qubits are successfully measured in the *Z* basis either directly or indirectly, as shown in Ref. . To understand this, let’s denote by *L* the logical qubit that is connected to a set *A* = {*A*1, *A*2, ...} of other qubits, represented by the green area in Fig. [figlogicalmeasurement](b). Because *X**L**Z**A* = *X**L* ⊗ *i**Z**A**i* is a stabilizer of the RGS at the logical level, the logical *X* measurement of qubit *L*, *M**X*, ℓ, has the same outcome as *Z**A*, i.e. the parity of the product of all *Z* measurements on the qubits in *A*. At the physical level, the logical qubit is encoded with a tree graph state where we consider a first-level qubit *B* and its set of neighbor qubits in the tree *C*(*B*). *X**B**Z**C*(*B*)*Z**A* is a stabilizer of the RGS at the physical level and thus *X**B**Z**C*(*B*) (represented by either of the light blue areas in Fig. [figlogicalmeasurement](b)) has the same parity as *Z**A* and therefore constitutes an *X* measurement of the logical qubit *L*.\nThe logical *X* and *Z* errors are thus given by:\n \n$$\\begin{aligned}\n \\bar{e}\\_X & = e\\_{I\\_0}, \\\\\n \\bar{e}\\_Z & = \\sum\\_{n = 0}^{b\\_{0}} \\binom{b\\_{0}}{n} \\mathrm{Pr}(I\\_{1}|Z\\_{1})^{b\\_{0}-n} (1 - \\mathrm{Pr}(I\\_{1}|Z\\_{1}))^{n} e\\_{n}, \\\\\n \\mathrm{with} \\; e\\_{n} & = \\sum\\_{i = 0}^{n}{\\left( \\binom{n}{i}{\\epsilon}^i (1- \\epsilon)^{n - i} \\sum\\_{\\substack{j = 0,\\\\ i+j=1[2]}}^{b\\_{0} - n} \\binom{b\\_{0} - n}{j} {e\\_{I\\_{1}}}^j (1- e\\_{I\\_{1}})^{b\\_{0} -n - j}\\right). }\n \\end{aligned}$$ \nThe derivation of *ē**Z* follows the same idea as *e**I**k*∣1 (for “*k* = − 1”) except that no *X* measurements are needed here.\nFidelity of the full protocol\n-----------------------------\nWe have used the results already derived in the supplementary materials of Ref. to calculate the global fidelity of the generated entangled pair of photons shared by Alice and Bob. The total fidelity *F**A**B* is given by\n \n*F**A**B* = 1 − (*E**x* + *E**y* + *E**z*), \nwith\n \n$$\\begin{aligned}\n E\\_x & = E\\_z = \\frac{1}{4} \\left(1 - (1 - 2 \\epsilon)^{2(N\\_{QR} + 1)} (1 - 2 \\bar{e}\\_X)^{2N\\_{QR}} \\right),\\\\\n E\\_y & = \\frac{1}{4} \\left( 1 + (1 - 2 \\epsilon)^{2(N\\_{QR} +1)} (1 - 2 \\bar{e}\\_X)^{2 N\\_{QR}} -2 (1 - 2 \\epsilon)^{2 (N\\_{QR} + 1)} (1 - 2 \\bar{e}\\_X)^{N\\_{QR}} (1 - 2 \\bar{e}\\_Z)^{(2m - 2) N\\_{QR}} \\right).\n \\end{aligned}$$ \n\n**Supplementary Materials: Resource requirements for efficient quantum communication using all-photonic graph states generated from a few matter qubits**\n=========================================================================================================================================================\nQuantum communication technologies show great promise for applications ranging from the secure transmission of secret messages to distributed quantum computing. Due to fiber losses, long-distance quantum communication requires the use of quantum repeaters, for which there exist quantum memory-based schemes and all-photonic schemes. While all-photonic approaches based on graph states generated from linear optics avoid coherence time issues associated with memories, they outperform repeater-less protocols only at the expense of a prohibitively large overhead in resources. Here, we consider using matter qubits to produce the photonic graph states and analyze in detail the trade-off between resources and performance, as characterized by the achievable secret key rate per matter qubit. We show that fast two-qubit entangling gates between matter qubits and high photon collection and detection efficiencies are the main ingredients needed for the all-photonic protocol to outperform both repeater-less and memory-based schemes.\nThe ability to share entangled states over long distances is a major milestone for the realization of a fully-functional quantum internet . Beyond secure communications via quantum key distribution (QKD) , the implementation of such a quantum internet would also have various applications ranging from distributed quantum computing , secure access to a remote quantum computer , accurate clock synchronization , and improved telescope observations .\nHowever, enabling world-wide quantum communication requires addressing the major problem of photonic losses, which significantly reduces the range of quantum information transfer. Even though direct amplification of a quantum state is made impossible due to the no-cloning theorem , this exponential photon loss can still be overcome through the realization of quantum repeaters (QR) .\n [figboth]\nSeveral QR approaches have been proposed and can be divided into two main categories depending on the method used to propagate quantum information between two adjacent repeater nodes . The first approach proposed, referred to here as a “memory-based” approach, relies on heralded-entanglement generation between two quantum memories (QM) situated at adjacent repeater nodes. Examples of memory-based schemes are depicted schematically in Figs. [figboth](a),(b). The heralded entanglement generally succeeds when a detector situated at an intermediate node measures photons emitted by the QMs . A classical signal carrying the outcome of an entanglement generation attempt must be sent back to the QMs, thus limiting the repetition rate of these protocols. When successful, entanglement swapping transfers the entanglement through these repeater nodes.\nThe second category of repeater relies on logically-encoded multi-photon states , resistant both to photonic losses and errors, to transfer quantum information across a network. One particularly promising approach of this type was put forward in Ref. , which proposed an all-optical QR protocol based on repeater graph states (RGS). An example of an RGS with logical encoding is shown in Fig. [figboth](c), while the corresponding communication protocol is summarized in Fig. [figboth](d) and will be detailed later. Ref. also proposed a method to construct these states probabilistically using single-photon sources, linear optics, and detectors . However, this generation procedure requires about 106 single-photon sources per repeater node to just barely outperform direct fiber transmission . These findings suggest that the RGS protocol may be well beyond the reach of current and near-future technological capabilities.\nHowever, it remains unclear whether the RGS protocol can become more feasible if alternative methods to create the states are used. Ref. proposed a deterministic generation procedure to produce a linear graph state using one quantum emitter which emits spin-entangled photons ; this procedure was later demonstrated experimentally . Ref. showed theoretically that emitters that undergo entangling gates can also be used to generate two-dimensional cluster states. Refs. built on this protocol and demonstrated that entanglement between emitters can be harnessed for the generation of more complex photonic graph states. These ideas have been extended further to general prescriptions and to protocols tailored to specific physical systems . Ref. introduced a protocol for producing an arbitrary-sized RGS using only a few matter qubits, thus significantly decreasing the resource overhead required for RGS generation, making it deterministic in principle. Using this generation technique, the RGS protocol only necessitates a few-qubit processor at each repeater node, which would ease its practical implementation compared to other error-correction-based proposals that generally require several hundreds of qubits per repeater node (with the notable exception of Ref. which uses techniques introduced in Ref. and requires deterministic spin-photon Bell measurements). Although the RGS protocol with deterministic state generation seems promising, a systematic and detailed evaluation of its performance and resource requirements has not been carried out.\nIn this paper, we compare the resource-efficiency—characterized by the achievable secret key rate per matter qubit—of this protocol to direct fiber transmission and to QR schemes based on memories and heralded entanglement generation. We first show that the rate per matter qubit has a fundamental upper bound in the case of memory-based QRs. We then review the RGS protocol and how RGSs can be generated using a few matter qubits. We evaluate the performance of this scheme, show that its rate per matter qubit does not have a theoretical upper-bound, and find the conditions under which it outperforms both the repeater-less and the memory-based QR approaches. These conditions depend sensitively on the speed with which two-qubit gates between the matter qubits can be executed and on the collection and detection efficiencies of the photons emitted by these matter qubits.\nUpper bound on rate for memory-based repeater schemes\n=====================================================\nIn this section, we show that there is a theoretical upper bound, *R*max(*Q**M*), on the rate per matter qubit for protocols based on quantum memories and heralded entanglement generation. In such protocols, the total distance *L* between Alice and Bob is divided into smaller distances *L*0 by *N**Q**R* = *L*/*L*0 − 1 repeater nodes. Quantum memories at adjacent repeater nodes are entangled via a heralded entanglement procedure (see Fig. [figboth](a)). When a repeater node shares entanglement connections with its two adjacent nodes, entanglement swapping is performed on the two memory qubits within that node to create a direct entanglement connection between memories on the adjacent nodes. This procedure can be repeated until Alice and Bob share an entangled qubit pair.\nIt is clear that creating an entangled pair between Alice and Bob requires generating an entanglement connection between each adjacent pair of repeaters. This means that the overall protocol rate *R*(*Q**M*) is limited by the entanglement generation rate ⟨*T*ent⟩− 1 between two adjacent repeaters. Here, we use this fact to derive an upper bound on the rate per matter qubit for QR protocols that use quantum memories and heralded entanglement generation.\nTo determine an upper bound on ⟨*T*ent⟩− 1, we focus for concreteness on the protocol presented in Fig. [figboth](a), which uses two quantum memories per node. A quantum memory should emit a photon that is maximally entangled with one of its degrees of freedom. Two photons generated at adjacent repeater nodes arrive at the same measurement node situated halfway between the two repeaters, where they are measured in a Bell state basis. Because a photon Bell state measurement using only linear optics succeeds with probability at best 1/2 (without ancillary qubits or QND measurements ), the overall success probability of the distant heralded entanglement generation is *P*ent ≤ 1/2. It is worth mentioning that a method for achieving heralded entanglement generation with higher success probability has been proposed , but its efficacy is restricted to qubits separated by a short distance, so we exclude this from our analysis (see Supplementary Materials for more details). A classical signal must then inform the repeater nodes of the success or failure of the Bell state measurement. Because the distance from the measurement node to the repeater nodes is *L*0/2, this means that the overall heralded entanglement generation attempt takes total time *T*trial ≥ *L*0/*c*. This includes the time *L*0/(2*c*) for the single-photon transfer from the repeater to the measurement node and also the time *L*0/(2*c*) for the classical signaling in the opposite direction. Here, we are neglecting the time it takes to prepare and pump the quantum memories. Throughout this protocol, a QM can be maximally entangled with at most one other qubit (either a photon or another QM). Therefore, it cannot emit another spin-entangled photon before receiving the classical signal carrying the information about the success or failure of the Bell measurement, hence limiting the repetition rate of the protocol. To generate an entanglement connection between the two adjacent nodes, the procedure must be repeated on average *P*ent− 1 times. The entanglement generation rate is therefore ⟨*T*ent⟩− 1 = *P*ent/*T*trial < *c*/(2*L*0). We derived this result for a specific heralded entanglement generation protocol but it also holds for all known protocols (see Supplementary Materials).\n |p1.6cmp7.0cm| Notation & Definition \n*L* & Total distance between Alice and Bob. \n*L*0 & Distance between adjacent nodes. \n*N**Q**R* & Number of repeater nodes. \n2*m* & Number of arms of the RGS. \n*b⃗* & Branching vector of an error-correction tree (*b⃗* = (*b*0, *b*1, …, *b**n* − 1)). \n*T**C**Z* & *C**Z* gate time. \n*η**t*(*l*) & Transmission of a fiber of length *l* (*η**t*(*l*) = exp( − *l*/*L*att)). \n*η**c*, *η**d* & In-fiber collection and detection efficiencies of photons. \n*L*att, *c* & Attenuation distance of the fiber and speed of light in a fiber. ($L\\_{\\mathrm{att}} \\approx 20 \\kilo\\meter$) \n*t*att & Average time of flight of photons in the fiber (*t*att = *L*att/*c*). \n*T**R**G**S* & Generation time of an RGS. \n*R*, *R**m* & Rate and rate per matter qubit of the protocol. \n*ε* & Single-photon error rate. \n[table3]\nFrom these results, we can show that the rate per matter qubit (where the number of matter qubits is *N**m* = 2(*N**Q**R* + 1) = 2*L*/*L*0) has a theoretical upper bound, *R*max(*Q**M*):\n \n$$\\frac{R^{(QM)}}{N\\_m} \\leq \\frac{\\langle T\\_{\\mathrm{ent}} \\rangle^{-1}}{N\\_m} \\leq \\frac{c}{4L} =R\\_{\\mathrm{max}}^{(QM)}.$$ \nThis theoretical upper bound also holds if there are more than two QMs at each repeater node (see Fig. [figboth](b)) as the rate would linearly increase with the number of matter qubits. Therefore, we have derived a general theoretical upper bound for memory-based protocols based on heralded entanglement generation. It is worth noting that the fundamental reason for this upper bound comes from the need for classical signaling in these protocols. Such classical signaling is not required for RGS protocols, enabling them to surpass this limit, as we show below. In the Supplementary Materials, we also show that a tighter bound, $R\\_{\\rm max}^{(QM)}=c/7L$, can be obtained for memory-based schemes in which there are two quantum memories per repeater node, and heralded entanglement swapping is used.\nWe emphasize that the upper bound derived in this section holds for QR protocols that are based on quantum memories and distant heralded entanglement generation. This corresponds to the first and second generations of QRs, as categorized in Ref. . Consequently, in the following, the RGS-based protocol will be compared only to these categories of QRs, for which the performance is limited by classical signaling.\nRate of the RGS protocol with deterministic graph state generation\n==================================================================\nIn this section, we review the RGS protocol as introduced in Ref. and the deterministic generation of RGSs using a few matter qubits as proposed in Ref. . We show how the rate of the RGS protocol depends on various parameters in the case where deterministic state generation methods are used.\nRGS protocol and rate\n---------------------\nAn RGS is a quantum state $\\ket{G}$ that can conveniently be represented in the form of a graph *G* = (*V*, *E*) with *V* vertices and *E* edges. Each vertex corresponds to a photonic qubit prepared in the $\\ket{+}$ state, and each edge corresponds to the application of a *C**Z* gate between the two qubits it connects:\n \n$$\\ket{G} = \\prod\\_{(i,j) \\in E} CZ\\_{ij} \\ket{+}^{\\otimes V}.$$ \nAn example of the graph representing an RGS is shown in Fig. [figboth](c). These states include 2*m* inner photonic qubits that are referred to as the first-leaf qubits. All the first-leaf qubits are fully connected to each other and each of them is also connected to one additional qubit, referred to as a second-leaf qubit. The first-leaf qubits are logically-encoded using tree graph states; further details on this are given below.\nIn an RGS protocol (see Fig. [figboth](d)), the distance *L* separating Alice and Bob is also divided into smaller steps *L*0 by *N**Q**R* = *L*/*L*0 − 1 source nodes where the RGSs are created. The RGS is divided into two equal parts, each containing *m* arms, and one part is sent to the left adjacent measurement node and the other to the right. Thus, half of one RGS meets half of another RGS at each measurement node, where each second-leaf qubit from one of the half-RGSs undergoes a Bell measurement with its counterpart from the other half-RGS. Further details about this entanglement swapping procedure at the measurement node are given later, but it is important to note that the RGS protocol does not use quantum memories at all and thus cannot store the information. This implies that an entanglement connection between Alice and Bob should be realized in only one trial with a probability *P**A**B* much higher than direct fiber transmission: *P**A**B* ≫ *η**t*(*L*). The generation of an entanglement connection between Alice and Bob requires the realization of successful entanglement connections between all the adjacent RGSs. It is important to note that the measurements performed at each measurement node do not require information from other measurement nodes so that, in contrast to memory-based approaches, the RGS protocol does not require any classical signaling while entanglement is being extended through the network. Classical signaling is needed only once at the end of the protocol to recover the Pauli frame of the Bell pair shared by Alice and Bob, i.e. to determine which local Pauli rotations Alice and Bob’s qubits should undergo. This means that in the RGS protocol, it is not necessary to wait for any classical signaling before proceeding to generate the next batch of RGSs needed to create the next Bell pair shared between Alice and Bob (see Supplementary Materials for more details). Consequently, unlike the memory-based scheme, the rate of the RGS protocol does not depend on the time it takes for the photon to get from one node to the next. It is limited only by the generation time *T**R**G**S* of an RGS:\n \n$$R^{(RGS)} = \\frac{{P\\_{RGS \\leftrightarrow RGS}}^{L/L\\_0}}{T\\_{RGS}},\n \\label{eq\\_rate}$$ \nwith *P**R**G**S* ↔ *R**G**S* the probability to generate an entanglement connection between two RGSs. The main notations used in this work are defined in Table [table3].\n[!ht] [figgeneration]\nA successful entanglement link can be generated if at least one of the Bell measurements at a measurement node succeeds. In that case, the two first-leaf qubits attached to the second-leaf qubits that underwent the successful Bell measurement are measured in the *X* basis, while the remaining 2*m* − 2 first-leaf qubits are measured in the *Z* basis. The *X* measurements transfer the entanglement connection to the next two adjacent measurement nodes, while the *Z* measurements disentangle all the excess qubits associated with failed Bell measurements. All these first-leaf qubit measurements must be successful in order to reliably create an entanglement link. Therefore, the probability to successfully create an entanglement link between two RGSs is given by:\n \n$$\\begin{aligned}\n P\\_{RGS \\leftrightarrow RGS} & = \\left(1 - \\left(1 - P\\_{\\mathrm{Bell}}\\right)^m\\right) \\\\\n & \\times \\mathrm{Pr}(M\\_{X,\\ell})^2 \\mathrm{Pr}(M\\_{Z,\\ell})^{2m-2},\\label{eq:PRGStoRGS}\n \\end{aligned}$$ \nwhere *P*Bell = *P*ph2/2 is the probability of a successful Bell measurement. This depends on *P*ph = *η**c**η**d**η**t*(*L*0/2), which is the probability that a single photon is emitted and collected into the fiber (*η**c*), is transmitted to the measurement node (*η**t*(*L*0/2)) and detected (*η**d*). Pr(*M**X*, ℓ) and Pr(*M**Z*, ℓ) are the probabilities that the logical *X* and *Z* measurements on the first-leaf qubits succeed. Note that if the first-leaf qubits were not logically encoded, we would have Pr(*M**X*)2Pr(*M**Z*)2*m* − 2 = *P*ph2*m* ≤ *η**t*(*L*0), and it would be impossible to have an advantage over direct fiber transmission. Therefore, the loss-tolerance of logically-encoded qubits is crucial for this protocol. Next, we review this encoding, which was introduced in Refs. and .\nLoss-tolerance with tree graph states\n-------------------------------------\nWe review how the probabilities Pr(*M**X*, ℓ) and Pr(*M**Z*, ℓ) depend on the single-photon transfer probability *P*ph and on the shape of the tree graph state used for the logical encoding. Ref. demonstrated that this encoding remains loss-tolerant as long as *P*ph is above 50%.\nWe consider the calculation of the probabilities of successful measurements of the logically encoded qubits in the presence of loss errors on a tree graph state. A tree is characterized by its branching vector *b⃗* = (*b*0, *b*1, ..., *b**n* − 1) (see Fig. [figboth](c)), which describes the connectivity between the different levels of the tree. To perform a *Z* measurement *M**Z*, *k* on a qubit at level *k*, it is possible to either perform a direct measurement on this qubit (with success probability *P*ph) or, if it fails (with probability 1 − *P*ph), perform an indirect measurement (with probability *r**k*). Thus, the overall success probability of a *Z* measurement at level *k* is:\n \nPr(*M**Z*, *k*) = *P*ph + (1 − *P*ph)*r**k*. \nTo perform an indirect measurement on a qubit (call it A) at level *k*, one can use the stabilizing property of a graph state . It is possible to deduce the outcome of the *Z* measurement on A by performing an *X* measurement on another qubit (B) at level *k* + 1 and a *Z* measurement on all the qubits, *C**i*, that are in the neighborhood of B at level *k* + 2 (see Fig. [figgeneration](a)). This works because of the invariance of graph states when they are acted upon by their stabilizers:\n \n$$\\begin{aligned}\n \\ket{G}& = X\\_B {\n \\mathbin{\\mathop{\\otimes}\\limits\\_{j \\in N(B)}}\n } Z\\_j \\ket{G} \\\\\n & = X\\_B \\otimes Z\\_A {\n \\mathbin{\\mathop{\\otimes}\\limits\\_{i\\in \\{1, b\\_{k+1}\\}}}\n } Z\\_{C\\_i} \\ket{G},\n \\end{aligned}$$ \nso we have:\n \n$$Z\\_A \\ket{G} =X\\_B {\n \\mathbin{\\mathop{\\otimes}\\limits\\_{i\\in \\{1, b\\_{k+1}\\}}}\n } Z\\_{C\\_i} \\ket{G}.$$ \nA single indirect measurement has a success probability *s**k*. Note, however, that the tree structure allows *b**k* indirect measurement attempts, and only one needs to succeed to indirectly measure a qubit at level *k*. So the probability that at least one indirect measurement succeeds is\n \n*r**k* = 1 − (1 − *s**k*)*b**k*, \nwith\n \n*s**k* = *P*phPr(*M**Z*, *k* + 2)*b**k* + 1. \nIt is thus possible to derive the success probability of a measurement recursively, given that the qubits at the lowest level can only be measured directly: Pr(*M**Z*, *n*) = *P*ph. Logical measurements in the *X* or *Z* basis are given by :\n \n$$\\begin{aligned}\n \\mathrm{Pr}(M\\_{X,\\ell}) & = r\\_0, \\\\\n \\mathrm{Pr}(M\\_{Z,\\ell}) & = \\left(P\\_{\\mathrm{ph}} + (1 - P\\_{\\mathrm{ph}}) r\\_1 \\right)^{b\\_0} \\\\\n & = \\mathrm{Pr}(M\\_{Z,1})^{b\\_0}.\n \\end{aligned}$$ \nIt is interesting to note that logical encoding with tree graph states can also correct single-qubit errors, as shown in Ref. and described in the Supplementary Materials. This will be used later when we evaluate the sensitivity of the RGS protocol to errors.\nGeneration of an RGS\n--------------------\nThe achievable rate between Alice and Bob also depends on the repetition rate of the protocol, which is given by the generation time of the RGS. Because it is impossible to realize deterministic two-qubit gates on photons with linear optics, such a graph state can either be generated probabilistically by the recursive fusion of smaller graphs using linear optics and Bell state measurements as shown in Ref. and , or deterministically using a few matter qubits as shown in Ref. . We now review the latter.\nAn arbitrary-sized RGS can be generated deterministically by following a given sequence based on four operations on matter qubits: the emission of a photon maximally entangled with the matter qubit *E*ph, the Hadamard gate *H*, measurements in the Pauli bases *M**X*, *M**Y*, *M**Z* and the *C**Z* gate. The generation of an RGS with 2*m* arms and a tree graph encoding with branching vector *b⃗* = (*b*0, *b*1, ..., *b**n* − 1) requires *n* + 1 matter qubits *Q*1,..., *Q**n* + 1, and is given by the sequence (see also Fig. [figgeneration](b))\n \n$$\\begin{aligned}\n & M\\_{Y,{Q\\_1}}(M\\_{X,Q\\_3} M\\_{X,Q\\_2} CZ\\_{Q\\_1,Q\\_3} CZ\\_{Q\\_2,Q\\_3} E\\_{\\mathrm{ph},Q\\_3} {G\\_3}^{b\\_0})^{2m} \\\\\n& \\mathrm{with} \\; G\\_{k} = M\\_{Z,Q\\_{k}} H\\_{Q\\_{k}} E\\_{\\mathrm{ph},Q\\_{k}} CZ\\_{Q\\_{k-1}, Q\\_{k}} {G\\_{k+1}}^{b\\_{k-2}} \\\\\n& \\mathrm{and} \\; G\\_{n+ 2} = E\\_{\\mathrm{ph},Q\\_{n+ 1}},\n \\end{aligned}$$ \nwhere, for simplicity, we have omitted the single photonic qubit rotations.\nThe overall generation time of an RGS using this procedure is therefore\n \n$$\\begin{aligned}\n T\\_{RGS} & = 2m \\left(1 + f(\\vec{b}, n-1)\\right) T\\_{E\\_{\\mathrm{ph}}} +T\\_M\\\\\n & + 2m \\left(2 + f(\\vec{b}, n-2)\\right) (T\\_M + T\\_{CZ}) \\\\\n & + 2m f(\\vec{b}, n-2) T\\_{H}, \\\\\n \\end{aligned}$$ \nwith *f*(*b⃗*, *k*) = ∑*i* = 0*k*∏*j* = 0*i**b**j* and with *T**E*ph, *T**M*, *T**H* and *T**C**Z* the times for photon emission, matter qubit measurement, Hadamard and *C**Z* gates, respectively.\nIn the following, we make the realistic assumption that the *C**Z* gate time *T**C**Z* is much longer than the durations of the other operations, and so we set *T**H* = *T**M* = *T**E*ph = 0 for simplicity. With this assumption, the generation time *T**R**G**S* only depends on the number of *C**Z* gates and their duration:\n \n*T**R**G**S* = 2*m*(2 + ∑*k* = 0*n* − 2∏*j* = 0*k**b**j*)*T**C**Z*. \nThis is for an RGS with 2*m* arms and a logical tree encoding with branching vector *b⃗* = (*b*0, ..., *b**n* − 1). In the following, we will assume a depth-two tree graph state (*b⃗* = (*b*0, *b*1)), so that the full RGS can be generated with only three matter qubits (*n* = 2). The total number of matter qubits in the network is therefore *N**m* = (*L*/*L*0 − 1)(*n* + 1) = 3(*L*/*L*0 − 1).\nPerformance comparisons\n=======================\nWe now evaluate the performance of the RGS protocol when it is generated deterministically by a few matter qubits. We compare our results to the direct fiber transmission limit derived by Refs. and to the memory-based upper bound *R*max(*Q**M*) = *c*/4*L* found in Sec. [secQM].\nOptimizing the RGS protocol\n---------------------------\n [figoptimizationtreeRGS]\n |p1.2cm|p1.2cm|p0.6cm|p0.6cm|p0.6cm|p1.8cm| *L*/*L*att & *L*0/*L*att & *m* & *b*0 & *b*1 & $R\\_{\\rm max}^{(RGS)} T\\_{CZ}$ \n10 & 0.23 & 11 & 8 & 4 & 2.4 × 10− 5 \n25 & 0.21 & 13 & 10 & 5 & 6.8 × 10− 6 \n50 & 0.19 & 14 & 10 & 5 & 2.8 × 10− 6 \n100 & 0.17 & 15 & 10 & 5 & 1.1 × 10− 6 \n150 & 0.15 & 16 & 10 & 5 & 6.8 × 10− 7 \n[table]\nIn the following, we show how the RGS protocol parameters can be optimized to maximize the overall rate per matter qubit, *R**m*(*R**G**S*), or in the presence of errors, the secret key rate per matter qubit. For the moment, we assume perfect photon collection and detection efficiencies (*η**c**η**d* = 1) and no single-photon errors (*ε* = 0); we take these effects into account later on. From Eqs. ,, and , the achievable rate per matter qubit *R**m*(*R**G**S*) of the RGS protocol for a total distance *L* is inversely proportional to the *C**Z* gate time *T**C**Z*, but it depends non-linearly on the separation distance, *L*0, between two RGS nodes and the RGS shape (number of arms 2*m* and the tree branching vector *b⃗* = (*b*0, *b*1)). Therefore, for each choice of the total distance *L*, there is a certain node separation and RGS shape that maximize the achievable rate.\nThe optimization of the rate per matter qubit for a total distance *L* = 50*L*att ($\\approx 1000\\; \\kilo\\meter$) is shown in Fig. [figoptimizationtreeRGS]. The position of each point corresponds to a specific RGS shape, the color of the point indicates the node separation *L*0 that optimizes the rate for that shape, and the size of the point represents the maximal rate per matter qubit for these parameters. This optimization converges, allowing us to extract the optimal RGS shape and distance *L*0 for this particular choice of the total distance *L*. The optimization can be repeated for various choices of *L*, and the extracted optimal parameters are recorded in Table [table].\nWe compare the RGS protocol to direct fiber transmission in Fig. [figoptimizationL](a). The maximum achievable rate per matter qubit for the RGS protocol, *R*max(*R**G**S*), is shown as a function of the total distance $L/L\\_{\\rm att}$. In the case of direct transmission, we show seven different curves corresponding to the achievable rate for seven different values of the single-photon source repetition rate. We see that the RGS protocol outperforms direct transmission with the highest repetition rate for $L\\gtrsim30L\\_{\\rm att}$. In the same figure, we also show how well the protocol works if we keep the optimized parameters fixed and change the total distance *L*. To demonstrate this, we fix the RGS parameters and node separation *L*0 to the values that optimize the rate for a total distance *L*tar. We then adjust *L* away from *L*tar without changing the RGS parameters or *L*0, and we calculate the new rate for each value of *L*. In Fig. [figoptimizationL](a), we show the resulting rates as a function of *L*/*L*0 for five different choices of *L*tar. We see that the RGS protocol continues to work well over a broad range of total distance *L* when we use parameters that are optimized for a large total distance *L*tar.\n [figoptimizationL]\nFor long distances *L* in the range 50*L*att − 200*L*att, *R*max(*R**G**S*) scales approximately as *L*− 1.27, and thus the scaling with distance is slightly worse than the upper bound scaling for memory-based schemes, *R*max(*Q**M*) ∼ *L*− 1, obtained in Sec. [secQM]. However, contrary to the memory-based schemes, there are no fundamental upper limits imposed by classical signaling on the maximal achievable rate *R*max(*R**G**S*), as the latter is inversely proportional to the *C**Z* gate time, *T**C**Z*. If *T**C**Z* is sufficiently small, then the RGS protocol should surpass the upper bound of memory-based schemes. This is illustrated in Fig. [figoptimizationL](b), where the red region of the plot indicates the regime where the RGS protocol outperforms memory-based schemes.\nFrom our calculations, it seems that the maximum achievable rate per matter qubit, *R**m**a**x*(*R**G**S*), is in principle unbounded since it is inversely proportional to the *C**Z* gate time. We should recall however that these results are based on the assumption that the *C**Z* gate takes much longer than the other operations *O* made on the matter qubits: *T**C**Z* ≫ *T**O*. Decreasing *T**C**Z* initially increases the rate, but if *T**C**Z* becomes small enough, neglecting the durations of other operations eventually becomes invalid. Consequently, to increase the rate further, not only the *C**Z* gate time but all the operation times should be reduced simultaneously. If the durations of all these operations could be made arbitrarily small, then the rate per matter qubit would increase to infinity. In practice however, the operation times could also have an intrinsic lower bound that limits the performance of the RGS protocol, but these lower bounds would depend on the specific system on which the protocol is implemented, while the limit imposed by classical signaling is much more stringent and general.\nFor distances below $200L\\_{\\mathrm{att}} \\approx 4000\\;\\kilo\\meter$, a gate time *T**C**Z* below $6\\times10^{-4} t\\_{\\mathrm{att}} \\approx 60\\;\\nano\\second$ is sufficiently short to outperform any memory-based scheme. For this range of total distance, the optimal value of the node separation *L*0 ranges from $0.15-0.19 L\\_{\\mathrm{att}} \\approx 3-3.8\\;\\kilo\\meter$. As an illustration of the RGS performance, for $L = 1000 {\\rm \\; km}$ and $T\\_{CZ} = 10 {\\rm \\; ns}$, the total rate is $R = 220 {\\rm \\; kHz} $ for *N**m* = 786 matter qubits used, leading to $R\\_{\\rm max}^{(RGS)}=276 {\\rm \\; Hz}$ per matter qubit, while $R\\_{\\rm max}^{(QM)} = 50 {\\rm \\; Hz}$ per matter qubit for this distance.\nSensitivity to errors\n---------------------\nSo far, we have considered the generation of a perfect RGS, which is a pure entangled state of many photons. Generating such a perfect state is not feasible experimentally, and so we need to evaluate the sensitivity of the RGS protocol to errors. To do so, we now consider single-photon loss and single-photon errors in our optimization process.\nThe single-photon loss (other than the fiber losses) depends on the probability that an emitted photon is neither collected into the fiber nor detected; this corresponds to the case where *η**c**η**d* ≠ 1. Fig. [figoptimizationL](c) shows the optimized rate per matter qubit for different values of *η**c**η**d*. Similarly, Fig. [figoptimizationL](d) compares the performance of the RGS protocol with the upper bound for memory-based protocols, *R*max(*Q**M*), as a function of *T**C**Z* and *η**c**η**d* for a total distance *L* = 50*L*att. These results show that the photon losses must be below 15% (*η**c**η**d* > 85%) in order for the RGS protocol to outperform both the direct fiber transmission and memory-based protocols. This rather stringent requirement is mainly due to the loss-tolerance of the tree graph state, which only works when the total photon loss (including fiber losses) is below 50%.\nApart from the photon losses, the RGS protocol also depends on other kinds of errors that reduce the fidelity *F**A**B* of the final entangled photon pair shared by Alice and Bob. These errors include, but are not limited to, single-photon measurement errors, photon Bell-measurement errors, and depolarization errors. In the present case where the RGS is generated using matter qubits, the limited coherence time of the matter qubits should also limit the fidelity of the entangled photons. For QKD applications, the final secret key rate depends on the total rate *R* and the fidelity *F**A**B* :\n \n*R**s**k**r* = *R*(1 − 2*h*(*F**A**B*)), \nwhere *h* is the binary entropy function:\n \n*h*(*F*) = − *F*log2(*F*) − (1 − *F*)log2(1 − *F*). \nIn the remainder of this paper, we now use *R**m*(*R**G**S*) to refer to the secret key rate per matter qubit, which coincides with the rate per matter qubit in the absence of errors. We model these imperfections by using a single-qubit error probability *ε* that affects all the photonic qubits. This model works well for qubit measurement errors as well as depolarization errors. In Ref. , it was also shown that decoherence errors on quantum emitters induce only local errors in the emitted cluster state. The detailed derivation of *F**A**B* is given in the Supplementary Materials. Fig. [figoptimizationL](e) compares the optimized secret key rate of the RGS protocol against that of direct fiber transmission as a function of the single-qubit error probability and the total distance *L*. Fig. [figoptimizationL](f) compares the performance of the RGS protocol to the upper bound for memory-based protocols for *L* = 50*L*att. We find that the single-qubit error should be below *ε* < 10− 4 to outperform both the direct fiber transmission and the memory-based protocol upper bound, $R^{(QM)}\\_{\\rm max}$.\nOne might expect a better tolerance to single-qubit errors since they are also corrected by the logical encoding of the 1st-leaf qubits. This is not the case because the 2nd-leaf qubits are not logically encoded and thus are very sensitive to errors. The dependence on single-qubit errors is actually similar to the protocols using heralded entanglement generation. It is also possible to use entanglement purification to improve the fidelity *F**A**B* and maximize the secret key rate.\nFrom the maximum single-qubit error rate, we can derive an order-of-magnitude estimate for the coherence time requirements for the matter qubits. To do so, we use two strong assumptions. We first assume that the fidelity *F**R**G**S* of an RGS depends on the ratio between the generation time *T**R**G**S* of the RGS and the coherence time *T*2 of the matter qubits:\n \n$$F\\_{RGS} \\approx e^{- \\frac{3 T\\_{RGS}}{T\\_2}}$$ \nThe coefficient 3 corresponds to the number of matter qubits that degrade the RGS fidelity. Here, we have made the simplifying assumption that the main source of infidelity is the matter qubit decoherence, and we have assumed that the effect of this decoherence is to reduce the overall fidelity of the graph state by *e*− *T**R**G**S*/*T*2, which decays exponentially with the time during which the qubit is attached to the RGS (*T**R**G**S*) divided by its coherence time. This assumption may or may not be justified depending on the system used to implement the protocol, but we use it simply to get an order of magnitude estimate of the typical coherence time that is required for this protocol. We also assume that the errors are homogeneously distributed over the *N*ph photons of an RGS, such that *F**R**G**S* = (1 − *ε*)*N*ph. This assumption is consistent with Ref. , which shows that errors generated by a quantum emitter and imprinted onto the generated graph state are localized. With these assumptions, the coherence time should satisfy *T*2 ≥ 2500*T**C**Z*.\nDiscussion\n==========\n |p1.8cm|p1.2cm|p1.0cm|p1.2cm|p1.1cm| **Matter qubit** & $\\bm{T\\_{CZ}}$ & & $\\bm{T\\_2}$ & $\\bm{\\beta}$ \nCoupled QDs & & & & \ndefects (diamond) & & & & \nTrapped ions & & & & \n[table2]\nIn the previous section, we derived a set of requirements that need to be fulfilled in order for the RGS protocol to have a clear advantage over both memory-based QRs and repeater-less protocols:\n1. $T\\_{CZ}\\leq T\\_{lim}(L) \\approx 60 \\nano\\second$,\n2. *T*2 ≥ 2500*T**C**Z*,\n3. *η**c**η**d* > 85%.\nIt is important to note that these bounds were obtained using RGSs with a depth-2 logical tree encoding. We may expect an increase in the rate per matter qubit if deeper trees were used but the improvement would certainly not be large as *T**R**G**S* grows quickly with tree depth. Checking this directly is challenging because the process of optimizing the RGS protocol becomes significantly more complex. We leave this intensive numerical analysis to future work.\nTable [table2] shows the current state-of-the-art for three candidate systems that can be used to generate RGSs. So far, the characteristics of the current systems are not good enough yet to exceed the memory-based resource-efficient upper bound *R*max(*Q**M*). The main limitation comes from long two-qubit gate times, since fast and high-fidelity gates are critical for the performance of the RGS protocol. Currently, most of the two-qubit gates that have been experimentally demonstrated are slow because they rely on local interactions, such as the hyperfine coupling, which are generally weak. Efficient light-matter interfaces using cavities can also be used to realize deterministic, photon-mediated *C**Z* gates between two matter qubits , using spin-photon gates . The crucial advantage of this technique is that it only requires optical interactions, which are intrinsically much faster than hyperfine interactions. Progress in the experimental realization of strong spin-dependent phase shifts has been made recently . The second condition (ii) will also be easier to achieve if the *C**Z* gate time is reduced.\nThe third condition (iii) requires the efficient collection of single photons emitted by a single quantum emitter. The collection of single photons needs to be facilitated by increasing the emission of photons into a single optical mode. This is generally realized using waveguides or cavities with single-mode emission probabilities *β* that can reach 99.7% . This single mode should then be efficiently coupled to a single-mode fiber. Finally, superconducting photon detectors can be used to achieve high detection efficiencies .\nIn conclusion, we have identified the main requirements that must be met in order for all-photonic repeater protocols based on deterministically generated graph states to outperform memory-based protocols. We did this by finding the graph state structures that maximize the secret key rate and by comparing this to an upper bound on the rate for memory-based schemes that we derived. We found that two-qubit gate times and the efficient collection and detection of emitted photons are the most important factors in determining whether the all-photonic approach is superior to memory-based ones. An interesting future direction would be to improve the protocol for the emitter-based generation of photonic graph states by incorporating atom-photon *C**Z* gates, which would both simplify the procedure and decrease gate times, and thus enhance the performance of deterministic, all-photonic repeaters.\nAcknowledgements\n================\nWe thank Yuan Zhan and Shuo Sun for their contribution in solving issues in the code. This research was supported by the NSF (Grant No. 1741656) and by the EU Horizon 2020 programme (GA 862035 QLUSTER).\nMethods\n=======\nThe numerical model used for the results in this article is available here:\n<https://github.com/Paulhilaire/performance_repeater_graph_state>.\nBounds for memory-based repeater protocols\n==========================================\nAs discussed in the main text, the rate of a memory-based quantum repeater based on heralded entanglement generation is limited by the rate ⟨*T**Q**M*⟩− 1 at which a given memory qubit can be re-used. ⟨*T**Q**M*⟩ is given by the sum of the average time required for generating entanglement ⟨*T*ent⟩ and the time during which the entanglement is stored ⟨*T*store⟩. ⟨*T*ent⟩ is given by *T*trial/*P*ent, where *T*trial is the time required for a heralded entanglement generation attempt, and *P*ent is the probability that an attempt is successful. Let’s consider three heralded entanglement procedures :\n* In Ref. , *T*trial = *L*0/*c*, and the probability is *P*ent = *e*− *L*0/*L*att/2 (assuming perfect photon emission and detection probability).\n* In Ref. , *T*trial = 2*L*0/*c*, and the probability is *P*ent = *e*− *L*0/*L*att.\n* In Ref. , *T*trial = *L*0/*c*, and the probability is *P*ent ∝ *e*− *L*0/(2*L*att). However, the protocol needs to be implemented in a regime where the probability to emit a photon is very small to avoid two-photon emission, which degrades the entanglement fidelity. Consequently, *P*ent ≪ 1/2.\nThus in all these cases, if we consider small values of *L*0 ≪ *L*att, we have:\n \n$$\\langle T\\_{\\mathrm{ent}} \\rangle \\geq \\frac{2L\\_0}{c} e^{L\\_0/L\\_{\\mathrm{att}}} \\geq \\frac{2L\\_0}{c}.$$ \nThis limitation can be understood from the fact that the photonic qubit and the classical signal need to travel back and forth either to a measurement node (twice the distance *L*0/2) or to the next QR (twice the distance *L*0). If the photon travels to a quantum repeater, it is possible to deterministically entangle the QM and the photon, so the entanglement probability is only bounded by the photonic losses. If the photon travels to an intermediate node, the entanglement generation procedure uses linear optics and has a success probability that cannot reach more than 1/2 (unless ancillary photonic qubits are used, but this requires another quantum emitter to produce them ).\nFrom this limitation, we find the upper bound *R*max(*Q**M*) ≥ max(*R*/*N**m*) in the case of QR protocols with two QMs. In this case, the rate *R* is limited by ⟨*T**Q**M*⟩− 1 ≤ ⟨*T*ent⟩− 1 = *c*/(2*L*0). The number of matter qubits is given by twice the number of repeaters plus Alice and Bob’s matter qubits, thus *N**m* = 2(*N**Q**R* + 1) = 2*L*/*L*0, therefore:\n \n*R*max(*Q**M*) = *c*/4*L*. \nIn the case of multiplexed repeaters, by multiplying the number of memories per repeater by *N*, the entangling rate between two repeaters is bounded by *N**c*/(2*L*0), but the number of matter qubits is now *N**m* = 2*N**L*/*L*0, so that the total rate is still bounded by *R*max(*Q**M*).\nRef. also proposes a method to generate heralded entanglement deterministically by detecting photons. The idea is to excite the two quantum emitters (each consisting of two ground states and one excited state that couples to only one of the ground states) such that they both emit a photon whose which-path information is erased using linear optics. When the first photon is detected a signal is sent to each quantum emitter to perform a rotation between the ground states. This rotation needs to be performed before the second photon is emitted. Therefore, the latency between the photon detection and the rotation should be much smaller than the lifetime of the quantum emitter’s excited state. A direct implication of this is that the photon detectors should be placed very close to the quantum emitters, which is impractical. For example, for quantum emitters with a lifetime under $10 {\\rm \\; ns}$ (which is typical for QDs and defects in diamond), the position of the detectors should be much smaller than $1 {\\rm \\; m}$. In addition, to obtain high fidelities with this scheme, the two quantum emitters are excited for a typical time *T*1 which should be much larger than the classical signaling time *L*0/*c*, and thus the previous analysis should still hold in that case.\nIn the case of memory-based schemes using heralded entanglement and 2 QMs per QR, it is possible to find a stricter upper bound. To derive this bound, consider two repeater nodes labeled 1 and 2. Node 1 contains two memory qubits QM1 and QM1ʹ, while node 2 contains QM2 and QM2ʹ. Suppose that we succeed in entangling QM1 and QM2 via a Bell measurement at an intermediate measurement node. How long do we have to maintain this entanglement before further entanglement swappings at neighboring nodes free up QM1 and QM2, allowing them to be re-used for creating the next Bell pair to be shared between Alice and Bob? The answer to this question is what we called $\\langle T\\_{\\rm store}\\rangle$ above, the average amount of time we have to store the entanglement. The upper bound on the rate is then $\\langle T\\_{\\rm QM}\\rangle^{-1}$, where $\\langle T\\_{\\rm QM}\\rangle=\\langle T\\_{\\rm ent}\\rangle+\\langle T\\_{\\rm store}\\rangle$. We obtained a bound for $\\langle T\\_{\\rm ent}\\rangle$ above and used this to derive the bound on the rate given in Eq. . There, we neglected the role of $\\langle T\\_{\\rm store}\\rangle$. Here, we reinstate this quantity to obtain a stricter bound.\nTo determine $\\langle T\\_{\\rm store}\\rangle$, we first note that the entanglement between QM1 and QM2 needs to be maintained until QM1ʹ and QM2ʹ become entangled with memory qubits on other nodes. Once this happens, the entanglement can then be swapped, leaving QM1 and QM2 completely disentangled and ready for re-use. Of course, QM1ʹ and QM2ʹ could already be entangled with other nodes by the time QM1 and QM2 become entangled. Thus, there are four equally probable cases to consider:\n* If QM1ʹ and QM2ʹ are already connected to other nodes, we immediately swap the entanglement and thus do not wait any longer. In this case $\\langle T\\_{\\mathrm{store}}^{\\rm case 1}\\rangle = 0$.\n* If one QM (either QM1ʹ or QM2ʹ) is already connected to other nodes, we need to wait an average time of ⟨*T*ent⟩ for the second QM to become entangled as well. In this case, $\\langle T\\_{\\rm store}^{\\rm case 2}\\rangle=\\langle T\\_{\\rm store}^{\\rm case 3}\\rangle= \\langle T\\_{\\mathrm{ent}} \\rangle$.\n* If neither QM1ʹ or QM2ʹ are connected, we need to wait for both to become connected. In this case, $\\langle T\\_{\\mathrm{store}}^{\\rm case 4}\\rangle = \\frac{3-2P\\_{\\mathrm{ent}}}{2- P\\_{\\mathrm{ent}}}\\times \\langle T\\_{\\mathrm{ent}} \\rangle $.\nThe expression for $\\langle T\\_{\\mathrm{store}}^{\\rm case 4}\\rangle$ above requires some explanation. It comes from the fact that both QM1ʹ and QM2ʹ become successfully entangled with other nodes after *n* attempts with probability\n \n$$\\mathrm{Pr}(n, P\\_{\\mathrm{ent}}) = [1-(1-P\\_{\\rm ent})^n]^2.$$ \nThe average storage time in this case then depends on the average number of attempts needed before entanglement is successfully created and on the time $T\\_{\\rm trial}$ each attempt takes:\n \n$$\\langle T\\_{\\rm store}^{\\rm case 4}\\rangle = T\\_{\\rm trial}\\sum\\_{n=0}^{\\infty} n [\\mathrm{Pr}(n,P\\_{\\rm ent})- \\mathrm{Pr}(n-1,P\\_{\\rm ent})] = \\frac{3-2P\\_{\\mathrm{ent}}}{2- P\\_{\\mathrm{ent}}}\\langle T\\_{\\rm ent}\\rangle.$$ \nThis implies that the overall average storage time is\n \n$$\\langle T\\_{\\rm store}\\rangle=\\frac{\\langle T\\_{\\rm store}^{\\rm case 1}\\rangle+\\langle T\\_{\\rm store}^{\\rm case 2}\\rangle+\\langle T\\_{\\rm store}^{\\rm case 3}\\rangle+\\langle T\\_{\\rm store}^{\\rm case 4}\\rangle}{4}=\\frac{7/4-P\\_{\\rm ent}}{2-P\\_{\\rm ent}}\\langle T\\_{\\rm ent}\\rangle.$$ \nThe upper bound on the rate per matter qubit is therefore\n \n$$R\\_{\\mathrm{max}}^{(2QM)}=\\frac{1}{(\\langle T\\_{\\rm ent}\\rangle+\\langle T\\_{\\rm store}\\rangle)N\\_m}=\\frac{2-P\\_{\\rm ent}}{15-8P\\_{\\rm ent}}\\frac{c}{L},$$ \nwhere we set $\\langle T\\_{\\rm ent}\\rangle\\to2L\\_0/c$ and *N**m* = 2*L*/*L*0 as appropriate for the memory-based scheme. If we set $P\\_{\\rm ent} \\rightarrow 1$ , this becomes\n \n$$R\\_{\\mathrm{max}}^{(2QM)}=\\frac{c}{7L}\\quad\\hbox{for}\\quad P\\_{\\rm ent}=1.$$ \nClassical signaling does not limit the repetition rate of the RGS protocol\n==========================================================================\nIn this section, we show that classical signaling does not limit the repetition rate of the RGS protocol. This is important to consider because some of the *X* and *Z* measurements that are performed on the RGS will, depending on the measurement outcomes, leave the remaining photons in a state that is only locally equivalent to a graph state. In this case, additional single-qubit gates must be applied to bring the state to a proper graph state before the next measurements are performed. Otherwise, the bases for these subsequent measurements will be effectively rotated, and the measurements will not modify the state in the intended way. In the following, we show that the extra single-qubit gates only need to be applied on qubits located at the same measurement node where the measurements on which they depend occur. Thus, classical communication between nodes is not needed. The only exception to this occurs at the very end of the protocol, when we are left with a linear cluster state in which the two end qubits are those of Alice and Bob. The final step is to turn this state into a Bell pair shared by Alice and Bob by performing *X* measurements on the inner qubits of the linear cluster state. Afterward, Alice and Bob’s qubits form a Bell pair whose Pauli frame depends on the *X* measurement outcomes. This information therefore needs to be transferred to them using classical signaling. However, the protocol can be repeated before this classical signal is received and thus, the classical signaling does not limit the repetition rate. The latter is only limited by the generation time of RGSs.\nRecursive construction of a repeater graph state.\n-------------------------------------------------\n [figclassicalsignaling]\nWe denote the RGS with *k* arms by $\\ket{RGS\\_k}$. For the purposes of this argument, we allow *k* to be either odd or even. We want to define this quantum state recursively. As illustrated in Fig. [figclassicalsignaling](a), we can make this state by starting with an RGS with *k* − 1 arms and then adding one new 1st-leaf qubit and one new 2nd-leaf qubit by applying *C**Z* gates between these two new qubits and between the new 1st-leaf qubit and all the existing 1st-leaf qubits in $\\ket{RGS\\_{k-1}}$. In other words, we can start from a 2-qubit cluster state and $\\ket{RGS\\_{k-1}}$ in a product state: $(\\ket{0\\_{k\\_1},+\\_{k\\_2}} + \\ket{1\\_{k\\_1},-\\_{k\\_2}}) \\otimes \\ket{RGS\\_{k-1}}$. Here, *k*1 and *k*2 denote the 1st-leaf qubit and the 2nd-leaf qubit from the ktharm. We ignore the normalization factors troughout this section. The remaining step is to apply *C**Z* gates between *k*1 and all the 1st-leaf qubits in $\\ket{RGS\\"},"data_id":{"kind":"string","value":"arxiv_0182388"}}},{"rowIdx":45594,"cells":{"content":{"kind":"string","value":"0,2); at (0,0) [label=below:5] ; at (-1,0) [label=below:3] ; at (-2,0) [label=below:1] ; at (0,1) [label=left:3] ; at (0,2) [label=left:1] ; at (1,0) [label=below:3] ; at (2,0) [label=below:2] ; at (3,0) [label=below:1] ;\n \n$\\scriptsize{(5)}$& & 1 & 5 & 30 & (, ) &\n[x=.7cm,y=.7cm] (-3,0)–(4,0); (0,0)–(0,1); at (0,0) [label=below:7] ; at (-1,0) [label=below:5] ; at (-2,0) [label=below:3] ; at (-3,0) [label=below:1] ; at (0,1) [label=left:3] ; at (1,0) [label=below:5] ; at (2,0) [label=below:4] ; at (3,0) [label=below:2] ; at (4,0) [label=below:1] ;\n \n$\\scriptsize{(6)}$& & 3 & 8 & 39 & (, ) &\n[x=.7cm,y=.7cm] (-2,0)–(4,0); (0,0)–(0,2); at (0,0) [label=below:9] ; at (-1,0) [label=below:6] ; at (-2,0) [label=below:3] ; at (0,1) [label=left:5] ; at (0,2) [label=left:1] ; at (1,0) [label=below:7] ; at (2,0) [label=below:5] ; at (3,0) [label=below:3] ; at (4,0) [label=below:1] ;\n \n$\\scriptsize{(7)}$& & 3 & 5 & 41& (, ) &\n[x=.7cm,y=.7cm] (-3,0)–(3,0); (0,0)–(0,2); at (0,0) [label=below:10] ; at (-1,0) [label=below:7] ; at (-2,0) [label=below:5] ; at (-3,0) [label=below:2] ; at (0,1) [label=left:5] ; at (0,2) [label=left:1] ; at (1,0) [label=below:7] ; at (2,0) [label=below:4] ; at (3,0) [label=below:1] ;\n \n|c|c|c|c|c|c|c| $ \\scriptsize{(\\#)}$& & r & f&dH & & (4)$\\scriptsize{(8)}$& & 4 & 9 & 76 & (, ) &\n[x=.7cm,y=.7cm] (-3,0)–(5,0); (0,0)–(0,1); at (0,0) [label=below:16] ; at (-1,0) [label=below:11] ; at (-2,0) [label=below:6] ; at (-3,0) [label=below:1] ; at (0,1) [label=left:8] ; at (1,0) [label=below:13] ; at (2,0) [label=below:10] ; at (3,0) [label=below:7] ; at (4,0) [label=below:4] ; at (5,0) [label=below:1] ;\n \n$\\scriptsize{(9)}$& & 7 & 9 & 98 & (, ) &\n[x=.7cm,y=.7cm] (-2,0)–(5,0); (0,0)–(0,2); at (0,0) [label=below:21] ; at (-1,0) [label=below:14] ; at (-2,0) [label=below:7] ; at (0,1) [label=left:11] ; at (0,2) [label=left:1] ; at (1,0) [label=below:17] ; at (2,0) [label=below:13] ; at (3,0) [label=below:9] ; at (4,0) [label=below:5] ; at (5,0) [label=below:1] ;\n \n$\\scriptsize{(10)}$& & 10 & 9 & 115 & (, ) &\n[x=.7cm,y=.7cm] (-4,0)–(3,0); (0,0)–(0,2); at (0,0) [label=below:25] ; at (-1,0) [label=below:20] ; at (-2,0) [label=below:15] ; at (-3,0) [label=below:10] ; at (-4,0) [label=below:5] ; at (0,1) [label=left:13] ; at (0,2) [label=left:1] ; at (1,0) [label=below:17] ; at (2,0) [label=below:9] ; at (3,0) [label=below:1] ;\n \n|c|c|c|c|c|c|c| $ \\scriptsize{(\\#)}$& & r & f&dH & & (4)$\\scriptsize{(11)}$& & 3 & 8 & 29 & (, ) &\n[x=.7cm,y=.7cm] (-3,0)–(3,0); (0,0)–(0,2); at (0,0) [label=below:7] ; at (-1,0) [label=below:5] ; at (-2,0) [label=below:3] ; at (-3,0) [label=below:1] ; at (0,1) [label=left:4] ; at (0,2) [label=left:1] ; at (1,0) [label=below:5] ; at (2,0) [label=below:3] ; at (3,0) [label=below:1] ;\n \n$\\scriptsize{(12)}$& & 6 & 9 & 57 & (, ) &\n[x=.7cm,y=.7cm] (-3,0)–(4,0); (0,0)–(0,2); at (0,0) [label=below:13] ; at (-1,0) [label=below:9] ; at (-2,0) [label=below:5] ; at (-3,0) [label=below:1] ; at (0,1) [label=left:7] ; at (0,2) [label=left:1] ; at (1,0) [label=below:10] ; at (2,0) [label=below:7] ; at (3,0) [label=below:4] ; at (4,0) [label=below:1] ;\n \n$\\scriptsize{(13)}$& & 15 & 10 & 145 & (, ) &\n[x=.7cm,y=.7cm] (-3,0)–(5,0); (0,0)–(0,2); at (0,0) [label=below:31] ; at (-1,0) [label=below:21] ; at (-2,0) [label=below:11] ; at (-3,0) [label=below:1] ; at (0,1) [label=left:16] ; at (0,2) [label=left:1] ; at (1,0) [label=below:25] ; at (2,0) [label=below:19] ; at (3,0) [label=below:13] ; at (4,0) [label=below:7] ; at (5,0) [label=below:1] ;\n \nCase (*b**α*) = (*N*, *N* − 1, *N* − 1)\n---------------------------------------\nNext, consider the condition [Diophantine-eq3] Np1 +N-1p2+N-1p3 =N . There are two infinite series of solutions [series su ii] (p1, p2, p3)= (N,2,2), (2, 2, 2N-2) , N . For the first series, one needs to add marginal deformation terms in to obtain the following IHS of type VIII2(*N*, *N*, 2, 2) x1N+x2N+x1 x32+x1 x42+x2 x3 x4=0. In fact, this isolated singularity is isomorphic to a singularity of type IV(*N*, 2, *N*, 2) x1N+x1 x22+x3N+x3 x42=0, which gives rise to AD[*D**N* + 1, *D**N* + 1] theories. We list its geometric data in table [tab:trinion series]. For *N* = 2*k* − 1, the 5d SCFT is the UV fixed point of the 5d $\\CN=1$ gauge theory SU(k)+(2k+1), which has flavor symmetry algebra is g*F* = so(4*k* + 2) ⊕ u(1). For *N* = 2*k*, the theory is the same SCFT coupled to 2*k* free hypermultiplets, which correspond to 2*k* conifold singularities in the crepant resolution. The EQ(4) for these infinite sequences are (4)=\n[x=.6cm,y=.6cm] (-2,0)–(2.4,0); (2.6,0)–(3.4,0); (3.6,0)–(5,0); (0,0)–(0,2); at (0,0) [label=below:*N*] ; at (-1,0) [label=above:*k*] ; at (-2,0) [label=below:1] ; at (0,1) [label=left:*k*] ; at (0,2) [label=above:1] ; at (1,0) [label=above:2*k* − 2] ; at (2,0) [label=below:2*k* − 3] ; at (4,0) [label=above:2] ; at (5,0) [label=below:1] ;\n& N= 2k-1 , \n[x=.6cm,y=.6cm] (-2,0)–(2.4,0); (2.6,0)–(3.4,0); (3.6,0)–(5,0); (0,0)–(0,2); at (0,0) [label=below:*N*] ; at (-1,0) [label=above:*k*] ; at (-2,0) [label=below:1] ; at (0,1) [label=left:*k*] ; at (0,2) [label=above:1] ; at (1,0) [label=above:2*k* − 1] ; at (2,0) [label=below:2*k* − 2] ; at (4,0) [label=above:2] ; at (5,0) [label=below:1] ;\n& N= 2k ,\n. The MQ(5) is obtained by replacing all the *S**U*(*n*) nodes in EQ(4) by *U*(*n*) nodes. The quiver for *N* = 2*k* is ugly, and can be dualized to extract the free sector, so that we obtain [R2N5d] (5)=\n[x=.6cm,y=.6cm] (-2,0)–(2.4,0); (2.6,0)–(3.4,0); (3.6,0)–(5,0); (0,0)–(0,2); at (0,0) [label=below:2*k* − 1] ; at (-1,0) [label=above:*k*] ; at (-2,0) [label=below:1] ; at (0,1) [label=left:*k*] ; at (0,2) [label=above:1] ; at (1,0) [label=above:2*k* − 2] ; at (2,0) [label=below:2*k* − 3] ; at (4,0) [label=above:2] ; at (5,0) [label=below:1] ;\n& N= 2k-1 , \n[x=.6cm,y=.6cm] (-2,0)–(2.4,0); (2.6,0)–(3.4,0); (3.6,0)–(5,0); (0,0)–(0,2); at (0,0) [label=below:2*k* − 1] ; at (-1,0) [label=above:*k*] ; at (-2,0) [label=below:1] ; at (0,1) [label=left:*k*] ; at (0,2) [label=above:1] ; at (1,0) [label=above:2*k* − 2] ; at (2,0) [label=below:2*k* − 3] ; at (4,0) [label=above:2] ; at (5,0) [label=below:1] ; at (7,0) [] $\\, \\oplus \\quad \\mb{H}^{2k} $ ;\n& N= 2k ,\nOne can check that, for *N* = 2*k* − 1, the magnetic quiver is indeed the ones of $SU(k)\\_{\\pm {3{\\over}2}}+(2k+1)\\bm{F}$ . This gives a nice consistency check on our geometric approach. The second series in is equivalent to, *i.e.* we have the equivalences (2, 2, N, 2N-2) (2, 2N, N, 2) . In addition, we find the sporadic solutions\n| | | | | |\n| --- | --- | --- | --- | --- |\n| | *N* | *p*1 | *p*2 | *p*3 |\n| | 5 | 3 | 2 | 3 |\n| | 7 | 2 | 3 | 4 |\n| | 9 | 3 | 2 | 4 |\n| | 10 | 4 | 2 | 3 |\n| | 16 | 2 | 3 | 5 |\n| | 21 | 3 | 2 | 5 |\n| | 25 | 5 | 2 | 3 |\nHere, we omitted the solution *N* = 4, (*p**α*) = (2, 3, 3) because it is equivalent to the model I(2, 4, 4, 4), the rank-1 *E*7 theory.\nFor the model $\\scriptstyle{(4)}$, we deduce from the resolution geometry that it is a rank-1 *E*6 theory coupled to a rank-0 theory *E*8( − 4) plus another conifold (a free hypermultiplet). From EQ(4), after gauging and rebalancing ugly nodes, we indeed get\n[x=.7cm,y=.7cm] (-2,0)–(2,0); (0,0)–(0,2); at (-4,1) [] MQ(5) = ; at (0,0) [label=below:3] ; at (-1,0) [label=below:2] ; at (-2,0) [label=below:1] ; at (0,1) [label=left:2] ; at (0,2) [label=left:1] ; at (1,0) [label=below:2] ; at (2,0) [label=below:1] ; at (4,1) [] $\\oplus \\quad \\mb{H}^7$;\n.\nSimilarly, the model $\\scriptstyle{(5)}$ is a rank-1 *E*7 theory coupled to a rank-0 theory *E*8( − 3) and another conifold. From gauging and rebalancing EQ(4), we get\n[x=.7cm,y=.7cm] (-3,0)–(3,0); (0,0)–(0,1); at (-5,1) [] MQ(5) = ; at (0,0) [label=below:4] ; at (-1,0) [label=below:3] ; at (-2,0) [label=below:2] ; at (-3,0) [label=below:1] ; at (0,1) [label=left:2] ; at (1,0) [label=below:3] ; at (2,0) [label=below:2] ; at (3,0) [label=below:1] ; at (5,1) [] $\\oplus \\quad \\mb{H}^{13}$;\n.\nThe model $\\scriptstyle{(6)}$ has a smooth resolution, and correspondingly ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}$ is a Lagrangian theory, with the *S**U*(*n*) quiver shown in table [tab:trinion series 3]. The corresponding 5d SCFT ${{\\mathcal{T}\\_{{\\mathbf X}}^{\\rm 5d}}}$ has *r* = 3 and *d**H* = 39. The structure of its Higgs branch can be studied using the magnetic quiver\n[x=.7cm,y=.7cm] (-2,0)–(4,0); (0,0)–(0,2); at (0,0) [label=below:9] ; at (-1,0) [label=below:6] ; at (-2,0) [label=below:3] ; at (0,1) [label=left:5] ; at (0,2) [label=left:1] ; at (1,0) [label=below:7] ; at (2,0) [label=below:5] ; at (3,0) [label=below:3] ; at (4,0) [label=below:1] ;\nIn particular, the 5d flavor symmetry is g*F* = e7 ⊕ u(1).\nThe model $\\scriptstyle{(7)}$ has a crepant resolutions with a residual terminal singularity *E*8( − 3). After gauging the *U*(1)*f* of EQ(4) and rebalancing, we get\n[x=.7cm,y=.7cm] (-3,0)–(3,0); (0,0)–(0,2); at (-5,1) [] MQ(5) = ; at (0,0) [label=below:7] ; at (-1,0) [label=below:5] ; at (-2,0) [label=below:3] ; at (-3,0) [label=below:1] ; at (0,1) [label=left:4] ; at (0,2) [label=left:1] ; at (1,0) [label=below:5] ; at (2,0) [label=below:3] ; at (3,0) [label=below:1] ; at (5,1) [] $\\oplus\\quad \\mb{H}^{12}$;\n. Note that the magnetic quiver of the interacting sector is exactly the same as the model $\\scriptstyle{(11)}$ in table [tab:trinion series 4], which corresponds to a *r* = 3, *f* = 8 5d SCFT. Thus, we conclude that the 5d SCFT for model $\\scriptstyle{(7)}$ is equivalent to the one of model $\\scriptstyle{(11)}$ plus 12 hypermultiplets.\nThe model $\\scriptstyle{(8)}$, $\\scriptstyle{(9)}$ and $\\scriptstyle{(10)}$ all have smooth crepant resolutions, and correspondingly ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}$ is a Lagrangian theory. From the magnetic quivers, we can show that they each have 5d global symmetry g*F* = e8 ⊕ su(2).\nCase (*b**α*) = (*N* − 1, *N* − 1, *N* − 1)\n-------------------------------------------\nLastly, consider the condition [Diophantine-eq4] N-1p1 +N-1p2+N-1p3 =N . For any *N*, we have the solution (*p**α*) = (2, 2, *N* − 1), but it turns out this is equivalent to the second series in, namely 2 (N, 2,2,N-1) 2(N, N, 2, 2) (N, 2, N, 2) . We also find the sporadic solutions\n| | | | | |\n| --- | --- | --- | --- | --- |\n| | *N* | *p*1 | *p*2 | *p*3 |\n| | 7 | 2 | 3 | 3 |\n| | 13 | 2 | 3 | 4 |\n| | 31 | 2 | 3 | 5 |\nThese three models have smooth resolutions and ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}$ is Lagrangian. The 5d SCFT for model $\\scriptstyle{(11)}$ has *r* = 3 and g*F* = e6 ⊕ su(2). The model $\\scriptstyle{(12)}$ has rank *r* = 6 and g*F* = e7 ⊕ su(2). Finally, the model $\\scriptstyle{(13)}$ has *r* = 15 and g*F* ⊃ e8. In appendix [app:Diophantine], we prove that there are no more solutions to the equations, and. This completes our classification.\n*D**p**b*(*G*)-Trinions SCFTs for *G* = *D*, *E*\n================================================\nIn this section, we generalize the four-dimensional trinion SCFTs to the cases with a central gauge group of type *D* or *E*. We focus on the *D**p**b*(*G*) trinions that can be engineered by threefold singularities in Type IIB. This leads us to discover a number of new S-dualities between SCFTs with different gauge groups, which we discuss in section [sec:dualities].\nTrinion Geometries for Generic *G*\n----------------------------------\nThe analysis of section [secquiverinesSU] can be generalized to the case of gaugings of three *D**p**b*(*G*) theories with *G* of special orthogonal or exceptional type. In the general case, however, we just have a Landau-Ginzburg description instead of a threefold singularity. The LG superpotential reads [superpot] = w2+f(t,x)+=13 zp fb(x,t) , where *W**G*(*w*, *x*, *t*) ≡ *w*2 + *f*(*t*, *x*) = 0 is the ADE singularity of type *G* and *f**b**α*(*x*, *t*)*z**α**p**α* denotes the *z*-dependent part of the threefold singularity describing *D**p**α**b**α*(*G*), shown in table [sing]. From a LG model point of view, *w* in is a massive field and can be integrated out, leaving five fields. The conformality condition reads [beta0] ++=h(G) . Importantly, if the singularity $\\CW=0$ is non-isolated, one needs to add marginal terms to in order to have an isolated singularity at the origin, so that the LG model is well-defined.\nWe will consider the special case when the LG model is equivalent to a model with at most four fields. This happens whenever at least one of the fields *z**α* is massive. For *p* = 2 or *p* = 1 and an appropriate choice of *b*, we have: [mass terms G] zp fb(x,t) = z2 zp fb(x,t) = t z, x z. Such massive fields can be integrated out. If we choose *p* = 2 for one of the legs, we are therefore considering a theory ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}$ of the form\n[x=.6cm,y=.6cm] at (-3,0) *D**p*2*b*2(*G*); at (0,0) *G*; at (3,0) *D**p*3*b*3(*G*); at (0,2) *D*2*h*∨(*G*); (-1.7,0)–(-.5,0); (.5,0)–(1.7,0); (0,0.5)–(0,1.3);\nThe constraint ([beta0]) now implies that the other two *D**p**b*(*G*) sectors should satisfy the relation [D2G cond] +=. Since we are now left with only four fields, the system can be understood as a threefold compactification in Type IIB. The second option is *p* = 1, which gives a mass term for *b* ≠ *h*∨(*G*), since in that case the *f**b*(*x*, *t*) function appearing in table [sing] is always linear in either *x* or *t*, we can simply choose the *D**p*1*b*1(*G*) theory to be *D*1*b*(*G*) with *b* ≠ *h*∨(*G*) (we remind the reader that *D*1*b*(*G*) is trivial for *b* = *h*∨(*G*)) and, due to, we take the other two *D**p**b*(*G*) models to satisy the constraint [D1G cond] +=h(G)-b. Under these conditions, we can integrate out both *z*1 and either *x* or *t* (the variable appearing in *f**b*1(*x*, *t*)). We can now simply introduce a new massive field *y* which enters quadratically in the superpotential. The threefold singularity is now given by a hypersurface in the four variables *z*2, *z*3, *y*, and *x* or *t* (the variable we have not integrated out). We can therefore also consider the family of trinions\n[x=.6cm,y=.6cm] at (-3,0) *D**p*2*b*2(*G*); at (0,0) *G*; at (3,0) *D**p*3*b*3(*G*); at (0,2) *D*1*b*(*G*); (-1.7,0)–(-.5,0); (.5,0)–(1.7,0); (0,0.5)–(0,1.3);\nsatisfying the condition. In the following, we list all such models by solving equations and and integrating out the massive LG fields. We collect the resulting theefold singularities and their basic properties in various tables in appendix [App:Tables].\n*D**p**b*(*G*)-Trinions with *b*1 = *h*∨ and *p*1 = 2\n-----------------------------------------------------\n### *b*2 = *b*3 = *h*∨\nIf we choose *b*1 = *b*2 = *h*∨, we have the universal solutions: (p2, p3) = (4,4), (3,6) , for any *G*. They are the only solutions, in this case. For *G* = *S**U*(*N*), these are the *E*7(*N* − 4) and *E*8(*N* − 6) series. For *S**O*(2*N*), the IHS equation reads xN-1+x t2+z2p2+z3p3=0. The basic properties of these models, for *G* = *S**O*(2*N*) and *G* = *E**n*, are listed in table [tab:b2b3h].\n### Other Sporadic Models\nFor *G* = *S**O*(2*N*) (so that *h*∨ = 2*N* − 2), we also have the sporadic models in table [tab:SporadicSO]. For *G* = *E*6, we have *h*∨ = 12 and the allowed values of *b* are 12, 9, 8. The sporadic models are in table [tab:sporadicE6]. For *G* = *E*7, we have *h*∨ = 18 and the allowed values of *b* are 18 and 14. We then find the sporadic models in table [tab:sporadicE7]. For *G* = *E*8, we have *h*∨ = 30 and the allowed values of *b* are 30, 24, 20, with the sporadic models in table [tab:sporadicE8].\n*D**p**b*(*G*)-Trinions with *b*1 ≠ *h*∨ and *p*1 = 1\n-----------------------------------------------------\nLet us also consider models with *b*1 ≠ *h*∨ and *p*1 = 1. For *G* = *S**O*(2*N*), we have to assign *b*1 = *N*, and we have the models when *b*2 = *b*3 = 2*N* − 2 shown in table [tab:Bounty]. If *b*2 = 2*N* − 2, *b*3 = *N*, then one needs to take =xN-1+x t2+z1 t+z2p2+z3p3t, add marginal terms to make it isolated, and then integrate out *z*1 and *t* (a similar procedure should be applied to the *b*2 = *b*3 = *N* cases as well). We list the resulting singular equations and the data of the corresponding theories in table [tab:Bounty]. Note that many of these models are equivalent to *E**k*(*n*) theories or to other models introduced above.\nFor *G* = *E*6, we can choose either *b*1 = 9 or 8, and the models are shown in table [tab:Twix]. Likewise, for *G* = *E*7 we need to set *b*1 = 14, and the corresponding models can be found in table [tab:Snickers]. Finally, for *G* = *E*8, we can choose either *b*1 = 24 or 20, as shown in table [tab:Ferrero].\nMatching the CB Spectrum to the Spectrum of the Singularity\n-----------------------------------------------------------\nWe can provide detailed checks of the above results by comparing the 4d CB spectrum obtained from the threefold singularity in Type IIB to the CB spectrum of the individual *D**p**b*(*G*) legs.\nLet us illustrate this by presenting one example in more detail. Let us choose the first model in table [tab:Twix]. The trinion takes the form [E6 trinion expl]\n[x=.6cm,y=.6cm] at (-2.3,0) *D*5; at (0,0) *E*6; at (2.3,0) *D*20; at (0,2) MN6; (-1.6,0)–(-0.6,0); (0.6,0)–(1.6,0); (0,0.5)–(0,1.3);\n. Here we used the fact that *D*19(*E*6) is the *E*6 MN theory. The three legs have the CB spectrum &D19(E6) : && = {3} , \n&D512(E6) : &&= {,,,,,,,,,,, } , \n&D2012(E6): &&= {,,,,,,,,,2,2,,,,,,3,3,3,,,. \n&&&,,,,,,,,,,,,,5,,,,, \n&&&. 6,6,,,,,,,,,,,,9,,,,} , the union of these spectra, plus the *E*6 Casimirs, which give Δ = {2, 5, 6, 8, 9, 12}, result in the spectrum of the *D**p**b*(*E*6) trinion, which is a theory of rank ${\\widehat}r= 76$. On the other hand, the LG description of the trinion is = x3 + t4 + t z1 + z25 +z320 . Integrating out the massive fields *t* and *z*1, we obtain the threefold singularity y2+ x3 + z25 +z320=0 . Its singularity spectrum precisely reproduce the CB spectrum spelled out above.\nS-Dualities between 4d *D**p**b*(*G*)-Trinions\n==============================================\nS-Dualities from Geometric Equivalence\n--------------------------------------\nBy comparing the results of sections [sec:SUN] and [sec7:DandE], we observe that, in a number of examples, the same singularity simultaneously describes more than one gauging of three *D**p**b*(*G*) theories, sometimes involving different gauge groups *G*. The reason for this is mundane from the 2d LG perspective. By the 2d/4d correspondence, the LG superpotential $\\CW$ corresponds to a *G*-gauging of three *D**p**b*(*G*) theories. It can then happen that the $\\CW$ for two distinct *D**p**b*(*G*) trinions are equal, up to massive fields that can be decoupled. For simplicity, we have focussed on LG models with at most 4 non-trivial fields, that can be interpreted as canonical singularities ${{\\mathbf X}}$ in Type IIB.\nThe consequence of this simple observation for the 4d $\\CN=2$ SCFTs ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}$ is rather interesting. Since the 4d theory is determined unambigiously by the singularity (or by $\\CW$), the distinct-looking *D**p**b*(*G*) trinions associated to a single ${{\\mathbf X}}$ are actually exactly equivalent. We have thus discovered a new type of S-duality among such 4d SCFTs. The equivalent *D**p**b*(*G*) trinions describe two S-duality frames of the same theory ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}$, and the gauge groups manifest in each description become weakly coupled at different cusps of the conformal manifold. From the field-theoretic point of view, these dualities appear rather miraculous. By inspection of the list of theories in section [sec:SUN] and section [sec7:DandE], we then identify a large number of these S-dualities. We list them below and briefly comment on some of their properties. We hope to return to an in-depth analysis of these theories in the future.\nThe list of S-dualities can be simplified by observing that, for a fixed gauge group *G*, we have the following equivalence relation between *D**p**b*(*G*) theories [leg-equiv] Dp1b1(G)Dp2b2(G)=. This follows from the geometric engineering of section [subsec:DpSummary], since the scaling dimensions of *D**p**b*(*G*) only depend on ${b{\\over}p}$.\nIn order to describe the structure of the conformal manifold of ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}$, we will need a class of isolated (*i.e.* with trivial conformal manifold) 4d SCFTs dubbed *R*2, *N* (for *N* odd) in. This is a class $\\CS$ trinion with punctures [*k* − 1, *k* − 1, 1], [*k* − 1, *k* − 1, 1] and [1*N*], with *N* = 2*k* − 1. This theory has CB operators of dimension 3, 5, …, *N* − 2, *N* and global symmetry *S**O*(2*N* + 4)2*N* × *U*(1), where the subscript 2*N* denotes the flavor central charge. We also notice that the 3d mirrors (or magnetic quivers) of *R*2, *N* theories appear in ([R2N5d]), which tells us that *R*2, *N* is the dimensional reduction of the 5d SCFT which UV completes $SU(k)\\_{\\pm\\frac{3}{2}}+(2k+1)\\bm{F}$. This family of theories is relevant because it arises in the description of the conformal manifold of *D*2*n*(*S**O*(2*n* + 2)) theories. More precisely, we find that *D*2*n*(*S**O*(2*n* + 2)) is equivalent to [lagrSO] -SU(2)-SU(3)-SU(4)R2,5SO(8)…SO(2n)R2,2n-1,where the notation *A* ← *G* → *B* means that we glue together the SCFTs *A* and *B* by gauging a *G* subgroup of their global symmetry.\nA List of New S-dualities\n-------------------------\nWe will now discuss in detail several examples of S-dualities we can obtain with this method.We find it convenient to organise the list according to *r*, the rank of the 5d SCFT ${{\\mathcal{T}\\_{{\\mathbf X}}^{\\rm 5d}}}$ associated with the same singularity.\n#### Rank-1 *E*8 model.\nAs our first example, the rank-1 *E*8 theory in 5d is described by the singularity x6+z12+z23+z36=0 . Modulo the trivial equivalences, there are actually two *D**p**b*(*G*) trinions that describe the same theory ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}$\n[x=.7cm,y=.7cm] at (-1.8,0.5) ${{\\mathscr{T}\\_{E\\_8}^{\\rm 4d}}}=$; (1,0)–(8,0); (3,0)–(3,1); at (1,0) [label=above:2] ; at (2,0) [label=above:4] ; at (3,0) [label=below:6] ; at (4,0) [label=above:5] ; at (5,0) [label=above:4] ; at (6,0) [label=above:3] ; at (7,0) [label=above:2] ; at (8,0) [label=above:1] ; at (3,1) [label=above:3] ;\n[x=.6cm,y=.6cm] at (-4, 0) ≡ ; at (-3,0) *D*6; at (0,0) *S**O*(8); at (3,0) *D*6; at (0,2) *D*14; (-2.3,0)–(-1.3,0); (1.3,0)–(2.3,0); (0,0.5)–(0,1.3);\nIn the first S-duality frame, which is fully Lagrangian, we clearly see that the conformal manifold has dimension 8. By using the equivalence $D\\_6(SO(8))\\equiv\\boxed{1}-SU(2)-SU(3)-SU(4)\\rightarrow R\\_{2,5}$ and the fact that *D*14(*S**O*(8)) is equivalent to *S**U*(2) *N**f* = 4 SQCD, we easily see that the *S**O*(8) trinions as well have a eight-dimensional conformal manifold, in which the *S**U*(5) and *S**U*(6) groups of the unitary trinion are replaced by *S**U*(2) and *S**O*(8). In the *S**O*(8) trinion description the theory is not fully Lagrangian, as it includes two copies of the strongly-coupled *R*2, 5 theory. Indeed the gauge couplings of the two trinions tend to zero at different cusps of the conformal manifold.\n#### Rank-2 *E*6 model.\nThe singularity that realizes the rank-2 *E*6 5d theory with (*r*, *f*, *d**H*) = (2, 6, 23) leads in 4d to the duality &\n[x=.7cm,y=.7cm] at (-1.8,0.5) ${{\\mathscr{T}\\_{E\\_6^{(3)}}^{\\rm 4d}}}=$; (1,0)–(5,0); (3,0)–(3,2); at (1,0) [label=below:2] ; at (2,0) [label=below:4] ; at (3,0) [label=below:6] ; at (4,0) [label=below:4] ; at (5,0) [label=below:2] ; at (3,1) [label=left:4] ; at (3,2) [label=left:2] ; at (6,.6) ≡ ;\n[x=.6cm,y=.6cm] at (-3,0) *D*3; at (0,0) *S**O*(8); at (3,0) *D*6; at (0,2) *D*2; (-2.3,0)–(-1.3,0); (1.3,0)–(2.3,0); (0,0.5)–(0,1.3);\n. In both S-duality frames, the conformal manifold has dimension seven. In the special unitary trinion this is obvious, and in the *S**O*(8) trinion this follows from the fact that *D*2(*S**O*(8)) is isolated (it is the rank-1 *E*6 theory), *D*6(*S**O*(8)) has three marginal couplings, as we have already explained, and *D*3(*S**O*(8)) is the Lagrangian theory $SO(4)-Sp(2)-\\boxed{4}$.\n#### Rank-2 *E*7 model.\nA similar result holds for the singularity that realizes the rank-2 *E*7 theory with (*r*, *f*, *d**H*) = (2, 7, 35): &\n[x=.7cm,y=.7cm] at (-1.5,0.5) ${{\\mathscr{T}\\_{E\\_7^{(4)}}^{\\rm 4d}}}=$; (1,0)–(7,0); (4,0)–(4,1); at (1,0) [label=below:2] ; at (2,0) [label=below:4] ; at (3,0) [label=below:6] ; at (4,0) [label=below:8] ; at (5,0) [label=below:6] ; at (6,0) [label=below:4] ; at (7,0) [label=below:2] ; at (4,1) [label=left:4] ; at (8,.3) ≡ ;\n[x=.6cm,y=.6cm] at (-3,0) *D*4; at (0,0) *S**O*(10); at (3,0) *D*8; at (0,2) *D*15; (-2.3,0)–(-1.3,0); (1.3,0)–(2.3,0); (0,0.5)–(0,1.3);\nOnce again we can match the dimension of the conformal manifold on the two sides. The special unitary side is easy to understand since the theory is Lagrangian, whereas the *S**O*(10) trinion is more subtle: *D*15(*S**O*(10)) is the rank-1 *E*6 theory, *D*8(*S**O*(10)) ≡ *R*2, 7 ← *S**O*(8) → *D*6(*S**O*(8)) from ([lagrSO]) and $D\\_4(SO(10))=\\boxed{1}-SU(2)-SO(6)-Sp(3)-\\boxed{5}$.\n#### Rank-2 *E*8 model and triality.\nIn the case of the *r* = 2 *E*8 theory, *i.e.* the hypersurface *x*12 + *z*12 + *z*23 + *z*36 = 0, we interestingly find a triality &\n[x=.7cm,y=.7cm] at (-0.5,0.5) ${{\\mathscr{T}\\_{E\\_8^{(6)}}^{\\rm 4d}}}=$; (1,0)–(8,0); (3,0)–(3,1); at (1,0) [label=above:4] ; at (2,0) [label=above:8] ; at (3,0) [label=below:12] ; at (4,0) [label=above:10] ; at (5,0) [label=above:8] ; at (6,0) [label=above:6] ; at (7,0) [label=above:4] ; at (8,0) [label=above:2] ; at (3,1) [label=above:6] ;\n&\n[x=.6cm,y=.6cm] at (-5,.3) ≡ ; at (-3,0) *D*3; at (0,0) *S**O*(14); at (3,0) *D*12; at (0,2) *D*17; (-2.3,0)–(-1.3,0); (1.3,0)–(2.3,0); (0,0.5)–(0,1.3);\n[x=.6cm,y=.6cm] at (-3.2,0) ≡ ; at (-2.3,0) *D*6; at (0,0) *E*6; at (2.3,0) *D*12; at (0,2) *D*19; (-1.6,0)–(-0.6,0); (0.6,0)–(1.6,0); (0,0.5)–(0,1.3);\nIn the *S**O*(14) trinion, the nine-dimensional conformal manifold arises as follows: *D*17(*S**O*(14)) is isolated, being equivalent to *R*2, 5. *D*3(*S**O*(14)) is equivalent to the Lagrangian theory $SO(6)-Sp(4)-\\boxed{7}$ and finally we have *D*12(*S**O*(14)), which fits in the family ([lagrSO]) and has a six-dimensional conformal manifold.\n#### Sporadic model $\\scriptstyle{(2)}$ with *r* = 2.\nThe singularity associated with the 5d theory with (*r*, *f*, *d**H*) = (2, 8, 46) in table [tab:trinion series 1] leads to the following S-duality &\n[x=.7cm,y=.7cm] at (-5.8,0.5) ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}=$; (-4,0)–(3,0); (0,0)–(0,1); at (0,0) [label=below:10] ; at (-1,0) [label=below:8] ; at (-2,0) [label=below:6] ; at (-3,0) [label=below:4] ; at (-4,0) [label=below:2] ; at (0,1) [label=left:5] ; at (1,0) [label=below:7] ; at (2,0) [label=below:4] ; at (3,0) [label=below:1] ; at (4,0.3) ≡ ;\n[x=.6cm,y=.6cm] at (-3,0) *D*10; at (0,0) *S**O*(12); at (3,0) *D*26; at (0,2) *D*16; (-2.3,0)–(-1.3,0); (1.3,0)–(2.3,0); (0,0.5)–(0,1.3);\n.\n#### Rank-3 *E*7 model.\nWe also find an S-duality based on the *r* = 3 *E*7 theory &\n[x=.7cm,y=.7cm] at (-1.8,0.5) ${{\\mathscr{T}\\_{E\\_7^{(8)}}^{\\rm 4d}}}=$; (1,0)–(7,0); (4,0)–(4,1); at (1,0) [label=below:3] ; at (2,0) [label=below:6] ; at (3,0) [label=below:9] ; at (4,0) [label=below:12] ; at (5,0) [label=below:9] ; at (6,0) [label=below:6] ; at (7,0) [label=below:3] ; at (4,1) [label=left:6] ; at (8,0.3) ≡ ;\n[x=.6cm,y=.6cm] at (-2.3,0) *D*4; at (0,0) *E*6; at (2.3,0) *D*12; at (0,2) *D*18; (-1.6,0)–(-0.6,0); (0.6,0)–(1.6,0); (0,0.5)–(0,1.3);\n.\n#### Rank-3 *E*8 model.\nLikewise, the rank-3 *E*8 theory leads to the S-duality\n[x=.7cm,y=.7cm] at (-1.8,0.5) ${{\\mathscr{T}\\_{E\\_8^{(12)}}^{\\rm 4d}}}=$; (1,0)–(8,0); (3,0)–(3,1); at (1,0) [label=above:6] ; at (2,0) [label=above:12] ; at (3,0) [label=below:18] ; at (4,0) [label=above:15] ; at (5,0) [label=above:12] ; at (6,0) [label=above:9] ; at (7,0) [label=above:6] ; at (8,0) [label=above:3] ; at (3,1) [label=above:9] ; at (9,0.3) ≡ ;\n[x=.6cm,y=.6cm] at (-2.3,0) *D*6; at (0,0) *E*7; at (2.3,0) *D*18; at (0,2) *D*114; (-1.6,0)–(-0.6,0); (0.6,0)–(1.6,0); (0,0.5)–(0,1.6);\n.\n#### Rank-3 model.\nThe singularity *x*9 + *y*2 + *z*23 + *z*39 = 0 associated with the rank-3 theories appearing in table [tab:Bounty] and [tab:Snickers], respectively, with (*r*, *f*, *d**H*) = (3, 0, 64), implies the duality &\n[x=.6cm,y=.6cm] at (-5.8,0.5) ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}=$; at (-3,0) *D*3; at (0,0) *S**O*(20); at (3,0) *D*9; at (0,2) *D*110; (-2.3,0)–(-1.3,0); (1.3,0)–(2.3,0); (0,0.5)–(0,1.3); at (4,0.3) ≡ ;\n[x=.6cm,y=.6cm] at (-2.3,0) *D*9; at (0,0) *E*7; at (2.3,0) *D*9; at (0,2) *D*114; (-1.6,0)–(-0.6,0); (0.6,0)–(1.6,0); (0,0.5)–(0,1.6);\n. Notice that the *S**O*(20) trinion is a fully Lagrangian SCFT, given by the orthosymplectic quiver\n[x=0.8cm,y=0.8cm] (-2,0)–(8,0); (0,0)–(0,1); at (-2,0) [label=below:Spin(8)] ; at (-1,0) [label=above:*S**p*(6)] ; at (0,0) [label=below:Spin(20)] ; at (0,1) [label=above:*S**p*(4)] ; at (1,0) [label=above:*S**p*(8)] ; at (2,0) [label=below:Spin(16)] ; at (3,0) [label=above:*S**p*(6)] ; at (4,0) [label=below:Spin(12)] ; at (5,0) [label=above:*S**p*(4)] ; at (6,0) [label=below:Spin(8)] ; at (7,0) [label=above:*S**p*(2)] ; at (8,0) [label=below:Spin(4)] ;\nwith a conformal manifold of dimension 13. This quiver has an electric one-form symmetry $\\frak{f}= {\\mathbb{Z}}\\_2^7$, in agreement with the geometry. This is an interesting example of an S-duality that does not involve an *S**U*(*N*) trinion.\nWe also find three curious examples associated with 5d SCFTs of rank larger than three:\n#### Rank-4 model.\nThe rank-4 theory with (*r*, *f*, *d**H*) = (4, 5, 40) appears in table [tab:Bounty] and in table [tab:Twix], corresponding to the singularity *t*4 + *y*2 + *z*26 + *z*36 = 0 which results in the S-duality &\n[x=.6cm,y=.6cm] at (-5.8,0.5) ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}=$; at (-3,0) *D*4; at (0,0) *S**O*(14); at (3,0) *D*6; at (0,2) *D*17; (-2.3,0)–(-1.3,0); (1.3,0)–(2.3,0); (0,0.5)–(0,1.3); at (4,0.3) ≡ ;\n[x=.6cm,y=.6cm] at (-2.3,0) *D*6; at (0,0) *E*6; at (2.3,0) *D*6; at (0,2) *D*18; (-1.6,0)–(-0.6,0); (0.6,0)–(1.6,0); (0,0.5)–(0,1.6);\n. The conformal manifold is of dimension 7.\n#### Rank-5 *E*8 model.\nThe rank-5 *E*8 theory, associated to the singularity *x*3 + *y*2 + *z*26 + *z*330 = 0, leads to the duality\n[x=.7cm,y=.7cm] at (-1.8,0.5) ${{\\mathscr{T}\\_{E\\_8^{(24)}}^{\\rm 4d}}}=$; (1,0)–(8,0); (3,0)–(3,1); at (1,0) [label=above:10] ; at (2,0) [label=above:20] ; at (3,0) [label=below:30] ; at (4,0) [label=above:25] ; at (5,0) [label=above:20] ; at (6,0) [label=above:15] ; at (7,0) [label=above:10] ; at (8,0) [label=above:5] ; at (3,1) [label=above:15] ; at (9,0.3) ≡ ;\n[x=.6cm,y=.6cm] at (-2.3,0) *D*6; at (0,0) *E*8; at (2.3,0) *D*30; at (0,2) *D*124; (-1.6,0)–(-0.6,0); (0.6,0)–(1.6,0); (0,0.5)–(0,1.6);\nNote that *D*18(*E*8) is the rank-1 *E*8 MN theory. We thus find an S-dual Lagrangian theory of the *E*8 gauging of the *E*8 MN theory (with the other two *D**p*(*E*8) tails appended, to total rank being ${\\widehat}r=141$) as a *E*8-shaped special unitary quiver.\n#### Rank-6 model.\nFinally, the rank-6 theory with (*r*, *f*, *d**H*) = (6, 0, 77) appears in table [tab:Bounty] and table [tab:Ferrero], corresponding to the singularity *x*15 + *y*2 + *z*25 + *z*33*z*2 = 0. This implies the duality &\n[x=.6cm,y=.6cm] at (-5.8,0.5) ${{\\mathscr{T}\\_{{\\mathbf X}}^{\\rm 4d}}}=$; at (-3,0) *D*5; at (0,0) *S**O*(32); at (3,0) *D*216; at (0,2) *D*116; (-2.3,0)–(-1.3,0); (1.3,0)–(2.3,0); (0,0.5)–(0,1.3); at (4,0.3) ≡ ;\n[x=.6cm,y=.6cm] at (-2.3,0) *D*15; at (0,0) *E*8; at (2.3,0) *D*324; at (0,2) *D*120; (-1.6,0)–(-0.6,0); (0.6,0)–(1.6,0); (0,0.5)–(0,1.6);\nWe note that the *S**O*(32) trinion is actually a Lagrangian theory\n[x=0.8cm,y=0.8cm] (-3,0)–(4,0); (0,0)–(0,1); at (-3,0) [label=above:*S**p*(3)] ; at (-2,0) [label=below:Spin(16)] ; at (-1,0) [label=above:*S**p*(11)] ; at (0,0) [label=below:Spin(32)] ; at (0,1) [label=above:*S**p*(7)] ; at (1,0) [label=above:*S**p*(12)] ; at (2,0) [label=below:Spin(20)] ; at (3,0) [label=above:*S**p*(6)] ; at (4,0) [label=below:Spin(8)] ;\nwith a 9-dimensional conformal manifold and an electric one-form symmetry $\\frak{f}= {\\mathbb{Z}}\\_2^5$, in perfect agreement with the geometry.\n### Acknowledgements\nWe thank Fabio Apruzzi, Lakshya Bhardwaj, Antoine Bourget, Stefano Cremonesi, Michele Del Zotto, Antonella Grassi, Amihay Hanany, Sheldon Katz, Horia Magureanu, Wolfger Peelaers and Timo Weigand for discussions. The work of SG, SSN, Y-NW, is supported by the ERC Consolidator Grant number 682608 ``Higgs bundles: Supersymmetric Gauge Theories and Geometry (HIGGSBNDL)\". SSN acknowledges support also from the Simons Foundation. CC is a Royal Society University Research Fellow and a Birmingham Fellow.\nResolutions of Hypersurface Canonical Singularities\n===================================================\nIn this section, we present the detailed resolution sequences and resolved geometry of the isolated hypersurface (IHS) equations. We use the notation and conventions of, see also. In particular blowups are denoted by (x1(i1), x2(i2), x3(i3), x4(i4); ), which blows up the locus *x*1 = *x*2 = *x*3 = *x*4 = 0 with weights *i**j* and the exceptional divisor is *δ*. This blowup inserts a weighted projective space given by the weights. If there are not (*i**j*) labels, the weights are 1.\n*E**n*(*k*) Theories\n--------------------\nFor the *E**n*(*k*) theories with rank *r* > 0, we review the resolution sequences in. For *E*6(*k*) theories with the singular equation x13+x23+x33+x43+k=0, the resolution sequence is (x1,x2,x3,x4;1),(x1,x2,x3,i;i+1) (i=1,…,r-1). We use *S**i* to denote the exceptional divisor *δ**i* = 0, and the rank of this theory is $r= \\lfloor \\frac{k}{3}\\rfloor+1$. The triple intersection numbers among the divisors *S*1, …, *S**r* are\nll S13=S23=…=Sr-13=0 , & Sr3=3 \nSi2Si-1=-3 , &Si-12Si=3 ,(i=2,…,r).\nFor any *k* ≥ 3, the divisors *S*1, …, *S**r* − 1 are ruled surfaces over genus-1 curves, and all the intersection curves *S**i* ⋅ *S**i* − 1 (*i* = 2, …, *r*) are smooth too. For the topology of the divisor *S**r*, we have the following cases:\n1. *k* = 3*r* − 3: the divisor *S**r* is a smooth *d**P*6.\n2. *k* = 3*r* − 2: the resolved CY3 is smooth, but the divisor *S**r* is a singular cubic surface with the following equation: [E6-singular-surface] x13+x23+x33=0, which has a point singularity at *x*1 = *x*2 = *x*3 = 0.\n3. *k* = 3*r* − 1: the resolved CY3 has a terminal singularity at the point *x*1 = *x*2 = *x*3 = *δ**r* = 0: x13+x23+x33+r2=0. The divisor *S**r* also has the same singular equation in ([E6-singular-surface]).\nFor *E*7(*k*) theories (rank $r=\\lfloor \\frac{k}{4}\\rfloor+1$) with the singular equation x12+x24+x34+x44+k=0, the resolution sequence is (x1(2),x2(1),x3(1),x4(1);1),(x1(2),x2(1),x3(1),i(1);i+1),i=1,…,r-1. The triple intersection numbers among the divisors *S*1, …, *S**r* are\nll S13=S23=…=Sr-13=0 , & Sr3=2 \nSi2Si-1=-2 , &Si-12Si=2 ,(i=2,…,r).\nFor any *k* ≥ 4, the divisors *S*1, …, *S**r* − 1 are smooth genus-1 ruled surfaces. For the topology of the divisor *S**r*, we have the following cases:\n1. *k* = 4*r* − 4: the divisor *S**r* is a smooth *d**P*7.\n2. *k* = 4*r* − 3: the resolved CY3 is smooth, but the divisor *S**r* is a singular surface with the following equation: [E7-singular-surface] x12+x24+x34=0, which has a point singularity at *x*1 = *x*2 = *x*3 = 0.\n3. *k* = 4*r* − *s* (*s* = 1, 2): the resolved CY3 has a terminal singularity at the point *x*1 = *x*2 = *x*3 = *δ**r* = 0: x12+x24+x34+r4-s=0. The divisor *S**r* also has the same singular equation in ([E7-singular-surface]).\nFor *E*8(*k*) theories (rank $r=\\lfloor \\frac{k}{6}\\rfloor+1$) with the singular equation x12+x23+x36+x46+k=0, the resolution sequence is (x1(3),x2(2),x3(1),x4(1);1),(x1(3),x2(2),x3(1),i(1);i+1),i=1,…,r-1. The triple intersection numbers among the divisors *S*1, …, *S**r* are\nll S13=S23=…=Sr-13=0 , & Sr3=1 \nSi2Si-1=-1 , &Si-12Si=1 ,(i=2,…,r).\nFor any *k* ≥ 6, the divisors *S*1, …, *S**r* − 1 are smooth genus-1 ruled surfaces. For the topology of the divisor *S**r*, we have the following cases:\n1. *k* = 6*r* − 6: the divisor *S**r* is a smooth *d**P*8.\n2. *k* = 6*r* − 5: the resolved CY3 is smooth, but the divisor *S**r* is a singular surface with the following equation: [E8-singular-surface] x12+x23+x36=0, which has a point singularity at *x*1 = *x*2 = *x*3 = 0.\n3. *k* = 6*r* − *s* (*s* = 1, 2, 3, 4): the resolved CY3 has a terminal singularity at the point *x*1 = *x*2 = *x*3 = *δ**r* = 0: x12+x23+x36+r6-s=0. The divisor *S**r* also has the same singular equation in ([E8-singular-surface]).\n13 Sporadic Trinion Models\n--------------------------\nWe now determine the resolution for the 13 sporadic trinion theories from section [sec:SUN].\n#### Sporadic theory #(1)\nStarting from the singular equation x13 +x24+ x36+x42 x3=0, after the resolution (x1,x2,x3,x4;1), there is still a residual terminal singularity at *x*1 = *x*2 = *δ*1 = *x*4 = 0: x13+x241+13+x42=0, which is type II (2, 3, 3, 4), and equivalent to type I (2, 3, 3, 6) (*E*8( − 3)).\n#### Sporadic theory #(2)\nStarting from the singular equation x12+ x25+ x310+x43 x3=0, the resolution sequence is (x1(2),x2(1),x3(1),x4(1);1),(x1(3),x2(1),x4(2),1(1);2). After the resolution, the equation becomes x12+ x251+ x31016+x43 x3=0, the divisor *δ*1 = 0 is singular at *x*1 = *x*4 = 0: 1=0: x12+x43 x3=0. The detailed geometry was already discussed in, which eventually gives rise to 5d $SU(2)-SU(2)-5\\bm{F}$ gauge theory after a flop.\n#### Sporadic theory #(3)\nStarting from the singular equation x13+ x25+ x315+x42 x3=0, the resolution sequence is (x1,x2,x3,x4;1),(x1(2),x2(1),x4(3),1(1);3),(x1(2),x2(1),x4(3),3(1);4),(x1,x4,1;2). After the resolution, the equation becomes 2 x13+12 3 x25+11221036 x315+x3 x42=0. The divisor *δ*3 = 0 is singular *x*1 = *x*4 = 0 3=0: 2 x13+x3 x42=0.\n#### Sporadic theory #(4)\nStarting from the singular equation x13+ x25+x32 x2+x43 x2=0, after the resolution (x1,x2,x3,x4;1), there is a residual terminal singularity at *x*1 = *x*3 = *x*4 = *δ*1 = 0: x13+12+x32+x431+x1 x3 x4=0, which is Type I (2, 2, 3, 6) (corresponding to *E*8( − 4)), and another terminal singularity at *x*1 = *x*2 = *x*3 = *δ*1 = 0: x13+x2512+x32 x2+x21+x1 x3=0, which is the conifold singularity.\n#### Sporadic theory #(5)\nStarting from the singular equation x12+x27+x33 x2+x44 x2+x32 x42=0, after the resolution (x1(2),x2(1),x3(1),x4(1);1), there is a residual terminal singularity at *x*1 = *x*2 = *x*3 = *δ*1 = 0: x12+x2713+x33 x2+x21+x32=0, which is a conifold singularity. There is also a residual terminal singularity at *x*1 = *x*3 = *x*4 = *δ*1 = 0: x12+13+x33+x441+x32 x42=0, which is type I (2, 3, 3, 6), the same as *E*8( − 3).\n#### Sporadic theory #(6)\nFrom the singular equation x13+x29+x32 x2+x44 x2=0, the resolution sequence is (x1,x2,x3,x4;1),(x1(2),x3(3),x4(1),1(1);3),(x1,x3,1;2). The resolved equation and exceptional divisors are smooth.\n#### Sporadic theory #(7)\nFrom the singular equation x14+ x210+x32 x2+x43 x2+x1 x3 x4=0, after the resolution (x1,x2,x3,x4;1),(x1(1),x3(2),x4(1),1(1);3),(x3,1;2), the equation has a terminal singularity at *x*1 = *x*3 = *x*4 = *δ*3 = 0: x143+33+x32+x43+x1 x3 x4=0. It is type I (2, 3, 3, 6), which is the same as *E*8( − 3).\n#### Sporadic theory #(8)\nFrom the singular equation x12+x216+x33 x2 +x45 x2+x32 x42=0, the resolution sequence is (x1(2),x2(1),x3(1),x4(1);1),(x1(3),x3(2),x4(1),1(1);3),(x1(3),x3(2),x4(1),3(1);4),(x3,1;2). The resolved equation is x12+11221036 x216+2 x2 x33+12 3 x2 x45+x32 x42=0, and the divisor *δ*3 = 0 is singular at *x*1 = *x*3 = 0: x12+2 x2 x33+x32 x42=0.\n#### Sporadic theory #(9)\nFrom the singular equation x13+ x221+x32 x2+x45 x2+x1 x3 x4=0, the resolution sequence is &(x1,x2,x3,x4;1)&(x1(2),x3(3),x4(1),1(1);4)&(x1(2),x3(3),x4(1),4(1);6)&(x1(2),x3(3),x4(1),6(1);7)&(x1,x3,1;2),(x3,2;3),(x1,x3,4;5). The resolved equation is 2 5 x13+x1 x3 x4+118 216 315 412 510 66 x221+3 x2 x32+13 2 42 6 x2 x45=0. The divisor *δ*6 = 0 is singular at *x*1 = *x*3 = 0: 6=0:2 5 x13+x1 x3 x4+3 x2 x32=0.\n#### Sporadic theory #(10)\nFrom the singular equation x15+x225+x32 x2+x43 x2=0, the resolution sequence is &(x1,x2,x3,x4;1),(x1(1),x3(2),x4(1),1(1);4),(x1(1),x3(3),x4(2),4(1);7),&(x1(1),x3(3),x4(2),7(1);9),(x1(1),x3(3),x4(2),9(1);10),&(x3,1;2),(x4,2;3),(x3,x4,4;5),(x3,5;6),(x3,x4,7;8). The resolved equation is 12 2 43 5 72 9 x15+x1 x3 x4+122 221 320 418 516 615 712 810 96 x225+2 6 x2 x32+1 32 5 8 x2 x43=0. The divisor *δ*9 = 0 is singular at *x*3 = *x*4 = 0: 6=0: x1 x3 x4+2 6 x2 x32+1 32 5 8 x2 x43=0.\n#### Sporadic theory #(11)\nFrom the singular equation x17+x22 x1+x33 x1+x43 x1+x2 x32+x2 x42+x2 x3 x4=0, the resolution sequence is (x1,x2,x3,x4;1),(x2(2),x3(1),x4(1),1(1);3),(x2,1;2).\nThe resolved equation and exceptional divisors are smooth.\n#### Sporadic theory #(12)\nFrom the singular equation x113+x22 x1+x33 x1+x44 x1+x3 x43+2 x2 x3 x4=0, the resolution sequence is &(x1,x2,x3,x4;1)&(x2(2),x3(1),x4(1),1(1);4)&(x2(3),x2(2),x4(1),4(1);6)&(x2,1;2),(x3,2;3),(x2,x3,4;5). The resolved equation and exceptional divisors are smooth.\n#### Sporadic theory #(13)\nFrom the singular equation x131+x22 x1+x33 x1+x45 x1+3 x2 x3 x4=0, the resolution sequence is &(x1,x2,x3,x4;1),(x2(2),x3(1),x4(1),1(1);5),(x2(3),x2(2),x4(1),5(1);9)&(x2(3),x2(2),x4(1),9(1);12),(x2(3),x2(2),x4(1),12(1);14),(x2(3),x2(2),x4(1),14(1);15)&(x2,1;2),(x3,2;3),(x2,3;4),&(x2,x3,5;6),(x2,6;7),(x3,7;8),&(x2,x3,9;10),(x2,10;11),(x2,x3,12;13). The resolved equation is &128 227 326 425 524 622 721 820 918 1016 1115 1212 1310 146 x131+2 4 7 11 x1 x22+&1 32 4 6 82 10 13 x1 x33+13 22 3 54 62 7 93 10 122 14 x1 x45+3 x2 x3 x4=0. The divisor *δ*14 = 0 is singular at *x*2 = *x*3 = 0: 14=0: 2 4 7 11 x1 x22+1 32 4 6 82 10 13 x1 x33+3 x2 x3 x4=0.\nType VIII2 (*N*, *N*, 2, 2) series\n----------------------------------\nConsider the type VIII2 (*N*, *N*, 2, 2) series in table [tab:trinion series], which are equivalent to type IV(*N*, 2, *N*, 2). We now resolve these models.\n#### *N* = 2*k* − 1.\nStart with the singular equation x12k-1+x22k-1+x1 x32+x1 x42+x2 x3 x4=0, the resolution sequence is (x1,x2,x3,x4;1),(x3,x4,j;j+1),j=1,…,k-2. The resolved CY3 and divisors are all smooth.\n#### *N* = 2*k*.\nStart with the singular equation x12k+x22k+x1 x32+x1 x42+x2 x3 x4=0, the resolution sequence is (x1,x2,x3,x4;1),(x3,x4,j;j+1),j=1,…,k-2. The resolved CY3 is (x12k+x22k)i=1k-1i2k-2i-1+x1 x32+x1 x42+x2 x3 x4=0. There are in total 2*k* conifold singularities at the loci k-1=x3=x4=x1+x2 e=0, j=1,…,2k.\nProof of the Completeness of the *S**U*(*N*) Trinions\n=====================================================\nIn this section, we prove the completeness of our list of solutions to the conditions, and in section [sec:SUN].\nFirst of all, it is trivial to see that when *p*1, *p*2, *p*3 ≥ 3, the l.h.s. of,, is always strictly smaller than the right hand side. Hence one of *p**α*’s has to be smaller than 3 (equal to 2). We start with the equation ([Diophantine-eq2]) ++=N, we assume that *p*1 ≤ *p*2.\nIf *p*1 = 2, we have +=. If *p*2, *p*3 > 3, then $\\frac{N}{p\\_2}+\\frac{N-1}{p\\_3}<N/2$, and the above equation cannot hold. If *p*2 = 3, we have p3=<6, which leads to sporadic solutions (*p*1, *p*2, *p*3) = (2, 3, 3), (2, 3, 4), (2, 3, 5).\nOn the other hand, if *p*3 = 2, we have the infinite sequence (*p*1, *p*2, *p*3) = (2, 2*N*, 2). If *p*3 = 3, we have p2=<6. This leads to sporadic solutions (*p*1, *p*2, *p*3) = (2, 3, 3), (2, 4, 3), (2, 5, 3).\nFinally, if *p*3 = 2 and *p*1 > 2, we have +=. When *p*1, *p*2 > 3, this equation cannot hold. Hence one has to set *p*1 = 3 and p2=<6. It leads to sporadic solutions (*p*1, *p*2, *p*3) = (3, 3, 2), (3, 4, 2), (3, 5, 2). This concludes the enumeration of solutions to ([Diophantine-eq2]).\nNext we consider ([Diophantine-eq3]) ++=N, we assume that *p*2 ≤ *p*3. Then if *p*1 = 2, we have +=> If *p*2, *p*3 > 3, then $\\frac{1}{p\\_2}+\\frac{1}{p\\_3}\\leq 1/2$ and the above inequality cannot be satisfied. So we have to set *p*2 = 2 or *p*2 = 3. For the first case, we get the infinite sequence (*p*1, *p*2, *p*3) = (2, 2, 2*N* − 2). For the second case, we have p3=<6. Thus we have the sporadic solutions (*p*1, *p*2, *p*3) = (2, 3, 3), (2, 3, 4), (2, 3, 5).\nIf *p*1 > 2 and *p*2 = 2, then we have +=. It is easy to see that if *p*1, *p*3 > 3, the l.h.s. above is strictly smaller than the r.h.s.. If *p*1 = 3, then we have p3=<6. There are the sporadic solutions (*p*1, *p*2, *p*3) = (3, 2, 3), (3, 2, 4), (3, 2, 5). If *p*3 = 2, we have the infinite sequence (*p*1, *p*2, *p*3) = (*N*, 2, 2). If *p*3 = 3, we have p1=<6. This leads to the sporadic solutions (*p*1, *p*2, *p*3) = (3, 2, 3), (4, 2, 3), (5, 2, 3).\nFor ([Diophantine-eq4]) ++=N, we assume that *p*1 ≤ *p*2 ≤ *p*3, and hence we have to set *p*1 = 2. Now the equation is simplified to +=> . If *p*2, *p*3 > 3, the above inequality cannot be satisfied. So if *p*2 = 2, we have the infinite sequence (*p*1, *p*2, *p*3) = (2, 2, *N* − 1). If *p*2 = 3, we have the equation p3=<6, which leads to the sporadic solutions (*p*1, *p*2, *p*3) = (2, 3, 3), (2, 3, 4), (2, 3, 5).\nThis concludes the enumeration of solutions.\nTables of Trinions with *G* = *D*, *E*\n======================================\nIn this appendix we summarize the trinion theories for general *G* = *D*, *E* found in section [sec7:DandE]. These are given in the following tables [tab:b2b3h], [tab:SporadicSO], [tab:sporadicE6], [tab:sporadicE7], [tab:sporadicE8], [tab:Bounty], [tab:Twix], [tab:Snickers], [tab:Ferrero].\n[hb]\n|c|c||c|c|c||c|c||c|c|*S**O*(2*N*) : *N* & (*p*2, *p*3) & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n2*k* & (4, 4) & 3*k* − 3 & 6 & 9*k* + 3 & 9*k* − 3 & 3*k* + 3 & 0 & 0 \n4*k* + 1 & (4, 4) & 6*k* − 3 & 3 & 18*k* + 6 & 18*k* + 3 & 6*k* & 6 & $\\mb{Z}\\_2^3$ \n4*k* + 3 & (4, 4) & 6*k* + 1 & 7 & 18*k* + 17 & 18*k* + 10 & 6*k* + 8 & 0 & 0 \n6*k* & (3, 6) & 6*k* − 2 & 4 & 30*k* + 2 & 30*k* − 2 & 6*k* + 2 & 0 & 0 \n6*k* + 1 & (3, 6) & 6*k* − 2 & 2 & 30*k* + 6 & 30*k* + 4 & 6*k* & 8 & $\\mb{Z}\\_2^4$ \n6*k* + 2 & (3, 6) & 6*k* & 4 & 30*k* + 12 & 30*k* + 8 & 6*k* + 4 & 0 & 0 \n6*k* + 3 & (3, 6) & 6*k* & 2 & 30*k* + 16 & 30*k* + 14 & 6*k* + 2 & 0 & $\\mb{Z}\\_2$ \n6*k* + 4 & (3, 6) & 6*k* + 2 & 6 & 30*k* + 23 & 30*k* + 17 & 6*k* + 8 & 2 & $\\mb{Z}\\_2$ \n6*k* + 5 & (3, 6) & 6*k* + 2 & 2 & 30*k* + 26 & 30*k* + 24 & 6*k* + 4 & 0 & $\\mb{Z}\\_2$ \n|c|c|c||c|c|c||c|c||c|c|*G* & (*p*2, *p*3) & *F* & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n*E*6 & (4, 4) & *x*3 + *t*4 + *z*24 + *z*34 & 3 & 0 & 27 & 27 & 3 & 12 & $\\mb{Z}\\_3^3$ \n*E*6 & (3, 6) & *x*3 + *t*4 + *z*23 + *z*36 & 5 & 4 & 32 & 28 & 9 & 2 & $\\mb{Z}\\_2$ \n*E*7 & (4, 4) & *x*3 + *x**t*3 + *z*24 + *z*34 & 9 & 3 & 33 & 30 & 12 & 0 & 0 \n*E*7 & (3, 6) & *x*3 + *x**t*3 + *z*23 + *z*36 & 8 & 2 & 36 & 34 & 10 & 4 & $\\mb{Z}\\_3$ \n*E*8 & (4, 4) & *x*3 + *t*5 + *z*24 + *z*34 & 12 & 0 & 36 & 36 & 12 & 0 & 0 \n*E*8 & (3, 6) & *x*3 + *t*5 + *z*23 + *z*36 & 12 & 0 & 40 & 40 & 12 & 8 & $\\mb{Z}\\_5$ \n[hb]\n| c||c |c c|c |ccc|cc|cc| (*b*2, *b*3)& *G* & *p*2 & *p*3 & *F* & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n (2*N* − 2, *N*)&*S**O*(8)& 3 & 4 & *x*3 + *x**t*2 + *z*23 + *z*34*t* & 2 & 6 & 23 & 17 & 8 & 2 & $\\mb{Z}\\_2$ \n&*S**O*(8)& 6 & 2 & *x*3 + *x**t*2 + *z*26 + *z*32*t* & 2 & 6 & 23 & 17 & 8 & 2 & $\\mb{Z}\\_2$ \n&*S**O*(12)& 5 & 2 & *x*5 + *x**t*2 + *z*25 + *z*32*t* & 4 & 4 & 30 & 26 & 8 & 4 & $\\mb{Z}\\_2^2$ \n(*N*, *N*)&*S**O*(8) & 2 & 4 & *x*3 + *x**t*2 + *z*22*t* + *z*34*t* + *z*36 & 2 & 6 & 23 & 17 & 8 & 2 & $\\mb{Z}\\_2$ \n&*S**O*(12)& 2 & 3 & *x*5 + *x**t*2 + *z*22*t* + *z*33*t* + *z*35 & 4 & 4 & 30 & 26 & 8 & 4 & $\\mb{Z}\\_2^2$ \n| c||c |c c|c |ccc|cc|cc| (*b*2, *b*3)& *G* & *p*2 & *p*3 & *F* & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n (12, 9)&*E*6 & 4 & 3 & *x*3 + *t*4 + *z*24 + *z*33*t* & 3 & 0 & 27 & 27 & 3 & 12 & $\\mb{Z}\\_3^3$ \n&*E*6 & 8 & 2 & *x*3 + *t*4 + *z*28 + *z*32*t* & 3 & 0 & 35 & 35 & 3 & 4 & $\\mb{Z}\\_3^2$ \n(12, 8) &*E*6 & 3 & 4 & *x*3 + *t*4 + *z*23 + *z*34*x* & 5 & 4 & 32 & 28 & 9 & 2 & $\\mb{Z}\\_2$ \n&*E*6 & 6 & 2 & *x*3 + *t*4 + *z*26 + *z*32*x* & 5 & 4 & 32 & 28 & 9 & 2 & $\\mb{Z}\\_2$ \n(8, 8) &*E*6 & 2 & 4 & *x*3 + *t*4 + *z*22*x* + *z*34*x* + *z*36 & 5 & 4 & 32 & 28 & 9 & 0 & $\\mb{Z}\\_2$ \n(9, 9) &*E*6 & 2 & 6 & *x*3 + *t*4 + *z*22*t* + *z*36*t* + *z*38 & 3 & 0 & 35 & 35 & 3 & 2 & $\\mb{Z}\\_3^2$ \n&*E*6 & 3 & 3 & *x*3 + *t*4 + *z*23*t* + *z*33*t* + *z*34 & 3 & 0 & 27 & 27 & 3 & 4 & $\\mb{Z}\\_3^3$ \n[!htbp]\n| c||c |c c|c |ccc|cc|cc| (*b*2, *b*3)& *G* & *p*2 & *p*3 & *F* & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n (18, 14)&*E*7 & 9 & 2 & *x*3 + *x**t*3 + *z*29 + *z*32*t* & 6 & 2 & 45 & 43 & 8 & 6 & $\\mb{Z}\\_2^3$ \n(14, 14) &*E*7 & 2 & 7 & *x*3 + *x**t*3 + *z*22*t* + *z*37*t* + *z*39 & 6 & 2 & 45 & 43 & 8 & 6 & $\\mb{Z}\\_2^3$ \n[!htbp]\n| c||c |c c|c |ccc|cc|cc| (*b*2, *b*3)& *G* & *p*2 & *p*3 & *F* & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n (30, 24)&*E*8 & 10 & 2 & *x*3 + *t*5 + *z*210 + *z*32*t* & 10 & 0 & 54 & 54 & 10 & 8 & $\\mb{Z}\\_3^2$ \n(30, 20)&*E*8 & 3 & 4 & *x*3 + *t*5 + *z*23 + *z*34*x* & 12 & 0 & 40 & 40 & 12 & 8 & $\\mb{Z}\\_5$ \n&*E*8 & 6 & 2 & *x*3 + *t*5 + *z*26 + *z*32*x* & 12 & 0 & 40 & 40 & 12 & 8 & $\\mb{Z}\\_5$ \n(24, 24)&*E*8 & 2 & 8 & *x*3 + *t*5 + *z*22*t* + *z*38*t* + *z*310 & 10 & 0 & 54 & 54 & 10 & 4 & $\\mb{Z}\\_3^2$ \n(20, 20)&*E*8 & 2 & 4 & *x*3 + *t*5 + *z*22*x* + *z*34*x* + *z*36 & 12 & 0 & 40 & 40 & 12 & 0 & $\\mb{Z}\\_5$ \n[!htbp]\n| c||c |c c|c |ccc|cc|cc| (*b*2, *b*3)& *G* & *p*2 & *p*3 & *F* & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n (2*N* − 2, 2*N* − 2)&*S**O*(8)& 4 & 12 & *x*3 + *y*2 + *z*24 + *z*312 & 0 & 6 & 36 & 30 & 6 & 0 & 0 \n&*S**O*(8)& 6 & 6 & *x*3 + *y*2 + *z*26 + *z*36 & 1 & 8 & 29 & 21 & 9 & 0 & 0 \n&*S**O*(10)& 3 & 24 & *x*4 + *y*2 + *z*23 + *z*324 & 0 & 6 & 72 & 66 & 6 & 0 & 0 \n&*S**O*(10)& 4 & 8 & *x*4 + *y*2 + *z*24 + *z*38 & 2 & 7 & 35 & 28 & 9 & 2 & $\\mb{Z}\\_2$ \n&*S**O*(12)& 3 & 15 & *x*5 + *y*2 + *z*23 + *z*315 & 0 & 0 & 56 & 56 & 0 & 0 & $\\mb{Z}\\_2^4$ \n&*S**O*(12)& 5 & 5 & *x*5 + *y*2 + *z*25 + *z*35 & 2 & 0 & 32 & 32 & 2 & 12 & $\\mb{Z}\\_2^6$ \n&*S**O*(14)& 3 & 12 & *x*6 + *y*2 + *z*23 + *z*312 & 2 & 8 & 59 & 51 & 10 & 2 & $\\mb{Z}\\_2$ \n&*S**O*(14)& 4 & 6 & *x*6 + *y*2 + *z*24 + *z*36 & 4 & 5 & 40 & 35 & 9 & 4 & $\\mb{Z}\\_2^2$ \n&*S**O*(20)& 3 & 9 & *x*9 + *y*2 + *z*23 + *z*39 & 3 & 0 & 64 & 64 & 3 & 14 & $\\mb{Z}\\_2^7$ \n&*S**O*(22)& 4 & 5 & *x*10 + *y*2 + *z*24 + *z*35 & 6 & 4 & 56 & 52 & 10 & 4 & $\\mb{Z}\\_2^2$ \n&*S**O*(26)& 3 & 8 & *x*12 + *y*2 + *z*23 + *z*38 & 5 & 6 & 80 & 74 & 11 & 2 & $\\mb{Z}\\_2$ \n&*S**O*(44)& 3 & 7 & *x*21 + *y*2 + *z*23 + *z*37 & 6 & 0 & 120 & 120 & 6 & 12 & $\\mb{Z}\\_2^6$ \n(2*N* − 2, *N*)&*S**O*(8) & 4 & 8 & *x*3 + *y*2 + *z*24 + *z*312 & 0 & 6 & 36 & 30 & 6 & 0 & 0 \n&*S**O*(8) & 5 & 5 & *x*3 + *y*2 + *z*25 + *z*35*x* & 0 & 0 & 26 & 26 & 0 & 0 & $\\mb{Z}\\_2^2$ \n&*S**O*(8) & 6 & 4 & *x*3 + *y*2 + *z*26 + *z*36 & 1 & 8 & 29 & 21 & 9 & 0 & 0 \n&*S**O*(8) & 9 & 3 & *x*3 + *y*2 + *z*29 + *z*33*x* & 0 & 0 & 28 & 28 & 0 & 0 & $\\mb{Z}\\_2^3$ \n&*S**O*(10) & 3 & 15 & *x*4 + *y*2 + *z*23 + *z*324 & 0 & 6 & 72 & 66 & 6 & 0 & 0 \n&*S**O*(10) & 4 & 5 & *x*4 + *y*2 + *z*24 + *z*38 & 2 & 7 & 35 & 28 & 9 & 2 & $\\mb{Z}\\_2$ \n&*S**O*(10) & 6 & 3 & *x*4 + *y*2 + *z*26 + *z*34*z*2 & 2 & 3 & 30 & 27 & 5 & 0 & 0 \n&*S**O*(10) & 16 & 2 & *x*4 + *y*2 + *z*216 + *z*33*z*2 & 1 & 7 & 53 & 46 & 8 & 0 & 0 \n&*S**O*(12) & 3 & 9 & *x*5 + *y*2 + *z*23 + *z*315 & 0 & 0 & 56 & 56 & 0 & 0 & $\\mb{Z}\\_2^4$ \n&*S**O*(12) & 4 & 4 & *x*5 + *y*2 + *z*24 + *z*35*z*2 & 2 & 4 & 36 & 32 & 6 & 0 & 0 \n&*S**O*(12) & 5 & 3 & *x*5 + *y*2 + *z*25 + *z*35 & 2 & 0 & 32 & 32 & 2 & 12 & $\\mb{Z}\\_2^6$ \n&*S**O*(12) & 10 & 2 & *x*5 + *y*2 + *z*210 + *z*33*z*2 & 2 & 8 & 46 & 38 & 10 & 0 & 0 \n&*S**O*(14) & 3 & 7 & *x*6 + *y*2 + *z*23 + *z*312 & 2 & 8 & 59 & 51 & 10 & 2 & $\\mb{Z}\\_2$ \n&*S**O*(14) & 8 & 2 & *x*6 + *y*2 + *z*28 + *z*33*z*2 & 3 & 5 & 45 & 40 & 8 & 0 & 0 \n&*S**O*(16) & 3 & 6 & *x*7 + *y*2 + *z*23 + *z*39*x* & 1 & 0 & 57 & 57 & 1 & 0 & $\\mb{Z}\\_2^3$ \n&*S**O*(16) & 7 & 2 & *x*7 + *y*2 + *z*27 + *z*33*x* & 3 & 0 & 45 & 45 & 3 & 12 & $\\mb{Z}\\_2^6$ \n&*S**O*(18) & 4 & 3 & *x*8 + *y*2 + *z*24 + *z*34*z*2 & 5 & 5 & 48 & 43 & 10 & 2 & $\\mb{Z}\\_2$ \n&*S**O*(20) & 3 & 5 & *x*9 + *y*2 + *z*23 + *z*39 & 3 & 0 & 64 & 64 & 3 & 14 & $\\mb{Z}\\_2^7$ \n&*S**O*(20) & 6 & 2 & *x*9 + *y*2 + *z*26 + *z*33*z*2 & 5 & 6 & 55 & 49 & 11 & 0 & 0 \n&*S**O*(32) & 3 & 4 & *x*15 + *y*2 + *z*23 + *z*34*z*2 & 2 & 4 & 72 & 68 & 6 & 4 & $\\mb{Z}\\_5$ \n&*S**O*(32) & 5 & 2 & *x*15 + *y*2 + *z*25 + *z*33*z*2 & 6 & 0 & 77 & 77 & 6 & 10 & $\\mb{Z}\\_2^5$ \n(*N*, *N*)&*S**O*(8) & 3 & 6 & *x*3 + *y*2 + *z*24*z*3 + *z*39 & 0 & 0 & 28 & 28 & 0 & 0 & $\\mb{Z}\\_2^3$ \n&*S**O*(8) & 4 & 4 & *x*3 + *y*2 + *z*26 + *z*36 & 1 & 8 & 29 & 21 & 9 & 0 & 0 \n&*S**O*(10) & 2 & 10 & *x*4 + *y*2 + *z*23*z*3 + *z*316 & 1 & 7 & 53 & 46 & 8 & 0 & 0 \n&*S**O*(12) & 2 & 6 & *x*5 + *y*2 + *z*23*z*3 + *z*310 & 2 & 8 & 46 & 38 & 10 & 0 & 0 \n&*S**O*(12) & 3 & 3 & *x*5 + *y*2 + *z*25 + *z*35 & 2 & 0 & 32 & 32 & 2 & 12 & $\\mb{Z}\\_2^6$ \n&*S**O*(16) & 2 & 4 & *x*7 + *y*2 + *z*23*z*3 + *z*37 & 3 & 0 & 45 & 45 & 3 & 12 & $\\mb{Z}\\_2^6$ \n&*S**O*(24) & 2 & 3 & *x*11 + *y*2 + *z*23*z*3 + *z*34*z*2 & 5 & 0 & 60 & 60 & 5 & 10 & $\\mb{Z}\\_2^5$ \n[!htbp]\n| c||c c|c |ccc|cc|cc| (*b*1, *b*2, *b*3)& *p*2 & *p*3 & *F* & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n(9, 12, 12) & 5 & 20 & *x*3 + *y*2 + *z*25 + *z*320 & 0 & 0 & 76 & 76 & 0 & 0 & $\\mb{Z}\\_3^2$ \n& 6 & 12 & *x*3 + *y*2 + *z*26 + *z*312 & 2 & 8 & 59 & 51 & 10 & 2 & $\\mb{Z}\\_2$ \n& 8 & 8 & *x*3 + *y*2 + *z*28 + *z*38 & 2 & 0 & 49 & 49 & 2 & 6 & $\\mb{Z}\\_3^3$ \n(9, 12, 9) & 5 & 15 & *x*3 + *y*2 + *z*25 + *z*320 & 0 & 0 & 76 & 76 & 0 & 0 & $\\mb{Z}\\_3^2$ \n& 6 & 9 & *x*3 + *y*2 + *z*26 + *z*312 & 2 & 8 & 59 & 51 & 10 & 2 & $\\mb{Z}\\_2$ \n& 7 & 7 & *x*3 + *y*2 + *z*27 + *z*38*z*2 & 1 & 0 & 50 & 50 & 1 & 0 & $\\mb{Z}\\_3$ \n& 8 & 6 & *x*3 + *y*2 + *z*28 + *z*38 & 2 & 0 & 49 & 49 & 2 & 6 & $\\mb{Z}\\_3^3$ \n& 10 & 5 & *x*3 + *y*2 + *z*210 + *z*36*z*2 & 1 & 0 & 51 & 51 & 1 & 0 & $\\mb{Z}\\_3$ \n& 16 & 4 & *x*3 + *y*2 + *z*216 + *z*35*z*2 & 1 & 0 & 65 & 65 & 1 & 0 & $\\mb{Z}\\_3^2$ \n(9, 12, 8) & 6 & 8 & *x*3 + *y*2 + *z*26 + *z*38*x* & 2 & 8 & 59 & 51 & 10 & 2 & $\\mb{Z}\\_2$ \n& 12 & 4 & *x*3 + *y*2 + *z*212 + *z*34*x* & 2 & 8 & 59 & 51 & 10 & 2 & $\\mb{Z}\\_2$ \n& 36 & 3 & *x*3 + *y*2 + *z*236 + *z*33*x* & 0 & 7 & 126 & 119 & 7 & 0 & 0 \n(9, 9, 9) & 4 & 12 & *x*3 + *y*2 + *z*25*z*3 + *z*316 & 1 & 0 & 65 & 65 & 1 & 0 & $\\mb{Z}\\_3^2$ \n& 6 & 6 & *x*3 + *y*2 + *z*28 + *z*38 & 2 & 0 & 49 & 49 & 2 & 6 & $\\mb{Z}\\_3^3$ \n(9, 9, 8) & 9 & 4 & *x*3 + *y*2 + *z*212 + *z*34*x* & 2 & 8 & 59 & 51 & 10 & 2 & $\\mb{Z}\\_2$ \n& 27 & 3 & *x*3 + *y*2 + *z*236 + *z*33*x* & 0 & 7 & 126 & 119 & 7 & 0 & 0 \n(9, 8, 8) & 3 & 24 & *x*3 + *y*2 + *z*23*x* + *z*324*x* + *z*336 & 0 & 7 & 126 & 119 & 7 & 0 & 0 \n& 4 & 8 & *x*3 + *y*2 + *z*24*x* + *z*38*x* + *z*312 & 2 & 8 & 59 & 51 & 10 & 2 & $\\mb{Z}\\_2$ \n(8, 12, 12) & 4 & 12 & *t*4 + *y*2 + *z*24 + *z*312 & 3 & 7 & 53 & 46 & 10 & 4 & $\\mb{Z}\\_3$ \n& 6 & 6 & *t*4 + *y*2 + *z*26 + *z*36 & 4 & 5 & 40 & 35 & 9 & 4 & $\\mb{Z}\\_2^2$ \n(8, 12, 9) & 4 & 9 & *t*4 + *y*2 + *z*24 + *z*39*t* & 3 & 7 & 53 & 46 & 10 & 4 & $\\mb{Z}\\_3$ \n& 12 & 3 & *t*4 + *y*2 + *z*212 + *z*33*t* & 3 & 7 & 53 & 46 & 10 & 4 & $\\mb{Z}\\_3$ \n(8, 12, 8) & 4 & 8 & *t*4 + *y*2 + *z*24 + *z*312 & 3 & 7 & 53 & 46 & 10 & 4 & $\\mb{Z}\\_3$ \n& 5 & 5 & *t*4 + *y*2 + *z*25 + *z*36*z*2 & 3 & 2 & 40 & 38 & 5 & 0 & 0 \n& 6 & 4 & *t*4 + *y*2 + *z*26 + *z*36 & 4 & 5 & 40 & 35 & 9 & 4 & $\\mb{Z}\\_2^2$ \n& 9 & 3 & *t*4 + *y*2 + *z*29 + *z*34*z*2 & 4 & 4 & 44 & 40 & 8 & 0 & 0 \n(8, 9, 9) & 3 & 9 & *t*4 + *y*2 + *z*23*t* + *z*39*t* + *z*312 & 3 & 7 & 53 & 46 & 10 & 4 & $\\mb{Z}\\_3$ \n(8, 9, 8) & 3 & 8 & *t*4 + *y*2 + *z*23*t* + *z*312 & 3 & 7 & 53 & 46 & 10 & 4 & $\\mb{Z}\\_3$ \n(8, 8, 8) & 3 & 6 & *t*4 + *y*2 + *z*24*z*3 + *z*39 & 4 & 4 & 44 & 40 & 8 & 0 & 0 \n& 4 & 4 & *t*4 + *y*2 + *z*26 + *z*36 & 4 & 5 & 40 & 35 & 9 & 4 & $\\mb{Z}\\_2^2$ \n[!htbp]\n| c||c c|c |ccc|cc|cc| (*b*2, *b*3)& *p*2 & *p*3 & *F* & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n(18, 18) & 5 & 45 & *x*3 + *y*2 + *z*25 + *z*345 & 0 & 0 & 176 & 176 & 0 & 0 & $\\mb{Z}\\_2^4$ \n& 6 & 18 & *x*3 + *y*2 + *z*26 + *z*318 & 3 & 8 & 89 & 81 & 11 & 4 & $\\mb{Z}\\_3$ \n& 9 & 9 & *x*3 + *y*2 + *z*29 + *z*39 & 3 & 0 & 64 & 64 & 3 & 14 & $\\mb{Z}\\_2^7$ \n(18, 14) & 5 & 35 & *x*3 + *y*2 + *z*25 + *z*345 & 0 & 0 & 176 & 176 & 0 & 0 & $\\mb{Z}\\_2^4$ \n& 6 & 14 & *x*3 + *y*2 + *z*26 + *z*318 & 3 & 8 & 89 & 81 & 11 & 4 & $\\mb{Z}\\_3$ \n& 8 & 8 & *x*3 + *y*2 + *z*28 + *z*39*z*2 & 3 & 2 & 66 & 64 & 5 & 0 & 0 \n& 9 & 7 & *x*3 + *y*2 + *z*29 + *z*39 & 3 & 0 & 64 & 64 & 3 & 14 & $\\mb{Z}\\_2^7$ \n& 15 & 5 & *x*3 + *y*2 + *z*215 + *z*36*z*2 & 2 & 0 & 76 & 76 & 2 & 0 & $\\mb{Z}\\_2$ \n& 36 & 4 & *x*3 + *y*2 + *z*236 + *z*35*z*2 & 1 & 8 & 149 & 141 & 9 & 0 & 0 \n(14, 14) & 4 & 28 & *x*3 + *y*2 + *z*25*z*3 + *z*336 & 1 & 8 & 149 & 141 & 9 & 0 & 0 \n& 7 & 7 & *x*3 + *y*2 + *z*29 + *z*39 & 3 & 0 & 64 & 64 & 3 & 14 & $\\mb{Z}\\_2^7$ \n[!htbp]\n| c||c c|c |ccc|cc|cc| (*b*1, *b*2, *b*3)& *p*2 & *p*3 & *F* & *r* & *f* & *d**H* & ${\\widehat}r$ & ${\\widehat}d\\_H$ & *b*3 & $\\frak{f}$ \n(24, 30, 30) & 6 & 30 & *x*3 + *y*2 + *z*26 + *z*330 & 5 & 8 & 149 & 141 & 13 & 8 & $\\mb{Z}\\_5$ \n&10 & 10 & *x*3 + *y*2 + *z*210 + *z*310 & 5 & 0 & 81 & 81 & 5 & 16 & $\\mb{Z}\\_3^4$ \n(24, 30, 24) & 6 & 24 & *x*3 + *y*2 + *z*26 + *z*330 & 5 & 8 & 149 & 141 & 13 & 8 & $\\mb{Z}\\_5$ \n& 7 & 14 & *x*3 + *y*2 + *z*27 + *z*315*z*2 & 4 & 0 & 99 & 99 & 4 & 0 & 0 \n& 9 & 9 & *x*3 + *y*2 + *z*29 + *z*310*z*2 & 6 & 0 & 82 & 82 & 6 & 0 & $\\mb{Z}\\_2$ \n& 10 & 8 & *x*3 + *y*2 + *z*210 + *z*310 & 5 & 0 & 81 & 81 & 5 & 16 & $\\mb{Z}\\_3^4$ \n& 15 & 6 & *x*3 + *y*2 + *z*215 + *z*37*z*2 & 5 & 0 & 91 & 91 & 5 & 12 & $\\mb{Z}\\_2^6$ \n& 25 & 5 & *x*3 + *y*2 + *z*225 + *z*36*z*2 & 4 & 0 & 126 & 126 & 4 & 0 & 0 \n(24, 30, 20) & 6 & 20 & *x*3 + *y*2 + *z*26 + *z*320*x* & 5 & 8 & 149 & 141 & 13 & 8 & $\\mb{Z}\\_{5}$ \n& 15 & 5 & *x*3 + *y*2 + *z*215 + *z*35*x* & 5 & 0 & 91 & 91 & 5 & 12 & $\\mb{Z}\\_2^6$ \n& 30 & 4 & *x*3 + *y*2 + *z*230 + *z*34*x* & 5 & 8 & 149 & 141 & 13 & 8 & $\\mb{Z}\\_{5}$ \n(24, 24, 24) & 5 & 20 & *x*3 + *y*2 + *z*26*z*3 + *z*325 & 4 & 0 & 126 & 126 & 4 & 0 & 0 \n& 6 & 12 & *x*3 + *y*2 + *z*27*z*3 + *z*315 & 5 & 0 & 91 & 91 & 5 & 12 & $\\mb{Z}\\_2^6$ \n& 8 & 8 & *x*3 + *y*2 + *z*210 + *z*310 & 5 & 0 & 81 & 81 & 5 & 16 & $\\mb{Z}\\_3^4$ \n(24, 24, 20) & 6 & 10 & *x*3 + *y*2 + *z*27*z*3 + *z*310*x* & 5 & 0 & 91 & 91 & 5 & 12 & $\\mb{Z}\\_2^6$ \n& 9 & 6 & *x*3 + *y*2 + *z*210*z*3 + *z*36*x* & 6 & 0 & 82 & 82 & 6 & 0 & $\\mb{Z}\\_2$ \n& 12 & 5 & *x*3 + *y*2 + *z*215 + *z*35*x* & 5 & 0 & 91 & 91 & 5 & 12 & $\\mb{Z}\\_2^6$ \n& 24 & 4 & *x*3 + *y*2 + *z*230 + *z*34*x* & 5 & 8 & 149 & 141 & 13 & 8 & $\\mb{Z}\\_5$ \n(24, 20, 20) & 4 & 20 & *x*3 + *y*2 + *z*24*x* + *z*320*x* + *z*330 & 5 & 8 & 149 & 141 & 13 & 8 & $\\mb{Z}\\_{5}$ \n& 5 & 10 & *x*3 + *y*2 + *z*25*x* + *z*310*x* + *z*315 & 5 & 0 & 91 & 91 & 5 & 4 & $\\mb{Z}\\_2^6$ \n(20, 30, 30) & 4 & 12 & *t*5 + *y*2 + *z*24 + *z*312 & 8 & 0 & 66 & 66 & 8 & 4 & $\\mb{Z}\\_5$ \n& 6 & 6 & *t*5 + *y*2 + *z*26 + *z*36 & 6 & 0 & 50 & 50 & 6 & 16 & $\\mb{Z}\\_5^2$ \n(20, 30, 24) & 5 & 6 & *t*5 + *y*2 + *z*25 + *z*36*t* & 10 & 0 & 52 & 52 & 10 & 4 & $\\mb{Z}\\_2^2$ \n& 15 &3 & *t*5 + *y*2 + *z*215 + *z*33*t* & 6 & 0 & 77 & 77 & 6 & 10 & $\\mb{Z}\\_2^5$ \n(20, 30, 20) & 4 & 8 & *t*5 + *y*2 + *z*24 + *z*312 & 8 & 0 & 66 & 66 & 8 & 4 & $\\mb{Z}\\_5$ \n& 5 & 5 & *t*5 + *y*2 + *z*25 + *z*36*z*2 & 10 & 0 & 52 & 52 & 10 & 4 & $\\mb{Z}\\_2^2$ \n& 6 & 4 & *t*5 + *y*2 + *z*26 + *z*36 & 6 & 0 & 50 & 50 & 6 & 16 & $\\mb{Z}\\_5^2$ \n& 9 & 3 & *t*5 + *y*2 + *z*29 + *z*34*z*2 & 8 & 0 & 56 & 56 & 8 & 0 & 0 \n(20, 24, 24) & 3 & 12 & *t*5 + *y*2 + *z*23*t* + *z*312*t* + *z*315 & 6 & 0 & 77 & 77 & 6 & 10 & $\\mb{Z}\\_2^5$ \n& 4 & 6 & *t*5 + *y*2 + *z*24*t* + *z*36*t* + *z*25 & 10 & 0 & 52 & 52 & 10 & 4 & $\\mb{Z}\\_2^2$ \n(20, 24, 20) & 3 & 10 & *t*5 + *y*2 + *z*23*t* + *z*315 & 6 & 0 & 77 & 77 & 6 & 10 & $\\mb{Z}\\_2^5$ \n& 4 & 5 & *t*5 + *y*2 + *z*24*t* + *z*36*t* + *z*25 & 10 & 0 & 52 & 52 & 10 & 4 & $\\mb{Z}\\_2^2$ \n(20, 20, 20) & 3 & 6 & *t*5 + *y*2 + *z*24*z*3 + *z*39 & 8 & 0 & 56 & 56 & 8 & 0 & 0 \n& 4 & 4 & *t*5 + *y*2 + *z*26 + *z*36 & 6 & 0 & 50 & 50 & 6 & 16 & $\\mb{Z}\\_5^2$ \n---\n1. In 6d SCFTs engineered from elliptic threefolds in F-theory, these effects have been studied in, while in 5d and 4d SCFTs this was discussed in.[↩](#fnref1)\n2. This need not be the case for generic canonical singularities. If the 5d Higgs branch consists of several cones, they can intersect the 4d CB at distinct points due to instanton corrections (see *e.g.* ), but we should still be able to work locally near each point from which a quantum HB emanates. We thank the JHEP referee for pointing it out.[↩](#fnref2)\n3. More precisely, we may formulate the SW geometry in terms of a 2d $\\CN=(2,2)$ Landau-Ginzburg (LG) superpotential *W* = *x**N* + *x**N* − *b**z**p*, with (*x*, *z*) ∈ C × C\\*.[↩](#fnref3)\n4. Recall that, upon gauging a common *G* symmetry of several SCFTs $\\CT\\_k$, conformal invariance requires $2h^\\vee(G) -\\half \\sum\\_k k\\_G^{\\CT\\_k}=0$.[↩](#fnref4)\n5. In some instances, the 3d Lagrangian can be also obtained from a 4d N = 1 Lagrangian.[↩](#fnref5)\n6. For the case of *S**U*(*N*), we ignore the terms *x*1 and *x*2 in *F*(*x*1, *x*2, *x*3, *z*) since they do not appear in deformation terms, and we take *x* ≡ *x*3.[↩](#fnref6)\n7. Consider the N = (2, 0) theory of type *A**N* − 1 on a sphere with a regular full puncture and an irregular puncture of type III labelled by *s* + 1 partitions of *N*. In the notation of, the partitions are *Y**s* + 1 = … = *Y*2 = (*N* − 1, 1) and *Y*1 = (1*N*). It is easy to check that the CB spectra of this class-S theory and of the $\\CD\\_s(N, N-1)$ model are identical. We then find that the 3d mirror constructed following the algorithm of is precisely the mirror dual of the quiver.[↩](#fnref7)\n8. Recall that by defining *e* ≡ *N**f* − 2*N**c* the balance of a node, then quivers with all nodes with *e* = 0 are good, if there is *e* = − 1 they are ugly and *e* < − 1 bad.[↩](#fnref8)\n9. This is assuming that non-trivial 3d $\\CN=4$ fixed points with either an empty HB or an empty CB do not exist. (Here, a ‘trivial theory’ is a theory of free (twisted) hypermultiplets or a discrete gauging thereof.)[↩](#fnref9)\n10. It so happens that, for consistency with the 4d gauge-theory language, we have to call these wrapped M5-branes ‘electric’ and the M2-brane ‘magnetic’. This is only a matter of terminology and it should not cause any confusion.[↩](#fnref10)\n11. As mentioned above, in a quiver with only black nodes, the actual gauge group is ∏*i**U*(*n**i*)/*U*(1), where the *U*(1) is the overall *U*(1). It is customary to write the ‘all-black-dots’ quiver as shown here. One can then ‘ungauge’ the *U*(1) factor at any gauge node and get an equivalent gauge theory.[↩](#fnref11)\n12. Note that we present the fluxes m for the monopole operators in hypermultiplets, which are each comprised of two 3d $\\CN=2$ chiral-multiplet monopole operators with fluxes ± m.[↩](#fnref12)\n13. This Seiberg-like duality was first suggested by, and its meaning was clarified in. See also.[↩](#fnref13)\n14. Recall that, locally, each stack of 5-branes can be T-dualized (and S-dualized) to the D3-brane boundary condition described by the 3d $\\CN=4$ *T**λ*[*S**U*(*N*)] theory. This is how the magnetic quiver is most easily understood, in this setup.[↩](#fnref14)\n15. For instance, for the partitions of *N* = 5 appearing in, we have: $[3,2] \\rightarrow {\\scriptsize\\young(01,12,2)} \\, : \\, p\\_\\Delta= (1,1,2,2)$, $[2,2,1] \\rightarrow {\\scriptsize\\young(012,23)} \\, : \\, p\\_\\Delta= (1,2,2,3)$, and $[1,1,1,1,1] \\rightarrow {\\scriptsize\\young(01234)}\\, : \\, p\\_\\Delta= (1,2,3,4)$.[↩](#fnref15)\n16. More precisely, *S**U*(4) is embedded in the *S**U*(6) factor of the maximal subgroup *S**U*(6) × *S**U*(2) ⊂ *E*6.[↩](#fnref16)\n17. The other three are identified with Goldstone modes for the Higgsing and we can discard them.[↩](#fnref17)\n"},"data_id":{"kind":"string","value":"arxiv_0182394"}}},{"rowIdx":45595,"cells":{"content":{"kind":"string","value":"Quantum Tanner codes\n====================\nTanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the celebrated expander codes of Sipser and Spielman, which are asymptotically good classical LDPC codes.\nIn this work we apply this prescription to the left-right Cayley complex that lies at the heart of the recent construction of a *c*3 locally testable code by Dinur *et al.* Specifically, we view this complex as two graphs that share the same set of edges. By defining a Tanner code on each of those graphs we obtain two classical codes that together define a quantum code. This construction can be seen as a simplified variant of the Panteleev and Kalachev asymptotically good quantum LDPC code, with improved estimates for its minimum distance. This quantum code is closely related to the Dinur *et al.* code in more than one sense: indeed, we prove a theorem that simultaneously gives a linearly growing minimum distance for the quantum code and recovers the local testability of the Dinur *et al.* code.\nIntroduction\n============\nContributions\n-------------\nConstructing good linear error correcting codes has been a major endeavor of the information theory community since the early existence results of Shannon. In the past decades, codes have found new playgrounds in fields such as program checking, probabilistically checkable proofs or even quantum computing. Such applications require additional properties beside the capability to correct errors. For instance, a *locally testable code* (LTC) comes with the remarkable feature that it is possible to read a contant number of bits of a noisy message and estimate how far the message is from the code. In a recent breakthrough, Dinur *et al.* showed that LTCs exist even when requiring constant rate, constant distance and constant locality. For quantum computing, codes must be able to correct two types of errors: (classical) bit flips and (quantum) phase flips. For this reason, one can define *quantum codes* by giving a pair of linear codes ${\\EuScript C}\\_0, {\\EuScript C}\\_1$ that will each correct one type of errors. These codes cannot be chosen arbitrarily: they must satisfy some compatibility condition that reads ${\\EuScript C}\\_1 \\supset {\\EuScript C}\\_0^\\perp$. Of major interest — for theory and experiments — are quantum low-density parity-check (LDPC) codes corresponding to the case when both ${\\EuScript C}\\_0 = \\ker H\\_0$ and ${\\EuScript C}\\_1 = \\ker H\\_1$ are defined by *sparse* parity-check matrices *H*0 and *H*1. Here, the major open question concerned the existence of good quantum LDPC codes displaying a constant rate and a constant (relative) distance, and was answered positively in the breakthrough work of Panteleev and Kalachev, which also provided an alternative construction of classical LTC.\nA specific combinatorial object lies at the heart of the construction of the LTC of Dinur *et al.* : a *square complex*, and more specifically a *left-right Cayley complex*, which is a generalisation of expander graphs in higher dimension with cells of dimension 0 (vertices), 1 (edges) and 2 (squares, hence the name). We note that left-right Cayley complexes are also equal to balanced products of Cayley graphs, as introduced by Breuckmann and Eberhardt. Recalling the seminal expander codes of Sipser and Spielman building on the ideas of Tanner to obtain good classical LDPC codes by putting the bits on the edges of an expander graph and the parity-check constraints on its vertices *via* codes of constant size, it is tempting to apply a similar recipe with higher dimensional objects such as square complexes. In that case, the bits are naturally placed on the 2-dimensional cells of the complex (squares) and the constraints are again enforced at the vertices. The novelty then is that the local view of a vertex becomes a matrix of bits, and allows one to put constraints corresponding to a small tensor product code. The consequence is far reaching: the redundancy between the row constraints and column constraints of the small tensor codes translates into a form of robustness, which can then propagate to the entire code through the square complex, giving in particular the LTCs of Dinur *et al.* when the square complex is sufficiently expanding. \nIn the present paper, we apply the Tanner code prescription to the quantum case and construct quantum codes with qubits on the squares and constraints on the vertices of a left-right Cayley complex. The condition ${\\EuScript C}\\_1 \\supset {\\EuScript C}\\_0^\\perp$ prevents the constraints to be those of small tensor codes, but should come from the dual of such tensor codes. We show that if the complex is sufficiently expanding (*e.g.* the complex of ) and if the small dual tensor codes exhibit robustness (which is true with high probability for random codes ), then the construction gives rise to a family of asymptotically good LDPC codes displaying constant rate and linear minimal distance. The present construction borrows a lot from, indeed it uses the same ingredients, since also uses Tanner codes and has a left-right Cayley complex embedded in its construction: the proposed code family may therefore be seen as a simplified variant of. However, we wish to stress that it also involves a conceptual shift: the recent series of breakthrough asymptotic constructions of LPDC codes with large distances arguably relies upon increasingly refined notions of chain-complex products that improve upon the simple product idea of. Our approach breaks away from this paradigm by proposing to take a geometric (square) complex and directly apply to it the Tanner code strategy. Indeed, we show that the square complex can be viewed as two ordinary graphs that share the same set of edges, and the two classical codes ${\\EuScript C}\\_0,{\\EuScript C}\\_1$ that make up the quantum code are simply defined as classical Tanner codes on these two graphs. We obtain improved estimates for the minimum distance of the resulting quantum code, which for any given code rate scales like $n/\\Delta^{3/2+\\eps}$, where *n* is the code length, Δ is the degree of the underlying Cayley graphs, and $\\eps >0$ can be taken to be arbitrarily close to 0.\nInterestingly, Panteleev and Kalachev showed that there is often a classical LTC hiding behind a good quantum LDPC code, and it turns out that we can recover exactly the LTC of Dinur *et al.* from our quantum Tanner codes.[3](#fn3) It can indeed be argued that the present construction is the missing link between the quantum code construction of and the LTC construction of, for we show that the linear minimum distance of our quantum code and the local testability of the Dinur *et al.* code are direct consequences of a uniting Theorem that is the main technical result of the present work.\nContext and history\n-------------------\n#### Locally testable codes.\nA binary linear code is a subspace of ${\\mathbb F}\\_2^n$. A *κ*-locally testable code with *q* queries is a code *C* that comes with a tester: the tester requires access to at most *q* of bits of any given *n*-bit word *x*, accepts all words of the code, and rejects a word $x \\notin C$ with probability ${\\geqslant}\\frac \\kappa nd(x,C)$, where *d*(*x*, *C*) is the Hamming distance from *x* to the code, and *κ* is a constant called the detection probability. A *c*3-LTC, as constructed in, is such that the rate and distance of the code, as well as the number *q* of queries, are all constant. In that example, the test simply picks a vertex of the left-right Cayley complex and checks whether the local conditions corresponding to the small tensor code are satisfied.\nLTCs were defined in the early 90s and were first studied in the context of probabilistically checkable proofs (PCPs), before being investigated as mathematical objects of interest notably in. A good overview of the field can be found in. Let us note that a third construction of *c*3-LTC, besides those of and, was obtained by Lin and Hsieh through a method similar to that of, but relying on lossless expanders rather than spectral expanders.\n#### Quantum LDPC codes.\nDefining the distance of a quantum code is slightly more complicated than in the classical case (where it is the minimum Hamming weight of a nonzero codeword). It is the minimum of two distances, *d**X* and *d**Z*, that basically characterize how well the code behaves against the two possible types of errors occurring in the quantum case: *X*-type errors, also called bit flips (swapping the basis states ∣0⟩ and ∣1⟩) and *Z*-type errors, or phase flips (adding a phase − 1 to the state ∣1⟩ while acting trivially on ∣0⟩). Recall that a quantum CSS code is defined by a pair of classical codes ${\\EuScript C}\\_0 = \\ker H\\_0, {\\EuScript C}\\_1 = \\ker H\\_1$ such that ${\\EuScript C}\\_1 \\supset {\\EuScript C}\\_0^\\perp$ (or equivalently *H*0 ⋅ *H*1*T* = 0). The distance of the code is then given by *d* = min(*d**X*, *d**Z*) with \n$$ d\\_X = \\min\\_{w \\in {\\EuScript C}\\_0 \\setminus {\\EuScript C}\\_1^{\\perp}} |w|, \\quad d\\_Z = \\min\\_{w \\in {\\EuScript C}\\_1 \\setminus {\\EuScript C}\\_0^\\perp} |w|.$$ \n It is worth noting that for a quantum LDPC code, the sparsity of *H*0 and *H*1 implies that both ${\\EuScript C}\\_0^\\perp$ and ${\\EuScript C}\\_1^\\perp$, and therefore ${\\EuScript C}\\_0$ and ${\\EuScript C}\\_1$ contain (many) words of constant weight. In particular, the codes ${\\EuScript C}\\_0$ and ${\\EuScript C}\\_1$ are not asymptotically good, quite the opposite!\nThe study of quantum LDPC codes arguably started around 1997 with the paradigmatic surface code construction of Kitaev that encodes a constant number of logical qubits into *n* physical qubits and achieves a distance of $\\sqrt{n}$. Improving on this scaling turned out to be challenging: despite an early construction of achieving *n*1/2log1/4*n* in 2002, no further progress was made on this question until 2020. In the meantime, a major development was the idea of taking a special product of classical codes, which turned out to correspond to the tensor product of chain complexes that represent the two classical codes, and yielded quantum codes of contant rate and minimum distance $\\Theta(\\sqrt{n})$. Things accelerated quickly in 2020 when the logarithmic dependence of was first improved for constructions based on high-dimensional expanders, and then much more decisively in a series of works introducing various combinations of chain complex products together with graph lifts. Already well known in the classical case, lifts turned out to be crucial to significantly break the $\\sqrt{n}$ barrier on the distance of quantum LDPC codes. Finally, Panteleev and Kalachev proved the existence of asymptotically good quantum LDPC codes by considering non-abelian lifts of products of (classical) Tanner codes.\nWe also remark that it is possible to define quantum locally testable codes. The existence of such codes would have implications in Hamiltonian complexity, which is the quantum version of computational complexity. In particular, such codes would imply the NLTS conjecture formulated by Hastings, and which is itself implied by the quantum PCP conjecture. Current constructions of quantum LTC are still very weak at the moment and far from sufficient for such applications, however: they only encode a constant number of logical qubits with a minimum distance bounded by $O(\\sqrt{n})$.\nA very fruitful approach to prove the existence of certain objects is the probabilistic method and it is indeed very effective to prove that good classical LDPC codes and good quantum (non LDPC) codes exist. This method has failed, however, to produce *c*3-LTCs or good quantum LDPC codes. On the one hand, it is well known that a good LDPC code cannot be locally testable since removing a single constraint will yield another good code and therefore a word violating this single constraint will actually be far from the code; on the other hand, picking a good LDPC code for ${\\EuScript C}\\_0$ in the quantum case forces one to choose ${\\EuScript C}\\_1^\\perp$ to contain words of large weight and ${\\EuScript C}\\_1$ will then not be LDPC. For both problems, it seems essential to enforce some minimal structure, and left-right Cayley complexes have provided this fitting, long-awaited framework.\nThe paper is organised as follows. Section [sec:overview] gives an overview of the paper, describes the quantum code construction, states the main theorem and its consequences, with sketches of proofs. It is structured as a stand-alone extended summary and concludes with some open problems. Section [sec:prelim] is a preliminary to the detailed part of the paper and introduces the required technical material. Section [sec:main] is the core of the paper, giving the detailed construction of the quantum code, proving the main theorem and showing how it implies a linear distance for the quantum code and also how it implies that the Dinur *et al.* code is locally testable. An appendix is devoted to proving the required behaviour of random dual tensor codes.\nOverview\n========\n#### The left-right Cayley complex.\nLet us summarise the construction of the square complex of Dinur *et al.*. It is an incidence structure *X* between a set *V* of vertices, two sets of edges *E**A* and *E**B*, that we will refer to as *A*-edges and *B*-edges, and a set *Q* of squares (or quadrangles). The vertex-set *V* is defined from a group *G*: it will be useful for us that the complex is bipartite, *i.e.* the vertex set is partitioned as *V* = *V*0 ∪ *V*1, with *V*0 and *V*1 both identified as a copy of the group *G*. Formally, we set *V*0 = *G* × {0} and *V*1 = *G* × {1}. Next we have two self-inverse subsets *A* = *A*− 1 and *B* = *B*− 1 of the group *G*: a vertex *v*0 = (*g*, 0) ∈ *V*0 and a vertex *v*1 = (*g*, 1) are said to be related by an *A*-edge if *g*ʹ = *a**g* for some *a* ∈ *A*. Similarly, *v*0 and *v*1 are said to be related by a *B*-edge if *g*ʹ = *g**b* for some *b* ∈ *B*. The sets *E**A* and *E**B* make up the set of *A*-edges and *B*-edges respectively. In other words, the graph ${\\EuScript G}\\_A=(V,E\\_A)$ is the double cover of the *left* Cayley graph ${{\\sf Cay}}(G,A)$ and likewise ${\\EuScript G}\\_B=(V,E\\_B)$ is the double cover of the *right* Cayley graph ${{\\sf Cay}}(G,B)$.\nNext, the set *Q* of squares is defined as the set of 4-subsets of vertices of the form \n{(*g*, 0), (*a**g*, 1), (*g**b*, 1), (*a**g**b*, 0)}. \n A square is therefore made up of two vertices of *V*0, two vertices of *V*1 as represented on Figure [fig:square].\n(0,0) – (0,2) – (2,2) – (2,0) – (0,0); at (0,0) *g* ∈ *V*0; at (0,2) *g**b* ∈ *V*1; at (2,2) *a**g**b* ∈ *V*0; at (2,0) *a**g* ∈ *V*1;\nLet us remark that, if we restrict the vertex set to *V*0, every square is now incident to only two vertices (those in *V*0). The set of squares can now be seen as a set of edges on *V*0, and it therefore defines a graph that we denote by ${\\EuScript G}\\_0^{\\square}=(V\\_0,Q)$. Similarly, restricting to vertices of *V*1 defines the graph ${\\EuScript G}\\_1^{\\square}$, which is an exact replica of ${\\EuScript G}\\_0^{\\square}$: both graphs are defined over a copy of the group *G*, with *g*, *g*, ʹ ∈ *G* being related by an edge whenever *g*ʹ = *a**g**b* for some *a* ∈ *A*, *b* ∈ *B*. We assume for simplicity that *A* and *B* are of the same cardinality Δ.\n#### Tanner codes on the complex *X*.\nRecall the definition of a Tanner code, or expander code, on a graph,. For ${\\EuScript G}=(V,E)$ a regular graph of degree *n*0 and *C*0 a binary linear code of length *n*0, we define the Tanner code $T({\\EuScript G},C\\_0)$ on ${\\mathbb F}\\_2^E$ as the set of binary vectors indexed by *E* (functions from *E* to ${\\mathbb F}\\_2$), such that on the edge neighbourhood of every vertex *v* ∈ *V*, we see a codeword of *C*0[4](#fn4). Sipser and Spielman’s celebrated result is that if the graph ${\\EuScript G}$ is chosen from a family of *n*0-regular expander graphs, and if the base code *C*0 has sufficiently large minimum distance and sufficiently large rate, then the Tanner codes $T({\\EuScript G},C\\_0)$ form a family of asymptotically good codes.\nNow for every vertex *v* of the graph ${\\EuScript G}\\_0^\\square=(V\\_0,Q)$ (or of ${\\EuScript G}\\_1^\\square$) associated to the square complex *X*, there is a natural identification[5](#fn5) of its neighbourhood with the product set *A* × *B*. It therefore makes sense to consider codes *C*0 on the coordinate set *A* × *B* that are obtained from two small codes *C**A* and *C**B* of length Δ = ∣*A*∣ = ∣*B*∣, defined on coordinate sets *A* and *B*, respectively. We will refer to the restriction of an assignment $x \\in {\\mathbb F}\\_2^Q$ to the *Q*-neighbourhood *Q*(*v*) of some vertex *v* ∈ *V*0 as the *local view* of *x* in *v*. The Tanner code construction therefore consists in constraining the local views of *x* to belong to the code *C*0.\n#### The locally testable code of Dinur *et al.*\nLet *C*0 be defined as the tensor code *C**A* ⊗ *C**B* on the coordinate set *A* × *B*. In other words, this is the code such that for every fixed *b* ∈ *B*, we see a codeword of *C**A* on {(*a*, *b*), *a* ∈ *A*}, and for every fixed *a* we see a codeword of *C**B* on {(*a*, *b*), *b* ∈ *B*}. The Tanner code $T({\\EuScript G}\\_0^\\square,C\\_0)$ is exactly the locally testable code of Dinur *et al.* . If *C**A* = *C**B* is a linear code with parameters [Δ, *ρ*Δ, *δ*Δ], the resulting Tanner code $T({\\EuScript G}\\_0^\\square,C\\_0)$ has length Δ2∣*G*∣/2, a rate at least 2*ρ*2 − 1 and is shown to have a normalized minimum distance ≥ *δ*2(*δ* − *λ*/Δ), where *λ* is the (common) second largest eigenvalue of the Cayley graphs ${{\\sf Cay}}(G,A)$ and ${{\\sf Cay}}(G,B)$.\n#### Two Tanner codes that define a quantum LDPC code.\nBesides the base code *C*0 = *C**A* ⊗ *C**B* that we have defined over *A* × *B*, define the code *C*1 = *C**A*⊥ ⊗ *C**B*⊥. Now consider the two Tanner codes ${\\EuScript C}\\_0=T({\\EuScript G}\\_0^\\square,C\\_0^\\perp)$ and ${\\EuScript C}\\_1=T({\\EuScript G}\\_1^\\square,C\\_1^\\perp)$ that are defined over the same coordinate set *Q*. We claim that this pair of codes $({\\EuScript C}\\_0,{\\EuScript C}\\_1)$ satisfies the definition of a quantum CSS code, namely that ${\\EuScript C}\\_1\\supset {\\EuScript C}\\_0^\\perp$. Note crucially that we now enforce constraints corresponding to the dual of a tensor code at each vertex.\nThis last fact is best seen by looking at the generators (in quantum coding jargon) or parity-checks for these codes. Define a **C*0-generator* for ${\\EuScript C}\\_0$ (resp. a *C*1-generator for ${\\EuScript C}\\_1$) as a vector of ${\\mathbb F}\\_2^Q$ whose support lies entirely in the *Q*-neighbourhood *Q*(*v*) of a vertex *v* of *V*0 (resp. *V*1), and which is equal to a codeword of *C*0 (resp. *C*1) on *Q*(*v*). (The codes *C**i* and ${\\EuScript C}\\_i$ should not be confused!) The code ${\\EuScript C}\\_0$ (resp. ${\\EuScript C}\\_1$) is by definition the space of vectors orthogonal to all *C*0-generators (*C*1-generators). Now consider a *C*0-generator and a *C*1-generator on vertices *v*0 and *v*1. If the generators have intersecting supports then the vertices *v*0 and *v*1 must be neighbours. If they are connected by a *B*-edge, then their *Q*-neighbourhoods *A* × *B* share an *A*-set {(*a*, *b*), *a* ∈ *A*} for a fixed *b*, on which the *C*0-generator must equal a codeword of *C**A* and the *C*1-generator must equal a codeword of *C**A*⊥. The two generators must therefore be orthogonal to each other. We reach the same conclusion analogously if *v*0 and *v*1 are connected by an *A*-edge.\nFor reasons of symmetry, we will wish the base codes *C*0 and *C*1 to have the same rate: we will require *C**A* to have some rate *ρ* and *C**B* to have rate 1 − *ρ*. In this case, the length of the quantum code ${\\EuScript Q}=({\\EuScript C}\\_0,{\\EuScript C}\\_1)$ is given by the number of squares in the complex, namely Δ2∣*G*∣/2, and the number of parity check constraints is 2*ρ*(1 − *ρ*)Δ2∣*G*∣. We conclude that the quantum code has rate at least (2*ρ* − 1)2 which is non-zero for every *ρ* ≠ 1/2.\nWe will show that under the right conditions for the choice of the left-right Cayley complex *X* and the component codes *C**A* and *C**B*, we obtain an asymptotically good family of quantum codes ${\\EuScript Q}=({\\EuScript C}\\_0,{\\EuScript C}\\_1)$. The required conditions on the choice of *X* are the same as in, namely that the two Cayley graphs ${{\\sf Cay}}(G,A)$ and ${{\\sf Cay}}(G,B)$ should be Ramanujan graphs or almost Ramanujan graphs, *i.e.* with a second largest eigenvalue $\\lambda{\\leqslant}c\\sqrt{\\Delta}$ for some small constant *c*, and that a non-degeneracy condition (called Total No-Conjugacy or TNC) is satisfied by the sets *A*, *B*, ensuring that for all choices of *g* ∈ *G*, *a* ∈ *A* and *b* ∈ *B*, it holds that *a**g* ≠ *g**b*.\nIt is worth noting that the good asymptotic properties of classical expander codes follow as soon as the distance of the small code is larger than the second eigenvalue of the expander graph, since the normalized minimum distance is known to be at least *δ*(*δ* − *λ*/Δ) > 0. In the case we are considering however, the distance of the small code is upper bounded by Δ while the second eigenvalue scales like 2Δ. It is therefore necessary to inspect more closely the structure of the small dual codes $C\\_A \\otimes {\\mathbb F}\\_2^B + {\\mathbb F}\\_2^A \\otimes C\\_B$ in order to get a nontrivial bound on the distance of the quantum Tanner code.\n#### Robustness of the component codes *C**A*, *C**B*.\nThe required conditions for the codes *C**A*, *C**B* are that the minimum distances of *C**A*, *C**B*, *C**A*⊥ and *C**B*⊥ are all sufficiently large, and that the two dual tensor codes (*C**A* ⊗ *C**B*)⊥ and (*C**A*⊥ ⊗ *C**B*⊥)⊥ are sufficiently *robust.* Viewing the sets *A* and *B* as the row and column sets respectively of Δ × Δ matrices, let us say that a dual tensor code $(C\\_A^\\perp\\otimes\nC\\_B^\\perp)^\\perp=C\\_A\\otimes{\\mathbb F}\\_2^B + {\\mathbb F}\\_2^A\\otimes C\\_B$ is *w*-robust if any codeword *x* of weight ≤ *w* has its support included in the union of ∣*x*∣/*d**A* columns and ∣*x*∣/*d**B* rows, where *d**A* and *d**B* are the minimum distances of *C**A* and *C**B*. A similar notion is used both in and. In particular, *w*-robustness is equivalent to a notion of robustness for the tensor code *C**A* ⊗ *C**B*, which loosely speaking, says that if a vector of ${\\mathbb F}\\_2^{A\\times B}$ is close to the column code $C\\_A\\otimes{\\mathbb F}\\_2^B$ as well as to the row code ${\\mathbb F}\\_2^A\\otimes C\\_B$, then it must also be close to the tensor code *C**A* ⊗ *C**B*. We shall be more precise with this notion later on.\nTo obtain asymptotically good quantum codes, we are however not quite able to prove the existence of component codes *C**A*, *C**B* that yield robust dual tensor codes for a large enough parameter *w*. To overcome this problem, following, we introduce the following tweak: recall that if *C* is a code defined on the coordinate set *S*, and *T* ⊂ *S* is a subset of *S*, then we may define the punctured code that we will denote by (*C*)*T*, as the set of codewords of *C* restricted the set of coordinates *T*. Let us say that the dual tensor code $C\\_A\\otimes{\\mathbb F}\\_2^B +{\\mathbb F}\\_2^A\\otimes C\\_B$ has *w*-robustness *with resistance to puncturing* *p*, if, for any subsets *A*ʹ ⊂ *A* and *B*ʹ ⊂ *B* of cardinality ≥ Δ − *w*ʹ, *w*ʹ ≤ *p*, it remains *w*-robust when punctured outside of the set *A*ʹ × *B*ʹ. Equivalently, if the dual tensor code obtained from the punctured codes (*C**A*)*A*ʹ and (*C**B*)*B*ʹ is *w*-robust.\nUsing the method of, we will obtain that for any $\\eps \\in (0,1/2)$ and $\\gamma\\in (1/2+\\eps,1)$, and for random pairs of codes *C**A*, *C**B* of given fixed rates, the associated dual tensor code $C\\_A\\otimes{\\mathbb F}\\_2^B +{\\mathbb F}\\_2^A\\otimes C\\_B$ is, with high probability, *w*-robust with resistance to puncturing *w*ʹ, for $w = \\Delta^{3/2-\\eps}$ and *w*ʹ = Δ*γ*.\nOur main technical result will now take the following form.\ntheomain [thm:main] Fix $\\eps \\in (0,1/2)$, $\\gamma\\in (1/2+\\eps,1)$ and *δ* > 0. For any fixed large enough Δ, if the component codes *C**A* and *C**B* have minimum distance ≥ *δ*Δ and if the dual tensor code $C\\_A\\otimes{\\mathbb F}\\_2^B +{\\mathbb F}\\_2^A\\otimes C\\_B=C\\_1^\\perp$ is *w*-robust with *p*-resistance to puncturing for $w = \\Delta^{3/2-\\eps/2}$ and *p* = Δ*γ*, then there exists an infinite family of square complexes *X* for which the Tanner code ${\\EuScript C}\\_1=T({\\EuScript G}\\_1^\\square,C\\_1^\\perp)$ of length *n* = ∣*Q*∣ has the following property:\nfor any codeword $x\\in{\\EuScript C}\\_1$ of non-zero weight $< \\delta n/4\\Delta^{3/2+\\eps}$, there exists a vertex *v* ∈ *V*0, and a codeword *y* of ${\\EuScript C}\\_1$ entirely supported by the *Q*-neighbourhood of *v*, on which it coincides with a codeword of the tensor code *C**A* ⊗ *C**B*, and such that ∣*x* + *y*∣ < ∣*x*∣.\nWe recall from that there exists an infinite sequence of degrees Δ (namely *q* + 1, for *q* an odd prime power), such that for each fixed degree Δ, there exists an infinite family of left-right Cayley complexes (over the groups *G* = PSL2(*q**i*)), satisfying the TNC condition, for which both the (left) Cayley graph ${{\\sf Cay}}(G,A)$ and the (right) Cayley graph ${{\\sf Cay}}(G,B)$ are Ramanujan graphs. These complexes provide the infinite families of Theorem [thm:main]. Let us also mention that randomly chosen codes *C**A* and *C**B* will typically achieve the requirements of the above theorem. In the quantum case, the codeword *y* will in fact belong to ${\\EuScript C}\\_0^\\perp$.\n#### Sketch of proof of Theorem [thm:main].\nLet $x \\in {\\EuScript C}\\_1$ be a codeword of sufficiently low weight. It induces a subgraph ${\\EuScript G}\\_{1,x}^\\square$ of ${\\EuScript G}\\_1^\\square$ with vertex set *S* ⊂ *V*1. The local view for any *v* ∈ *S* corresponds to a codeword of $C\\_1^\\perp = C\\_A\\otimes{\\mathbb F}\\_2^B +{\\mathbb F}\\_2^A\\otimes C\\_B$. The *w*-robustness of this code guarantees that codewords of weight less than $w =\n\\Delta^{3/2-\\eps}$ have a support restricted to a small number of rows and columns: in particular, and this is the crucial consequence of the robustness property, it implies that the view restricted to any column (or row) is at distance $O(\\Delta^{1/2-\\eps})$ from a word of *C**A* (or *C**B*).\nLet us call *normal vertices* the vertices of *S* with degree less than $\\Delta^{3/2-\\eps}$ in ${\\EuScript G}\\_{1,x}^\\square$. Expansion in ${\\EuScript G}\\_1$ ensures that the set of exceptional (*i.e.* not normal) vertices is small compared to *S* and we will neglect it in this sketch. Dealing with exceptional vertices, however, is slightly technical since their number is not *that* small, and this is the reason why we will require robust codes that are resistant to puncturing (in order to discard rows or columns belonging to an exceptional vertex).\nDefine *T* ⊂ *V*0 to be vertices of *V*0 sharing a column (or a row) with a normal vertex in *S*, and such that the local view on this column (or row) is close to a nonzero codeword of *C**A* (or *C**B*). The global codeword *x* induces a subgraph of ${\\EuScript G}\\_A\\cup{\\EuScript G}\\_B$ in which the vertices of *T* have a degree Ω(Δ). Expansion in the graph ${\\EuScript G}\\_A \\cup {\\EuScript G}\\_B$ then implies that when ∣*x*∣ is too small, there can’t be too many vertices in *T*, which in turn implies that a typical vertex in *T* must be adjacent to a large number Ω(Δ) of vertices in *S*. In other words, the rows and columns that are close to codewords of *C**A* and *C**B* in the local views of normal vertices of *S*, must cluster around the vertices of *T*. So the local view of a typical vertex of *T* consist of many columns (or rows) containing almost undisturbed codewords of *C**A* (or *C**B*). The robustness of *C**A* ⊗ *C**B* then implies that the local view of such a vertex must be $\\Delta^{3/2-\\eps}$-close to a codeword *y* of the tensor code, which cannot be the zero codeword since the local view has weight Ω(Δ2). Adding this codeword will decrease the weight of *x*.\n#### Asymptotically good quantum codes.\nA straightforward consequence of Theorem [thm:main] is that the quantum code ${\\EuScript Q}=({\\EuScript C}\\_0,{\\EuScript C}\\_1)$ described above has constant rate and minimum distance linear in its length *n*.\n[thm:good-qLDPC] Let *X* be the infinite family of square complexes from Theorem [thm:main]. For any *ρ* ∈ (0, 1/2), $\\eps\\in(0,1/2)$ and *δ* > 0 satisfying − *δ*log2*δ* − (1 − *δ*)log2(1 − *δ*) < *ρ*, randomly choosing *C**A* and *C**B* of rates *ρ* and 1 − *ρ* yields, with probability > 0 for Δ large enough, an infinite sequence of quantum codes ${\\EuScript Q}=({\\EuScript C}\\_0,{\\EuScript C}\\_1)$ of rate (2*ρ* − 1)2, length *n* and minimum distance ${\\geqslant}\\delta n/4 \\Delta^{3/2+\\eps}$.\nFor a more precise statement see Theorems [thm:123] and [thm:goodcodes].\n[Sketch of proof of Theorem [thm:good-qLDPC]] Recall that the minimum distance of a quantum code $({\\EuScript C}\\_0,{\\EuScript C}\\_1)$ is the smallest weight of a vector that is either in ${\\EuScript C}\\_0\\setminus {\\EuScript C}\\_1^\\perp$ or in ${\\EuScript C}\\_1\\setminus C\\_0^\\perp$. The vector *y* in Theorem [thm:main] is in ${\\EuScript C}\\_0^\\perp$, so by applying repeatedly the existence a such a *y*, we obtain that any codeword *x* of ${\\EuScript C}\\_1$ of weight $<\\delta n/4\\Delta^{3/2+\\eps}$ must belong to ${\\EuScript C}\\_0^\\perp$. To similarly bound from below the weight of a vector in ${\\EuScript C}\\_0$ but not in ${\\EuScript C}\\_1^\\perp$, one must apply Theorem [thm:main] to the code ${\\EuScript C}\\_0$ instead of ${\\EuScript C}\\_1$, which just means that we need to ensure that the distance and robustness properties required of *C**A* and *C**B* are also satisfied by *C**A*⊥ and *C**B*⊥. We choose *C**A* and *C**B* randomly by picking a uniform random parity-check matrix for one code and a uniform random generator matrix for the other, so that properties typically satisfied by the pair *C**A*, *C**B* will also be satisfied by the pair *C**A*⊥, *C**B*⊥.\n#### Recovering the local testability of the construction of .\nRecall that the tester picks randomly a vertex *v* ∈ *V*0 and checks whether the local view *x**v* of the input vector $x\\in{\\mathbb F}\\_2^Q$ belongs to the small tensor code *C*0 = *C**A* ⊗ *C**B*. Proving local testability amounts to proving that the distance $d(x,{\\EuScript C})$ of *x* to the LTC ${\\EuScript C}$ is always proportional to the size ∣*S*∣ of the subset *S* ⊂ *V*0 of vertices for which *x**v* is not a codeword of *C*0. To this end we consider the collection Z = (*c**v*)*v* ∈ *V*0 of all the closest *C*0-codewords to the local views *x**v* of *x*. Conflating the small vector *c**v* ∈ *C*0 with a vector in ${\\mathbb F}\\_2^Q$ that equals *c**v* on the *Q*-neighbourhood of *v* and is zero elsewhere, we then define the vector $z=\\sum\\_{v\\in V\\_0}c\\_v\\in{\\mathbb F}\\_2^Q$ that we call the mismatch of the collection Z. If we have *z* = 0 then Z is the collection of local views of a global codeword $c\\in {\\EuScript C}$, and its distance to *x* must be proportional (up to a Δ2 factor) to ∣*S*∣. Otherwise the key observation is that the local views of the mismatch *z* at all the vertices of *V*1, must consist of codewords of $C\\_A\\otimes{\\mathbb F}\\_2^B+{\\mathbb F}\\_2^A\\otimes C\\_B$, in other words *z* must belong to the code ${\\EuScript C}\\_1$ of Theorem [thm:main]. The same Theorem [thm:main] then states that there exists a vector $y\\_v\\in{\\mathbb F}\\_2^Q$, that is equal to a codeword of *C*0 on the *Q*-neighbourhood of some vertex *v* ∈ *V*0, and such that ∣*z* + *y**v*∣ < ∣*z*∣. Consequently, if one replaces *c**v* by *c**v* + *y**v* in the collection Z, one reduces the weight of the mismatch, and by repeatedly applying the procedure (which we can think of as decoding the mismatch), we obtain a updated list Z than coincides with the set of local views of a codeword of ${\\EuScript C}$ whose distance to *x* is again easily shown to be proportional to the size of the original set *S*.\n#### Comments and open problems.\nA natural follow up problem is to devise an efficient decoding algorithm for quantum Tanner codes. At the moment, to the best of our knowledge, the only efficient decoder for quantum codes that corrects adversarial errors of weight larger than $\\sqrt{n}$ is that of which can correct errors of weight $\\Omega(\\sqrt{n} \\log n)$.\nWhile the construction of quantum Tanner codes described above is arguably conceptually simpler than the balanced product and lifted product constructions of, it comes at the price of larger weights for the generators, namely Θ(Δ2) instead of Θ(Δ). Even if they are constant, it would be very useful to decrease these weights as much as possible and it would be interesting to explore how the weight reduction technique of Hastings can help on this issue.\nIt would be naturally very interesting to find other complexes, besides left-right Caley complexes, on which the Tanner construction can be applied to yield good families of quantum LDPC codes.\nIt would also be desirable to have completely explicit constructions. Reed-Solomon codes (binarised versions) come close, but tensor codes of Reed-Solomon codes are only known to have the required robustness when the sum of the rates of the component codes is < 1, which is insufficient for the above constructions. In a similar vein, we remark that if one could improve the robustness of the component dual tensor codes to values above Δ3/2, we could improve the dependence on Δ of the relative minimum distance of the quantum code, potentially up to Ω(1/Δ). We also remark that we cannot hope to improve the dependence on Δ of the relative minimum distance above *O*(1/Δ). Indeed, we will prove the existence of words of ${\\EuScript C}\\_1\\setminus {\\EuScript C}\\_0^\\perp$ of weight less than *n*/Δ (see Section [sec:goodLDPC] for details). This shows that there is little room for improvement in our estimation of the quantum code minimum distance.\n#### Acknowledgements.\nWe would like to thank Benjamin Audoux, Alain Couvreur, Shai Evra, Omar Fawzi, Tali Kaufman, Jean-Pierre Tillich, and Christophe Vuillot for many fruitful discussions on quantum codes over the years. We also thank Max Hopkins for spotting a technical error in the application of Proposition [prop:tensor] in an earlier version of this manuscript. We acknowledge support from the Plan France 2030 through the project ANR-22-PETQ-0006. AL acknowledges support from the ANR through the QuantERA project QCDA, and GZ acknowledges support from the ANR through the project QUDATA, ANR-18-CE47-0010.\nPreliminaries\n=============\nExpander Graphs\n---------------\nLet ${\\EuScript G}=(V,E)$ be a graph. Graphs will be undirected but may have multiple edges. For *S*, *T* ⊂ *V*, let *E*(*S*, *T*) denote the multiset of edges with one endpoint in *S* and one endpoint in *T*[6](#fn6). Let ${\\EuScript G}$ be a Δ-regular graph on *n* vertices, and let Δ = *λ*1 ≥ *λ*2 ≥ … ≥ *λ**n* be the eigenvalues of the adjacency matrix of ${\\EuScript G}$. For *n* ≥ 3, we define $\\lambda({\\EuScript G}):= \\max\\{|\\lambda\\_i|,\n\\lambda\\_i\\neq \\pm \\Delta\\}$. The connected graph ${\\EuScript G}$ is said to be Ramanujan if $\\lambda({\\EuScript G}){\\leqslant}2\\sqrt{\\Delta-1}$.\nWe recall the following version of the expander mixing lemma (see e.g. ).\n[Expander mixing lemma] [lem:mixing] For a Δ-regular non-bipartite, connected graph $\\cG$ and any sets $S, T \\subset V(\\cG)$, it holds that \n$$ |E(S,T) | {\\leqslant}\\frac{\\Delta}{|V|} |S| |T| + \\lambda({\\EuScript G}) \\sqrt{|S| |T|}.$$ \n For a Δ-regular bipartite connected graph $\\cG$ over the vertex set *V* = *V*0 ∪ *V*1 and any sets *S* ⊂ *V*0, *T* ⊂ *V*1, it holds that \n$$ |E(S,T) | {\\leqslant}\\frac{\\Delta}{|V\\_0|} |S| |T| + \\lambda({\\EuScript G}) \\sqrt{|S| |T|}.$$ \nLeft-right Cayley complexes.\n----------------------------\nA *left-right Cayley complex* *X* is introduced in from a group *G* and two sets of generators *A* = *A*− 1 and *B* = *B*− 1. As in we will restrict ourselves, for the sake of simplicity, to the case ∣*A*∣ = ∣*B*∣ = Δ. The complex is made up of vertices, *A*-edges, *B*-edges, and squares. The vertex set is *G*, the *A*-edges are pairs of vertices of the form {*g*, *a**g*} and *B*-edges are of the form {*g*, *g**b*} for *g* ∈ *G*, *a* ∈ *A*, *b* ∈ *B*. A *square* is a set of group elements of the form {*g*, *a**g*, *g**b*, *a**g**b*}. The *Total No-Conjugacy* condition (TNC) requires that\n \n$$\\begin{aligned}\n\\label{eqn:TNC}\n\\forall a \\in A, b \\in B, g \\in G, \\quad ag \\ne gb.\\end{aligned}$$ \nThis condition ensures that a square contains exactly 4 distinct vertices and that every vertex is incident to exactly Δ2 squares. For a vertex *v*, the set of incident squares is called the *link* of *v*, and denoted *X**v*. The TNC condition implies that the link of any vertex is in bijection with the set *A* × *B* (see Claim 3.7 of ). We will naturally refer to sets of the form {*a*} × *B* as rows and sets *A* × {*b*} as columns.\nWe recall from that there exists an infinite sequence of degrees Δ (namely *q* + 1, for *q* an odd prime power) such that each fixed degree Δ, there exists an infinite family of left-right Cayley complexes (over the groups *G* = PSL2(*q**i*)), satisfying the TNC condition, for which both the (left) Cayley graph ${{\\sf Cay}}(G,A)$ and the (right) Cayley graph ${{\\sf Cay}}(G,B)$ are Ramanujan graphs.\nWhen the Cayley graphs ${{\\sf Cay}}(G,A)$ and ${{\\sf Cay}}(G,B)$ are not bipartite (as is the case of the above family[7](#fn7)), it will be convenient for us to make them so by replacing them by their double covers. So we make two copies *V*0 = *G* × {0} and *V*1 = *G* × {1} of *G* and define the graphs ${\\EuScript G}\\_A=(V=V\\_0\\cup V\\_1,E\\_A)$ and ${\\EuScript G}\\_B=(V,E\\_B)$ with the edge set *E**A* made up of pairs {(*g*, 0), (*a**g*, 1)}, *a* ∈ *A*, and *E**B* consisting of the pairs {(*g*, 0), (*g**b*, 1)}, *b* ∈ *B*.\nFinally, the set *Q* of squares of the square complex *X* is defined as the set of 4-subsets of vertices of the form \n{(*g*, 0), (*a**g*, 1), (*g**b*, 1), (*a**g**b*, 0)}. \nLet us introduce two further graphs that exist on the complex *X*. The first is just the union of ${\\EuScript G}\\_A$ and ${\\EuScript G}\\_B$, and we denote it by ${\\EuScript G}^\\cup = (V,E\\_A\\cup\nE\\_B)$. The second graph we denote by ${\\EuScript G}^\\square=(V,E^\\square)$: it puts an edge between all pairs of vertices of the form {(*g*, *i*), (*a**g**b*, *i*)}, *g* ∈ *G*, *a* ∈ *A*, *b* ∈ *B*, *i* = 0, 1. The graph ${\\EuScript G}^\\square$ is therefore made up of two connected components, on *V*0 and *V*1, that we denote by ${\\EuScript G}\\_0^\\square$ and ${\\EuScript G}\\_1^\\square$. We note that ${\\EuScript G}^\\square$ is regular of degree Δ2, and may have multiple edges.\nIf ${\\EuScript G}\\_A$ and ${\\EuScript G}\\_B$ are Ramanujan, then ${\\EuScript G}^\\cup$ and ${\\EuScript G}^\\square$ inherit most of their expansion properties. Specifically:\n[lem:lambda] Assume that ${\\EuScript G}\\_A$ and ${\\EuScript G}\\_B$ are Ramanujan graphs, then \n$$ \\lambda({\\EuScript G}^\\cup) {\\leqslant}4\\sqrt{\\Delta}, \\quad\n\\lambda({\\EuScript G}\\_0^\\square) {\\leqslant}4\\Delta, \\quad \\lambda({\\EuScript G}\\_1^\\square){\\leqslant}4\\Delta.$$ \nLet *M**A* and *M**B* be the adjacency matrices of ${\\EuScript G}\\_A$ and ${\\EuScript G}\\_B$. Since these graphs are bipartite, they admit two eigenvalues Δ and − Δ and the remaining eigenvalues have an absolute value less than $2\\sqrt{\\Delta}$. The adjacency matrices of ${\\EuScript G}^\\cup$ and ${\\EuScript G}^\\square$ are respectively *M**A* + *M**B* and *M**A**M**B* = *M**B**M**A*. Since *M**A* and *M**B* commute, they can be made to have the same eigenspaces: therefore the eigenvalues of *M**A* + *M**B* are the sum of the eigenvalues of *M**A* and *M**B* and the eigenvalues of *M**A**M**B* are the products of the eigenvalues of *M**A* and *M**B*. The eigenvalue − Δ has the same eigenspace in *M**A* and *M**B*, therefore the eigenvalue Δ2 has an eigenspace of dimension 2 in *M**A**M**B*, meaning that ${\\EuScript G}^\\square$ splits into the two connected components ${\\EuScript G}\\_0^\\square$ and ${\\EuScript G}\\_1^\\square$.\n#### The quadripartite version.\nA way of avoiding the somewhat cumbersome TNC condition is to make the complex quadripartite rather than simply bipartite. In this case we construct the vertex set *V* as the disjoint union of four copies of the group *G*: we set *V* = *V*0 ∪ *V*1 with *V*0 = *V*00 ∪ *V*11 and *V*1 = *V*10 ∪ *V*01, where *V**i**j* = *G* × {*i*, *j*}, *i*, *j* ∈ {0, 1}. The set *Q* of squares is then defined as the set of 4-subsets of vertices of the form \n{(*g*, 00), (*a**g*, 01), (*g**b*, 10), (*a**g**b*, 11)}. \n We see that this time the *Q*-neighbourhood of any vertex becomes naturally in bijection with *A* × *B* without requiring any special properties of *A* = *A*− 1 and *B* = *B*− 1. In the present case the edge set *E**A* of ${\\EuScript G}\\_A$ becomes the set of pairs of the form {(*g*, 00), (*a**g*, 01)} and of the form {(*g*, 10), (*a**g*, 11)}, and the edge set *E**B* of ${\\EuScript G}\\_B$ becomes the set of pairs {(*g*, 00), (*g**b*, 10)} and {(*g*, 01), (*g**b*, 11)}. The graphs ${\\EuScript G}\\_A$ and ${\\EuScript G}\\_B$ are therefore both made up of two connected components. As before, we may set ${\\EuScript G}^\\cup=(V,E\\_A\\cup E\\_B)$, and finally the graph ${\\EuScript G}^\\square$ over the vertex set *V* puts an edge between (*g*, 00) and (*a**g**b*, 11) as well as between (*g*, 01) and (*a**g**b*, 10) for all *g* ∈ *G*, *a* ∈ *A*, *b* ∈ *B*. The two connected components ${\\EuScript G}\\_0^\\square$ and ${\\EuScript G}\\_1^\\square$ have now become bipartite, the vertex set of ${\\EuScript G}\\_0^\\square$ being *V*00 ∪ *V*11 and that of ${\\EuScript G}\\_1^\\square$ being *V*01 ∪ *V*10.\nIt is readily seen that Lemma [lem:lambda] also holds in the quadripartite case with the eigenvalue analysis being similar. In the sequel we will not need the bipartite structure of ${\\EuScript G}\\_0^\\square$ and ${\\EuScript G}\\_1^\\square$, and to lighten notation we will just assume the complex to be bipartite and not necessarily quadripartite. The quantum code construction presented in Section [sec:construction] is however straightforwardly adapted to the quadripartite case, and its analysis is essentially unchanged.\nWe note that the quadripartite version of the square left-right Cayley complex appears in.\nTanner codes\n------------\nA binary linear code of length *n* is an ${\\mathbb F}\\_2$-linear subspace of ${\\mathbb F}\\_2^n$. For sets *E* of cardinality ∣*E*∣ = *n*, it will be convenient for us to identify ${\\mathbb F}\\_2^n$ with ${\\mathbb F}\\_2^E$, which we can think of as the space of functions from *E* to ${\\mathbb F}\\_2$. Identication with ${\\mathbb F}\\_2^n$ amounts to defining a one-to-one map between *E* and [*n*] = {1, 2, …, *n*}, *i.e.* a numbering of the elements of *E*.\nLet ${\\EuScript G}=(V,E)$ be a regular graph of degree Δ, and for any vertex *v* denote by *E*(*v*) the set of edges incident to *v*. Assume an identification of ${\\mathbb F}\\_2^{E(v)}$ with ${\\mathbb F}\\_2^\\Delta$ for every *v* ∈ *V*. Let $x\\in{\\mathbb F}\\_2^{E}$ be a vector indexed by (or a function defined on) the set *E*. Let us define the *local view* of *x* at vertex *v* as the subvector *x**v* = (*x**e*)*e* ∈ *E*(*v*), *i.e.* *x* restricted to the edge-neighbourhood *E*(*v*) of *v*.\nLet *C*0 be a linear code of length Δ, dimension *k*0 = *ρ*0Δ, and minimum distance *d*0 = *δ*0Δ. We define the Tanner code associated to ${\\EuScript G}$ and *C*0 as \n$$T({\\EuScript G},C\\_0) =\\{x\\in{\\mathbb F}\\_2^E : x\\_v\\in C\\_0\\;\\text{for all}\\;v\\in V\\}.$$ \n In words, the Tanner code is the set of vectors over *E* all of whose local views lie in *C*0. By counting the number of linear equations satisfied by the Tanner code, we obtain\n \n$$\\label{eq:dimtanner}\n\\dim T({\\EuScript G},C\\_0) {\\geqslant}(2\\rho\\_0-1)n.$$ \nWe also have the bound on the minimum distance *d* of the Tanner code: \n$$d{\\geqslant}\\delta\\_0 (\\delta\\_0- \\lambda({\\EuScript G})/\\Delta)n.$$ \n Therefore, if $({\\EuScript G}\\_i)$ is a family of Δ-regular expander graphs with $\\lambda({\\EuScript G}\\_i){\\leqslant}\\lambda <d\\_0$, and if *ρ*0 > 1/2, then the associated family of Tanner codes has rate and minimum distance which are both Ω(*n*), meaning we have an asymptotically good family of codes, as was first shown in.\nQuantum CSS codes\n-----------------\nA quantum CSS code is specific instance of a stabilizer code that can be defined by two classical codes ${\\EuScript C}\\_0$ and ${\\EuScript C}\\_1$ in the ambient space ${\\mathbb F}\\_2^n$, with the property that ${\\EuScript C}\\_0^\\perp \\subset {\\EuScript C}\\_1$. If both codes are defined by their parity-check matrix, ${\\EuScript C}\\_0 = \\ker H\\_0, {\\EuScript C}\\_1 =\\ker H\\_1$, then the condition is equivalent to *H*0*H*1*T* = 0. The resulting quantum code ${\\EuScript Q}= ({\\EuScript C}\\_0, {\\EuScript C}\\_1)$ is the following subspace of (C2)⊗ *n*, the space of *n* qubits: \n$$ {\\EuScript Q}:= \\mathrm{Span}\\left\\{ \\sum\\_{z \\in {\\EuScript C}\\_1^\\perp} |x+z\\rangle \\: : \\: x\\in {\\EuScript C}\\_0 \\right\\},$$ \n where $\\{ |x\\rangle \\: : \\: x\\in {\\mathbb F}\\_2^n\\}$ is the canonical basis of (C2)⊗ *n*.\nIn practice, it is convenient to describe the code *via* its *generators*. A CSS code admits *X*-type generators which correspond to the rows of *H*1 and and *Z*-type generators, corresponding to the rows of *H*0. The condition ${\\EuScript C}\\_0^\\perp \\subset {\\EuScript C}\\_1$ is simply that the rows of *H*0 are orthogonal to the rows of *H*1, where orthogonality is with respect to the standard inner product over ${\\mathbb F}\\_2^n$. A CSS code is called *LDPC* is both *H*0 and *H*1 are sparse matrices, *i.e.* each row and column has constant weight independent of the code length *n*. Equivalently, each generator acts nontrivially on a constant number of qubits, and each qubit is only involved in a constant number of generators.\nThe dimension *k* of the code counts the number of logical qubits and is given by \n$$k = \\text{dim} \\, ({\\EuScript C}\\_0/{\\EuScript C}\\_1^\\perp) = \\text{dim} \\, {\\EuScript C}\\_0 + \\text{dim} \\, {\\EuScript C}\\_1 - n.$$ \n Its minimum distance is *d* = min(*d**X*, *d**Z*) with \n$$ d\\_X = \\min\\_{w \\in {\\EuScript C}\\_0 \\setminus {\\EuScript C}\\_1^{\\perp}} |w|, \\quad d\\_Z = \\min\\_{w \\in {\\EuScript C}\\_1 \\setminus {\\EuScript C}\\_0^\\perp} |w|.$$ \n We denote the resulting code parameters by ⟦*n*, *k*, *d*⟧. We say that a code family $({\\EuScript Q}\\_n)\\_n$ is *asymptotically good* if its parameters are of the form \n⟦*n*, *k* = Θ(*n*), *d* = Θ(*n*)⟧. \nTensor codes and dual tensor codes: robustness\n----------------------------------------------\nLet *A* and *B* be two sets of size Δ. We define codes on the ambient space ${\\mathbb F}\\_2^{A\\times B}$ that we may think of the space of matrices whose rows (columns) are indexed by *A* (by *B*). If $C\\_A\\subset {\\mathbb F}\\_2^A$ and $C\\_B\\subset{\\mathbb F}\\_2^B$ are two linear codes, we define the *tensor* (or product) code *C**A* ⊗ *C**B* as the space of matrices *x* such that for every *b* ∈ *B* the column vector (*x**a**b*)*a* ∈ *A* belongs to *C**A* and for every *a* ∈ *A* the row vector (*x**a**b*)*b* ∈ *B* belongs to *C**B*. It is well known that dim(*C**A* ⊗ *C**B*) = dim(*C**A*)dim(*C**B*) and that the minimum distance of the tensor code is *d*(*C**A* ⊗ *C**B*) = *d*(*C**A*)*d*(*C**B*).\nConsider the codes $C\\_A\\otimes{\\mathbb F}\\_2^B$ and ${\\mathbb F}\\_2^A\\otimes C\\_B$ consisting respectively of the space of matrices whose columns are codewords of *C**A* and whose rows are codewords of *C**B*. We may consider their sum $C\\_A\\otimes{\\mathbb F}\\_2^B + {\\mathbb F}\\_2^A\\otimes C\\_B$ which is called a *dual tensor* code, since it is the dual code of the tensor code $C\\_A^\\perp\\otimes C\\_B^\\perp=(C\\_A^\\perp\\otimes{\\mathbb F}\\_2^B)\\cap({\\mathbb F}\\_2^A\\otimes\nC\\_B^\\perp)$. It is relatively straightforward to check that $d(C\\_A\\otimes{\\mathbb F}\\_2^B +\n{\\mathbb F}\\_2^A\\otimes C\\_B)=\\min(d(C\\_A),d(C\\_B))$.\n[def:robust] Let 0 ≤ *w* ≤ Δ2. Let *C**A* and *C**B* be codes of length Δ with minimum distances *d**A* and *d**B*. We shall say that the dual tensor code $C=C\\_A\\otimes{\\mathbb F}\\_2^B +\n{\\mathbb F}\\_2^A\\otimes C\\_B$ is *w*-*robust*, if for any codeword *x* ∈ *C* of Hamming weight ∣*x*∣ ≤ *w*, there exist *A*ʹ ⊂ *A*, *B*ʹ ⊂ *B*, ∣*A*ʹ∣ ≤ ∣*x*∣/*d**B*, ∣*B*ʹ∣ ≤ ∣*x*∣/*d**A*, such that *x**a**b* = 0 whenever $a\\notin A',\nb\\notin B'$.\nIn words, *w*-robustness means that any dual tensor codeword of weight at most *w* is entirely supported within the union of a set of at most ∣*c*∣/*d**A* columns and a set of at most ∣*c*∣/*d**B* rows. In fact, the following proposition shows that any such codeword is the sum of a word of $C\\_A\\otimes{\\mathbb F}\\_2^B$ and of a word of ${\\mathbb F}\\_2^A\\otimes C\\_B$ supported on a few columns or rows.\nLet *C**A* and *C**B* be codes of length Δ with minimum distances *d**A* and *d**B*, and suppose $C=C\\_A\\otimes{\\mathbb F}\\_2^B +\n{\\mathbb F}\\_2^A\\otimes C\\_B$ is *w*-robust with 0 < *w* < *d**A**d**B*. Then for any codeword *x* ∈ *C* such that ∣*x*∣ ≤ *w*, there exist *A*ʹ ⊂ *A*, *B*ʹ ⊂ *B*, ∣*A*ʹ∣ ≤ ∣*x*∣/*d**B*, ∣*B*ʹ∣ ≤ ∣*x*∣/*d**A* and a decomposition *x* = *c* + *r*, with $c \\in C\\_{A} \\otimes {\\mathbb F}\\_2^{B'}$ and $r \\in {\\mathbb F}\\_2^{A'}\n\\otimes C\\_B$.\nTo see this, apply the definition and write *x* = *r*ʹ + *c*ʹ, with *r*ʹ*a**b* = *c*ʹ*a**b* for any (*a*, *b*) ∈ (*A* \\ *A*ʹ) × (*B* \\ *B*ʹ). The restrictions of *r*ʹ and *c*ʹ to (*A* \\ *A*ʹ) × (*B* \\ *B*ʹ) both belong to the code obtained by tensoring *C**A*ʹ and *C**B*ʹ, the punctured codes deduced from *C**A* and *C**B* by throwing away coordinates of *A*ʹ and *B*ʹ. This code is the same as the punctured code obtained from *C**A* ⊗ *C**B* by throwing away the coordinates *A*ʹ × *B* ∪ *A* × *B*ʹ. Therefore, there exists a tensor codeword of $C\\_A\\otimes C\\_B=C\\_{A}\n\\otimes {\\mathbb F}\\_2^{B}\\cap {\\mathbb F}\\_2^A\\otimes C\\_B$ that coincides with *c*ʹ = *r*ʹ on (*A* \\ *A*ʹ) × (*B* \\ *B*ʹ): adding this tensor codeword to both *c*ʹ and *r*ʹ yields the required pair *r*, *c* such that *x* = *r* + *c*.\nOur notion of robustness for the dual tensor code also implies a form of robustness for the corresponding tensor code.\n[prop:tensor] Let *C**A* and *C**B* be codes of length Δ and minimum distances *d**A*, *d**B* such that the dual tensor code $C\\_A\\otimes{\\mathbb F}\\_2^B + {\\mathbb F}\\_2^A \\otimes C\\_B$ is *w*-robust with *w* ≤ *d**A**d**B*/2. Then, any word *x* close to both the column and row code is also close to the tensor code: precisely, if $d(x, C\\_A \\otimes {\\mathbb F}\\_2^{B})+ d(x, {\\mathbb F}\\_2^{A} \\otimes C\\_B) {\\leqslant}w$ then\n \n$$\\begin{aligned}\n\\label{eqn:robust}\nd(x, C\\_A \\otimes C\\_B) {\\leqslant}\\frac{3}{2} \\left( d(x, C\\_A \\otimes {\\mathbb F}\\_2^{B})+ d(x, {\\mathbb F}\\_2^{A} \\otimes C\\_B) \\right).\\end{aligned}$$ \nThe conclusion of Proposition [prop:tensor] is very close to a property called “robustly testable” in. The difference is that robustly testable tensor codes are required to satisfy without any condition on its right hand side (equivalently, with *w* = 2Δ2), at the expense of allowing a looser constant than 3/2. We note that the constant 3/2 in Proposition [prop:tensor] is tight[8](#fn8).\nLet *x* be an *A* × *B* matrix that we write as *x* = *C* + *e**C* = *R* + *e**R* with $C \\in C\\_A \\otimes {\\mathbb F}\\_2^{B}$, $R \\in {\\mathbb F}\\_2^{A} \\otimes C\\_B$ and \n$$|e\\_C| = d(x, C\\_A \\otimes {\\mathbb F}\\_2^{B}), \\quad |e\\_R| = d(x, {\\mathbb F}\\_2^{A}\\otimes C\\_B).$$ \n Let us suppose ∣*e**C*∣ + ∣*e**R*∣ ≤ *w*. From *C* + *R* = *e**C* + *e**R*, we obtain that ∣*C* + *R*∣ ≤ *w* and the robust testability of the dual product code implies that there exist $c \\in C\\_A \\otimes {\\mathbb F}\\_2^{B}$, $r \\in {\\mathbb F}\\_2^{A} \\otimes C\\_B$, supported respectively, since *w* ≤ *d**A**d**B*/2, on at most *d**B*/2 columns and at most *d**A*/2 rows, and such that *C* + *R* = *c* + *r*. Since *c* is supported by at most *d**B*/2 columns, we have that ∣*c* + *r*∣ ≥ ∣*c*∣, and similarly ∣*c* + *r*∣ ≥ ∣*r*∣.\nNote in particular that *R* + *r* = *C* + *c* is a codeword of the tensor code *C**A* ⊗ *C**B*. One can therefore write *x* = (*C* + *c*) + *c* + *e**C*, from which we have: \n*d*(*x*, *C**A* ⊗ *C**B*) ≤ ∣*c* + *e**C*∣ ≤ ∣*c*∣ + ∣*e**C*∣ ≤ ∣*c* + *r*∣ + ∣*e**C*∣ = ∣*e**C* + *e**R*∣ + ∣*e**C*∣ ≤ 2∣*e**C*∣ + ∣*e**R*∣. \n Writing *x* = (*R* + *r*) + *r* + *e**R*, we similarly get *d*(*x*, *C**A* ⊗ *C**B*) ≤ 2∣*e**R*∣ + ∣*e**C*∣, and adding the two inequalities we conclude that \n$$d(x,C\\_A\\otimes C\\_B) {\\leqslant}\\frac{ 3}{2} \\left(d(x, C\\_A \\otimes {\\mathbb F}\\_2^B) +\nd(x, {\\mathbb F}\\_2^{A}\\otimes C\\_B)\\right). \\qedhere$$ \nIf $C\\_A\\subset {\\mathbb F}\\_2^A$ is a code and *A*ʹ ⊆ *A*, let us denote by $C\\_{A'}\\subset{\\mathbb F}\\_2^{A'}$ the *punctured code* consisting of all subvectors (*c**a*)*a* ∈ *A*ʹ of all codewords (*c**a*)*a* ∈ *A* of *C**A*. Similarly, for a code *C**B* we denote by *C**B*ʹ the punctured code on *B*ʹ for *B*ʹ ⊆ *B*.\nLet us introduce the following twist on the above definition of robustness for dual tensor codes, which allows us to boost its potential.\nLet $C\\_A\\subset {\\mathbb F}\\_2^A$ and $C\\_B\\subset {\\mathbb F}\\_2^B$. For integers *w*, *p*, let us say that the dual tensor code $C\\_{A}\\otimes{\\mathbb F}\\_2^{B}+{\\mathbb F}\\_2^{A}\\otimes\nC\\_{B}$ is *w*-robust with *p*-resistance to puncturing, if for any *A*ʹ ⊂ *A* and *B*ʹ ⊂ *B* such that ∣*A*ʹ∣ = ∣*B*ʹ∣ = Δ − *w*ʹ, with *w*ʹ ≤ *p*, the dual tensor code \n$$C\\_{A'}\\otimes{\\mathbb F}\\_2^{B'}+{\\mathbb F}\\_2^{A'}\\otimes C\\_{B'}$$ \n is *w*-robust.\nWe shall need the following result on the robustness of random dual tensor codes.\ntheoProbust [thm:P-robust] Let 0 < *ρ**A* < 1 and 0 < *ρ**B* < 1. Let 0 < *ɛ* < 1/2 and $1/2+\\eps<\\gamma <1$. Let *C**A* be a random code obtained from a random uniform *ρ**A*Δ × Δ generator matrix, and let *C**B* be a random code obtained from a random uniform (1 − *ρ**B*)Δ × Δ parity-check matrix. With probability tending to 1 when Δ goes to infinity, the dual tensor code \n$$C\\_{A}\\otimes{\\mathbb F}\\_2^{B}+{\\mathbb F}\\_2^{A}\\otimes C\\_{B}$$ \n is Δ3/2 − *ɛ*-robust with Δ*γ*-resistance to puncturing.\nExcept for the fact that we allow a larger robustness parameter, (namely Δ1/2 − *ɛ*), this result is essentially in. We provide a proof in the appendix, which closely follows the approach of.\nAsymptotically good quantum Tanner codes\n========================================\nThe construction\n----------------\nLet *G* belong to an infinite family of groups with size ∣*G*∣ → ∞, together with two sets of generators *A* = *A*− 1 and *B* = *B*− 1 of fixed cardinality Δ. We form the associated family of left-right Cayley complexes *X*, and will assume throughout that they satisfy the TNC condition.\nRecall that the square-neighbourhood (or link) *Q*(*v*) of a vertex *v* = (*g*, *i*) ∈ *G* × {0, 1} is the set of squares incident to *v*. To lighten notation, let us write, for *a* ∈ *A*, *b* ∈ *B*, *a**v* = (*a**g*, 1 − *i*), *v**b* = (*g**b*, 1 − *i*), *a**v**b* = (*a**g**b*, *i*), so that {*v*, *a**v*, *v**b*, *a**v**b*} is a square incident to *v*. The TNC condition implies in particular that the map\n \n$$\\begin{aligned}\nA\\times B &\\to& Q(v)\\\\\n(a,b) & \\mapsto & \\{v,av,vb,avb\\}\\end{aligned}$$ \nis one-to-one, and let *ϕ**v* denote the inverse of this map. A crucial consequence of the TNC condition is that for two adjacent vertices *v*, *v*ʹ ∈ *V*, the intersection *Q*(*v*) ∩ *Q*(*v*ʹ) has cardinality Δ, and more specifically,\n \n$$\\begin{aligned}\n\\phi\\_v(Q(v)\\cap Q(vb))=A\\times\\{b\\}&\\quad&\\phi\\_{vb}(Q(v)\\cap\nQ(vb))=A\\times\\{b^{-1}\\}\\\\\n\\phi\\_v(Q(v)\\cap Q(av))=\\{a\\}\\times B&\\quad&\\phi\\_{av}(Q(v)\\cap\nQ(av))=\\{a^{-1}\\}\\times B.\\end{aligned}$$ \nFor a vector $x\\in{\\mathbb F}\\_2^Q$, we may define its local view *x**v* as the restriction (*x**q*)*q* ∈ *Q*(*v*) of *x* to the link *Q*(*v*) of *v*. Through the indexation map *ϕ**v*, the local view at any vertex *v* can be seen as an *A* × *B* array, all of its rows being shared by the local views of the *A*-neighbours of *v*, and all of its columns being shared by the local views of its *B*-neighbours. The situation is illustrated on Figure [fig:grid].\n(0,0) rectangle (3,3); (1,0) rectangle (1.5,3); (0,0) grid (3,3);\n(4,0) rectangle (7,3); (6,0) rectangle (6.5,3); (4,0) grid (7,3);\nat (-0.7,1.5) *Q*(*v*); at (7.8,1.5) *Q*(*v**b*); at (1.25,-0.4) *b*; at (6.35,-0.35) *b*− 1;\nWe define a quantum CSS code ${\\EuScript Q}= ({\\EuScript C}\\_0, {\\EuScript C}\\_1)$ by associating qubits to the squares of the complex *X* and enforcing constraints on the local view of each vertex. The idea is to associate *Z*-type generators with vertices of *V*0 and *X*-type generators with *V*1. Since their support can only overlap on a column or on a row, it suffices to impose orthogonality constraints between the two types of generators on each row and column to obtain a CSS code satisfying ${\\EuScript C}\\_0^\\perp \\subset {\\EuScript C}\\_1$. More precisely, we choose two codes $C\\_A\\subset{\\mathbb F}\\_2^A$ and $C\\_B\\subset{\\mathbb F}\\_2^B$ and define the two tensor codes *C*0 = *C**A* ⊗ *C**B* and *C*1 = *C**A*⊥ ⊗ *C**B*⊥. To define the two sets of generators, choose a basis *β*0 of *C*0 and a basis *β*1 of *C*1. For every vertex *v* ∈ *V*0 we define dim*C*0 generators $x\\in{\\mathbb F}\\_2^Q$ of type *Z* by requiring that their local views *x**v* at *v* be equal (through *ϕ**v*) to a basis element in *β*0 and that *x**q* = 0 for all $q\\notin Q(v)$. Similarly, we define ∣*V*1∣dim*C*1 generators of type *X* by imposing a local view equal to a basis element of *β*1 on a single vertex *v* ∈ *V*1 and requiring *x**q* = 0 for all values outside the neighbourhood *Q*(*v*) of *v*. We see that *X*-generators and *Z*-generators are orthogonal by design.\nEquivalently, the code ${\\EuScript C}\\_0$ (${\\EuScript C}\\_1$) orthogonal to all *Z*-generators (*X*-generators) is defined as the set of vectors $x\\in{\\mathbb F}\\_2^Q$ such that *x**v* is in *C*0⊥ (in *C*1⊥). In other words, the quantum code ${\\EuScript Q}= ({\\EuScript C}\\_0, {\\EuScript C}\\_1)$ is the pair of Tanner codes\n \n$$\\label{eq:tanner}\n{\\EuScript C}\\_0=T({\\EuScript G}\\_0^\\square,C\\_0^\\perp), \\qquad {\\EuScript C}\\_1=T({\\EuScript G}\\_1^\\square,C\\_1^\\perp).$$ \nTo have the same number of *X* and *Z*-type generators, we shall set *ρ* = dim*C**A*/Δ and require dim*C**B* = Δ − dim*C**A*: consequently we shall have dim*C*0 = dim*C*1 = *ρ*(1 − *ρ*)Δ2.\nIt is immediate that this code is LDPC since all its generators have weight at most Δ2, which is a constant, and any qubit is involved in at most 4*ρ*(1 − *ρ*)Δ2 ≤ Δ2 generators[9](#fn9). The code length is the number of squares in the complex, *n* = Δ2∣*G*∣/2. While computing the exact dimension *k* of the code would require to check for possible dependencies among generators, it is straightforward to get a lower bound by counting the number of generators, namely ∣*V*0∣dim*C*0 + ∣*V*1∣dim*C*1 = 2*ρ*(1 − *ρ*)Δ2∣*G*∣. This yields the following bound for the rate of the quantum code:\n \n$$\\label{eq:rate}\n \\frac 1n\\dim{\\EuScript Q}{\\geqslant}(2\\rho-1)^2,$$ \nwith equality when all the generators are independent. In particular, the rate is > 0 for any value of *ρ* ≠ 1/2.\nMost of the rest of this section is devoted to establishing a linear lower bound for the minimum distance of this quantum code, provided that the Cayley graphs associated to *X* are sufficiently expanding and the dual tensor codes *C*0⊥ and *C*1⊥ are sufficiently robust. This in turn requires Δ to be large enough and *G* to be non Abelian (since the Cayley graph of an Abelian group cannot be an expander if the degree is constant). But the quantum Tanner code construction is more general, and even small examples where *C**A* = *C**B*⊥ is a small Hamming code could display good performance. For instance, one could consider $G = \\mathbbm{Z}/8\\mathbbm{Z} \\times\n\\mathbbm{Z}/2\\mathbbm{Z}$, *A* = {(*a*, 0) : *a* ≠ 0}, *B* = {(*b*, 1) : *b* ≠ 1} and *C**A*, *C**B* to be the [7, 4, 3] Hamming code and its dual [7, 3, 4]. The associated quantum Tanner code has length 392, dimension at least 8, and generators of weight 12.\nProof of Theorem [thm:main]\n---------------------------\nLet us recall our main technical theorem:\nWe consider a left-right Cayley complex *X* from Section [subsec:LRCayley] over a group *G*, satisfying the TNC condition and such that ${{\\sf Cay}}(G,A)$ and ${{\\sf Cay}}(G,B)$ are Ramanujan graphs.\nLet $x \\in {\\EuScript C}\\_1$ be a codeword of weight $|x| < \\delta n/4\\Delta^{3/2+\\eps}$. It induces a subgraph ${\\EuScript G}\\_{1,x}^\\square$ of ${\\EuScript G}\\_1^\\square$ with vertex set *S* ⊂ *V*1. The local view for any *v* ∈ *S* corresponds to a codeword of $C\\_1^\\perp =\nC\\_A\\otimes{\\mathbb F}\\_2^B +{\\mathbb F}\\_2^A\\otimes C\\_B$. This code is *w*-robust so codewords of weight less than $w = \\Delta^{3/2-\\eps}$ have a support restricted to a small number ≤ *w*/(*δ*Δ) of rows and columns: in particular, the local view restricted to any column (or row) is at distance at most $\\Delta^{1/2-\\eps}/\\delta$ from a codeword of *C**A* (or *C**B*).\nLet us call *normal vertices* the vertices of *S* with degree less than $\\Delta^{3/2-\\eps}$ in ${\\EuScript G}\\_{1,x}^\\square$, and define *S**e*, the set of *exceptional vertices* with degree greater than $\\Delta^{3/2- \\eps}$. Expansion in ${\\EuScript G}\\_1^\\square$ ensures that the set of exceptional vertices is small compared to *S*.\n[claim:Se] The set of exceptional vertices has size\n \n$$\\begin{aligned}\n\\label{eqn:Se}\n|S\\_e| {\\leqslant}\\frac{64}{\\Delta^{1-2\\eps}} |S|.\\end{aligned}$$ \nThe degree of each vertex *v* ∈ *S* in the subgraph ${\\EuScript G}\\_{1,x}^\\square$ is at least *δ*Δ since the local view must correspond to a codeword of $C\\_1^\\perp = C\\_A\\otimes{\\mathbb F}\\_2^B +{\\mathbb F}\\_2^A \\otimes C\\_B$. It implies that\n \n$$\\label{eq:|S|}\n|S| {\\leqslant}\\frac{2|x|}{\\delta \\Delta} {\\leqslant}\\frac{1}{4\\Delta^{1/2+\\eps}}|V\\_1|.$$ \nApplying the Expander Mixing Lemma [lem:mixing] to *E*(*S**e*, *S*) in ${\\EuScript G}\\_1^\\square$, regardless of whether ${\\EuScript G}\\_1^\\square$ is bipartite or not, we obtain from Lemma [lem:lambda] that:\n \n$$\\begin{aligned}\n |E(S\\_e,S) | &{\\leqslant}\\frac{\\Delta^2}{|V\\_1|}|S\\_e||S| +4\\Delta \\sqrt{|S\\_e| |S|}\\\\\n &{\\leqslant}\\frac 12\\Delta^{3/2-\\eps}|S\\_e| + 4\\Delta \\sqrt{|S\\_e| |S|}\\end{aligned}$$ \nBy definition of *S**e*, it also holds that $|E(S\\_e, S)| {\\geqslant}\\Delta^{3/2- \\eps}|S\\_e|$. Combining both inequalities, we obtain $\\Delta^{1/2-\\eps} \\sqrt{|S\\_e|} {\\leqslant}8 \\sqrt{|S|}$ and the claim follows.\nThe support of the local view of any normal vertex of *S* decomposes into a small number of rows and columns, which are shared with vertices in *V*0. We now introduce the set *T* of vertices of *V*0 whose local views share with a normal vertex of *S* either a row or a column of large weight. Formally:\n#### Defining the subset *T* ⊂ *V*0.\nThe vector *x*, viewed as a set of squares, defines a subset *E**x* of edges of ${\\EuScript G}^\\cup$, namely the edges incident to a square in *x*. Let us say that an edge of *E**x* is *heavy,* if it is incident to at least $\\delta \\Delta- \\Delta^{1/2-\\eps}/\\delta$ squares of *x*. Let *T* be the set of vertices of *V*0 that are connected to (at least) one *normal* vertex of *S* *through a heavy edge*. Let us keep in mind that the local view of a normal vertex of *S* is supported by at most $\\Delta^{1/2-\\eps}/\\delta$ rows and at most $\\Delta^{1/2-\\eps}/\\delta$ columns, so a heavy edge between a normal vertex of *S* and a vertex of *T* corresponds to either a row or a column shared by the two local views, which is at distance at most $\\Delta^{1/2-\\eps}/\\delta$ from a nonzero codeword of *C**A* (or *C**B*).\n[claim:degreeT] The degree in *E**x* of any vertex of *T* is at least $\\delta\\Delta- \\Delta^{1/2-\\eps}/\\delta$.\nThis follows simply from the fact that a row of weight *w* in a local view is incident to *w* columns.\nFor Δ large enough, the size of the set *T* satisfies:\n \n$$\\label{eqn:T}\n|T|{\\leqslant}\\frac{64}{\\delta^2\\Delta}|S|.$$ \nFrom Claim [claim:degreeT] we have the following lower bound on the number of edges of ${\\EuScript G}^\\cup$ between *S* and *T*, \n$$ |E(S,T)| {\\geqslant}\\delta \\Delta \\Big(1- \\frac{1}{\\delta^2 \\Delta^{1/2+\\eps}}\\Big) |T|.$$ \n Applying the Expander mixing Lemma [lem:mixing] to *E*(*S*, *T*), we have, from Lemma [lem:lambda], \n$$|E(S,T)|{\\leqslant}\\frac{4\\Delta}{|V|}|S||T| + 4\\sqrt{\\Delta}\\sqrt{|S||T|}.$$ \n Recalling, we have $|S| {\\leqslant}|V|/8\\Delta^{1/2+\\eps}$, we have therefore \n$$|T|\\delta\\Delta\\left(1-\\frac{1}{\\delta^2\\Delta^{1/2+\\eps}}-\\frac{1}{2\\delta\\Delta^{1/2+\\eps}}\n\\right){\\leqslant}4\\sqrt{\\Delta}\\sqrt{|S||T|}$$ \n which implies \n$$\\delta\\Delta\\sqrt{|T|}{\\leqslant}8\\sqrt{\\Delta}\\sqrt{|S|}$$ \n for Δ large enough, from which the claim follows.\nFrom this last claim we infer that a typical vertex in *T* must be adjacent to a large number of vertices in *S*, linear in Δ, which means that its local view should consist of many columns (or rows) containing almost undisturbed codewords of *C**A* (or *C**B*). The robustness of *C**A* ⊗ *C**B* then implies that the local view of such a vertex must be close to a codeword of the tensor code, and that adding the corresponding word will decrease the weight of *x*. We now detail the argument.\nDefine *d̄**T* to be the average (over *T*) number of heavy edges incident to a vertex of *T*. The number of edges between *T* and *S* \\ *S**e* is ∣*E*(*T*, *S* \\ *S**e*)∣ = *d̄**T*∣*T*∣ ≥ ∣*S*∣ − ∣*S**e*∣. Applying and , we have \n$$ \\bar{d}\\_T {\\geqslant}\\frac{|S|-|S\\_e|}{|T|}{\\geqslant}\\frac{\\delta^2}{64}\\left(1-\\frac{64}{\\Delta^{1-2\\eps}}\\right)\\Delta=: 2\\alpha \\Delta.$$ \nLet *η* be the fraction of vertices of *T* with degree greater than *α*Δ. Since the maximum degree of a vertex in ${\\EuScript G}^{\\cup}$ is 2Δ, it holds that \n2*α*Δ ≤ *d̄**T* ≤ 2Δ*η* + (1 − *η*)*α*Δ \n and *η* ≥ *α*/(2 − *α*) ≥ *α*/2. We have just shown:\n[claim:heavy] At least a fraction *α*/2 of vertices of *T* are incident to at least *α*Δ heavy edges.\nWe will need to single out a vertex *v* of *T* whose existence is guaranteed by claim [claim:heavy], but we also need this vertex to not be incident to too many exceptional vertices of *S*. To this end we estimate the total number of edges of ${\\EuScript G}^\\cup$ between *T* and *S**e*.\nFrom the Expander mixing Lemma [lem:mixing],\n \n$$\\begin{aligned}\nE(S\\_e,T)&{\\leqslant}\\frac{4\\Delta}{|V|}|S\\_e||T| +\n4\\sqrt{\\Delta}\\sqrt{|T||S\\_e|}\\nonumber\\\\\n&{\\leqslant}\\frac{256\\Delta^{2\\eps}}{|V|}|T||S| +\n32\\Delta^{\\eps}\\sqrt{|T||S|}\\nonumber\\\\\n&{\\leqslant}\\frac{32}{\\Delta^{1/2-\\eps}}|T|+32\\Delta^{\\eps}\\sqrt{|T||S|}\\label{eq:32}\\end{aligned}$$ \nby first applying , and then . Now every vertex of *S* \\ *S**e* is, by definition of *T*, adjacent to a vertex of *T* in ${\\EuScript G}^\\cup$. Since the degree of ${\\EuScript G}^\\cup$ is 2Δ we get that ∣*S*∣ − ∣*S**e*∣ ≤ 2Δ∣*T*∣. From, we have that, for Δ large enough, ∣*S**e*∣ ≤ ∣*S*∣/2, whence ∣*S*∣ ≤ 4Δ∣*T*∣. From we therefore obtain \n$$E(S\\_e,T){\\leqslant}\\beta\\Delta^{1/2+\\eps}|T|$$ \n with *β* = 64 + 32/Δ. We therefore have that at most an *α*/4 proportion of vertices of *T* are adjacent to more than $\\frac\n4\\alpha\\beta\\Delta^{1/2+\\eps}$ vertices of *S**e*. Summarising, we have shown, together with claim [claim:heavy]\n[claim:alpha/4] At least a fraction *α*/4 of vertices of *T*\n* are incident to at least *α*Δ heavy edges.\n* are adjacent to at most $d\\_1=\\frac{4\\beta}{\\alpha}\\Delta^{1/2+\\eps}$ vertices of *S**e*.\nPick any vertex *v* whose existence is guaranteed by claim [claim:alpha/4]. The local view at *v* is illustrated on Figure [fig:local].\n(0,0) rectangle (3,3); (0.2,0) – (0.2,2.4) – (0,2.4) – (0,2.7) – (0.2,2.7) – (0.2,3) – (0.5,3) – (0.5,2.7) – (3,2.7) – (3,2.4) – (0.5,2.4) – (0.5,0) – (0.2,0); at (-0.5,1.5) *A*; at (1.5,3.5) *B*;\nWe may therefore define two subsets *A*ʹ ⊂ *A*, *B*ʹ ⊂ *B*, with ∣*A*ʹ∣ = ∣*B*ʹ∣ = Δ − *d*1, and such that in the view of *v*, all rows and columns indexed by *A*ʹ and *B*ʹ are normal, *i.e.* are not shared with the local views of vertices in *S**e*. Observe that for Δ large enough we have *d*1 ≤ Δ*γ*, since we have imposed $\\gamma>1/2+\\eps$. The theorem’s hypothesis on resistance to puncturing implies therefore that the punctured code $C\\_{A'} \\otimes {\\mathbb F}\\_2^{B'} + {\\mathbb F}\\_2^{A'} \\otimes C\\_{B'}$ is $\\Delta^{3/2-\\eps/2}$-robust.\nDenote by $x\\_v\\in{\\mathbb F}\\_2^{A\\times B}$ the local view of *x* at vertex *v*, and by *x**v*ʹ its restriction to coordinates in *A*ʹ × *B*ʹ. Proposition [prop:tensor], applied for large enough Δ, states that there exists a codeword *c*ʹ ∈ *C**A*ʹ ⊗ *C**B*ʹ of the punctured code *C**A*ʹ ⊗ *C**B*ʹ close to *x**v*ʹ: specifically, \n$$d(x\\_v', c') {\\leqslant}\\frac{3}{2} \\left(d(x\\_v', C\\_{A'} \\otimes {\\mathbb F}\\_2^{B'}) +\nd(x\\_v', {\\mathbb F}\\_2^{A'}\\otimes C\\_{B'})\\right) {\\leqslant}3 \\Delta \\frac{\\Delta^{1/2-\\eps}}{\\delta}$$ \n since each column (or row) of *x**v* is at distance at most $\\frac{\\Delta^{1/2-\\eps}}{\\delta}$ from *C**A* (or *C**B*), and there are not more than Δ columns (rows) altogether.\nSince *d*1 < *δ*Δ, there is a unique tensor codeword *c* ∈ *C**A* ⊗ *C**B* that coincides with *c*ʹ on *A*ʹ × *B*ʹ, and we have, since ∣(*A* × *B*) \\ (*A*ʹ × *B*ʹ)∣ ≤ 2*d*1Δ, \n*d*(*x**v*, *c*) ≤ *d*(*x**v*ʹ, *c*ʹ) + 2*d*1Δ. \n This last upper bound scales like $\\Delta^{3/2+\\eps}$, but the first point of Claim [claim:alpha/4] implies that the weight of *x**v* is a quantity that scales like Δ2, so for large enough values of Δ it must be that *d*(*x**v*, *c*) = ∣*x**v* + *c*∣ < ∣*x**v*∣ (implying in particular that *c* is non-zero). Defining the vector $y\\in{\\mathbb F}\\_2^Q$ to coincide with *c* on the *Q*-neighbourhood of *v* and to be zero elsewhere, we have just shown that ∣*x*∣ − ∣*x* + *y*∣ = ∣*x**v*∣ − ∣*x**v* + *c*∣ is positive, which concludes the proof of the theorem.\n[rem:a] The last part of the proof of Theorem [thm:main] shows that when Δ is large enough, not only do we have ∣*x*∣ − ∣*x* + *y*∣ > 0 for *y* thus constructed, but there is furthermore a constant *a* such that we have ∣*x*∣ − ∣*x* + *y*∣ > *a*Δ2.\n[rem:negative] Theorem [thm:main] stays valid for negative values of $\\eps$, *i.e.* when $\\eps\\in (-1/2,0)$. In this case however, there is no need for any resistance to puncturing, because the exceptional rows and columns at the local view of *v* from Claim [claim:alpha/4] will number *o*(Δ3/2) and contribute a negligible amount to the distance between *x**v* and the tensor codeword *c*.\nConsequences of Theorem [thm:main]: asymptotically good quantum LDPC codes\n--------------------------------------------------------------------------\nThe following theorem is a direct consequence of Theorem [thm:main].\n[thm:123] Fix *ρ* ∈ (0, 1/2), $\\eps \\in (0,1/2)$, $\\gamma\\in (1/2+\\eps,1)$ and *δ* > 0. If Δ is large enough and *C**A* and *C**B* are codes of length Δ such that\n1. 0 < dim*C**A* ≤ *ρ*Δ and dim*C**B* = Δ − dim*C**A*,\n2. the minimum distances of *C**A*, *C**B*, *C**A*⊥, *C**B*⊥ are all ≥ *δ*Δ,\n3. [cond3] both dual tensor codes *C*0⊥ = (*C**A* ⊗ *C**B*)⊥ and *C*1⊥ = (*C**A*⊥ ⊗ *C**B*⊥)⊥ are $\\Delta^{3/2-\\eps/2}$-robust with Δ*γ*-resistance to puncturing,\nthen the quantum code ${\\EuScript Q}=({\\EuScript C}\\_0,{\\EuScript C}\\_1)$ defined in has parameters \n$$\\llbracket n, k{\\geqslant}(1-2\\rho)^2n, d{\\geqslant}\\frac{\\delta}{4\\Delta^{3/2+\\eps}}n\n\\rrbracket.$$ \nThe statement on the dimension *k* is formula . Recall that the minimum distance of the quantum code ${\\EuScript Q}$ is the smallest weight of a vector that is either in ${\\EuScript C}\\_0\\setminus {\\EuScript C}\\_1^\\perp$ or in ${\\EuScript C}\\_1\\setminus\n{\\EuScript C}\\_0^\\perp$. Let $x\\in{\\EuScript C}\\_1$ be a non-zero vector of weight $<\\delta\nn/4\\Delta^{3/2+\\eps}$. The vector *y* guaranteed by Theorem [thm:main] is in ${\\EuScript C}\\_0^\\perp\\subset{\\EuScript C}\\_1$, and since ∣*x* + *y*∣ < ∣*x*∣, we may apply repeatedly Theorem [thm:main] to create a sequence of vectors of ${\\EuScript C}\\_1$ of decreasing weights, and ultimately obtain that *x* must be a sum of vectors of ${\\EuScript C}\\_0^\\perp$. The same argument applies symmetrically to the weight of vectors of ${\\EuScript C}\\_0\\setminus{\\EuScript C}\\_1^\\perp$, since we have imposed that *C**A*⊥ and *C**B*⊥ also satisfy the hypotheses of Theorem [thm:main].\nTo obtain asymptotically good families of quantum codes it remains to show that codes *C**A*, *C**B* satisfying the conditions 1,2,3 of Theorem [thm:123] exist. This is achieved through the random choice result of Theorem [thm:P-robust].\nSpecifically, we have:\n[thm:goodcodes] Fix *ρ* ∈ (0, 1/2), $\\eps \\in (0,1/2)$, $\\gamma\\in (1/2+\\eps,1)$ and *δ* > 0 such that − *δ*log2*δ* − (1 − *δ*)log2(1 − *δ*) < *ρ*. Fix some large enough Δ and let *r* = ⌊*ρ*Δ⌋. Let *C**A* be the random code defined by a random uniform *r* × Δ generator matrix and let *C**B* be the random code defined by a random uniform *r* × Δ parity-check matrix. With non-zero probability *C**A* and *C**B* satisfy conditions 1,2,3 of Theorem [thm:123] yielding an infinite family of quantum codes of parameters \n$$\\llbracket n, k{\\geqslant}(1-2\\rho)^2n, d{\\geqslant}\\frac{\\delta}{4\\Delta^{3/2+\\eps}}n\n\\rrbracket.$$ \nIt is well-known that for *r* < Δ, a random uniform *r* × Δ matrix has rank *r* with probability at least 1 − 1/2Δ − *r* so condition 1 will hold with probability tending to 1 when Δ goes to infinity.\nThe value *δ* has been chosen to be below the Gilbert-Varshamov bound for the imposed rates of all 4 codes *C**A*, *C**B*, *C**A*⊥, *C**B*⊥, so all their minimum distances will be ≥ *δ*Δ with probability tending to 1 when Δ goes to infinity.\nFinally, both the pair (*C**A*, *C**B*) and the pair (*C**A*⊥, *C**B*⊥) are chosen according to the same probability distribution, namely one random uniform generator matrix and one random uniform parity-check matrix, therefore, with probability tending to 1 when Δ goes to infinity, both dual tensor codes *C*0⊥ and *C*1⊥ will satisfy the conclusion of Theorem [thm:P-robust] in other words condition [cond3].\n#### An upper bound on the minimum distance of ${\\EuScript Q}$.\nTo evaluate the tightness of the lower bound on the minimum distance *d* of the quantum code ${\\EuScript Q}$ given by Theorem [thm:123], we now derive the following upper bound on *d*.\n[prop:n/Delta] The minimum distance *d* of the quantum code ${\\EuScript Q}$ is not more than *n*/Δ.\nRecalling Remark [rem:negative], we have that if we could find the required dual tensor codes *C*0⊥ and *C*1⊥ that are both $\\Delta^{3/2-\\eps}$-robust for $\\eps\\to -1/2$, we would obtain a lower bound on the quantum code minimum distance that practically closes the gap with the upper bound of Proposition [prop:n/Delta].\nTo prove Proposition [prop:n/Delta] we shall show the existence of small-weight ( ≤ *n*/Δ) codewords of ${\\EuScript C}\\_1\\setminus {\\EuScript C}\\_0^\\perp$. We do this by constructing many small-weight codewords of ${\\EuScript C}\\_1$, that are too numerous to all be in ${\\EuScript C}\\_0^\\perp$. Those codewords are best described when using the quadripartite version of the Left-Right Cayley complex described at the end of Section [subsec:LRCayley]. In this case the graph ${\\EuScript G}\\_1^\\square$ is bipartite, with its vertex set *V*1 split into the disjoint union of two sets *V*10 and *V*01. We have that for every vertex *v* ∈ *V*1, all squares in its local view that are indexed by row *a*, are indexed by row *a*− 1 in all neighbouring local views. Therefore, if for every vertex *v* ∈ *V*10 we restrict its *Q*-neighbourhood *Q*(*v*) to row *a* for a fixed *a*, and similarly restrict all *Q*-neighbourhoods of vertices of *V*01 to row *a*− 1, we obtain a subgraph ${\\EuScript G}\\_{1a}^\\square$ of ${\\EuScript G}\\_1^\\square$ that is a copy of the double cover of the graph ${{\\sf Cay}}(G,B)$. Furthermore, the edge set of ${\\EuScript G}\\_1^\\square$ is the disjoint union of the edge sets of the graphs ${\\EuScript G}\\_{1a}^\\square$ for *a* ranging in *A*.\nNow we see that any codeword of the Tanner code $T({\\EuScript G}\\_{1a}^\\square,C\\_B)$ yields a codeword of ${\\EuScript C}\\_1$, where every non-zero local view at any vertex of *V*10 must be entirely supported by row *a* (and row *a*− 1 for vertices of *V*01) and coincide with a codeword of *C**B*. Any such codeword has weight at most *n*/Δ: expanding $T({\\EuScript G}\\_{1a}^\\square,C\\_B)$ to the whole index set ${\\EuScript G}\\_1^\\square$ by padding its words with 0s, we may define the disjoint direct sum \n$${\\EuScript L}= \\sum\\_{a\\in A}T({\\EuScript G}\\_{1a}^\\square,C\\_B).$$ \n We claim that:\n[lem:eL] $\\dim{\\EuScript L}\\cap{\\EuScript C}\\_0^\\perp {\\leqslant}2\\dim C\\_A.\\dim T({\\EuScript G}\\_{1a}^\\square,C\\_B)$ for any given *a* ∈ *A*. In particular when dim*C**A* < Δ/2 we have ${\\EuScript L}\\not\\subset{\\EuScript C}\\_0^\\perp$.\nThe second claim of Lemma [lem:eL] follows from the first, since clearly $\\dim{\\EuScript L}=\\Delta\\ell$ where ℓ is the common dimension of the Δ isomorphic Tanner codes $T({\\EuScript G}\\_{1a}^\\square,C\\_B)$. Lemma [lem:eL] implies therefore that at least one non-zero Tanner codeword in some $T({\\EuScript G}\\_{1a}^\\square,C\\_B)$ (which must be of weight ≤ *n*/Δ) cannot be in ${\\EuScript C}\\_0^\\perp$ which proves Proposition [prop:n/Delta]. It remains therefore to prove Lemma [lem:eL]. We will need the following easy fact on Tanner codes on bipartite graphs:\n[lem:tannereasy] Let ${\\EuScript G}=(W,E)$ be a regular bipartite graph on the vertex set *W* = *W*0 ∪ *W*1, and let $C=T({\\EuScript G},C\\_0)$ be a Tanner code on it for some inner code *C*0. Let *x* = ∑*w* ∈ *W**c**v* be a vector in ${\\mathbb F}\\_2^E$ such that every *c**w* is a word supported by the edge-neighbourhood *E*(*w*) of *w* and that coincides with a codeword of *C*0 on *E*(*w*). Then if *x* is a Tanner codeword of *C*, then so are the partial sums ∑*w* ∈ *W*0*c**w* and ∑*w* ∈ *W*1*c**w*.\nWe have that *x* = ∑*w* ∈ *W*0*x**w* = ∑*w* ∈ *W*1*x**w* where *x**w* is the local view of *x* at vertex *w*. Therefore ∑*w* ∈ *W**c**w* + ∑*w* ∈ *W*1*x**w* = 0, so that \n∑*w* ∈ *W*0*c**w* = ∑*w* ∈ *W*1*c**w* + *x**w* \n and therefore ∑*w* ∈ *W*0*c**w* is a vector whose local views, both at vertices of *W*0 and at vertices of *W*1, are all in *C*0, so ∑*w* ∈ *W*0*c**w* is a Tanner codeword. Similarly, so is ∑*w* ∈ *W*1*c**w*.\n[Proof of Lemma [lem:eL]] Let *e*1, …*e**k* be a basis of *C**A* corresponding to some information set *a*1, …, *a**k*, for *k* = dim*C**A*. This means that *e**i* has value 1 on index *a**i* and has value 0 on all indices *a**j*, *j* ≠ *i*.\nLet $x\\in{\\EuScript L}\\cap{\\EuScript C}\\_0^\\perp$. Since $x\\in{\\EuScript C}\\_0^\\perp$, it decomposes as\n \n*x* = ∑*i* = 1*k*∑*v* ∈ *V*00*e**i* ⊗ *x**v**i* + ∑*i* = 1*k*∑*w* ∈ *V*11*e**i* ⊗ *x**w**i*. \nwhere *x**v**i*, *x**w**i* are all codewords of *C**B*. We have abused notation somewhat by identifying the local view of *x* at a vertex *v* with its decomposition expressed as a tensor codeword. But the point is that every term of the form *e**i* ⊗ *x**v**i* contributes to the sum a codeword of *C**B* to the local view at *v* on row *a**i*. Since we must see on column *a**i* a Tanner codeword of $T({\\EuScript G}\\_{1a\\_i}^\\square,C\\_B)$ (because $x\\in{\\EuScript L}$), we have from Lemma [lem:tannereasy] that the whole sum \n∑*v* ∈ *V*00*e**i* ⊗ *x**v**i* \n must be equal, when restricted to row *a**i*, to a codeword of $T({\\EuScript G}\\_{1a\\_i}^\\square,C\\_B)$. Therefore, the left part of the sum must belong to a sum of dim*C**A* codes all isomorphic to a $T({\\EuScript G}\\_{1a}^\\square,C\\_B)$ Tanner code, and by a similar argument, so must the right part of the sum, which concludes the proof.\nConsequences of Theorem [thm:main]: classical locally testable codes\n--------------------------------------------------------------------\nLet *X* be the left right Cayley complex of Theorem [thm:main], and let *C**A* = *C**B* be a code of length Δ, rate *ρ* ∈ (0, 1) and distance *δ*Δ and suppose that the associated dual tensor code $C\\_A\\otimes{\\mathbb F}\\_2^B+{\\mathbb F}\\_2^A\\otimes C\\_B$ satisfies Theorem [thm:main].\nLet *C*0 = *C**A* ⊗ *C**B*: the Tanner code ${\\EuScript C}= T({\\EuScript G}\\_0^\\square, C\\_0)$ is precisely the locally testable code defined of Dinur *et al.* and is shown () to have parameters \n[*n*, *k* ≥ (2*ρ*2 − 1)*n*, *d* ≥ *δ*2(*δ* − *λ*/Δ)*n*] \n where *λ* is the (common) second largest eigenvalue of the constituent Ramanujan Cayley graphs.\nRecall from the tester for this code: given a word $x \\in {\\mathbb F}\\_2^Q$, pick a random vertex *v* ∈ *V*0, accept if the local view *x**v* belongs to the tensor code *C*0 and reject if $x\\_v \\notin C\\_0$. Let *T* ⊂ *V*0 be the set of vertices for which the local test rejects,\n \n$$\\begin{aligned}\n T = \\{ v \\in V\\_0 \\: : \\: x\\_v \\notin C\\_0\\}.\\end{aligned}$$ \nThe fraction of rejecting local tests is therefore $\\zeta(x) := \\frac{|T|}{|V\\_0|}$.\nRecall that a code is said to be locally testable with *q* queries and detection probability *κ*, if the tester accesses at most *q* bits from *x*, always accepts when *x* is a codeword, and otherwise satisfies\n \n$$\\label{eq:testable}\n \\zeta(x) {\\geqslant}\\kappa \\frac 1n d(x,{\\EuScript C}).$$ \nIn the present case, the number of queries is *q* = Δ2, the size of the *Q*-neighbourhood of a vertex. The goal is to establish for some constant *κ*.\nThe strategy to establish the local testability of the code ${\\EuScript C}$ is to define a decoder that is guaranteed to always find a codeword close to *x* if *ζ*(*x*) (or ∣*T*∣) is sufficiently small. The difference with a classical decoder is that we make no assumption on how far *x* actually is from the code ${\\EuScript C}$.\nTheorem [thm:main] can be converted into such a decoding algorithm. Let $x\n\\in {\\mathbb F}\\_2^Q$ be an initial vector. Our goal is to find a close enough codeword $c \\in {\\EuScript C}$. For each vertex *v* ∈ *V*0, let *c**v* ∈ *C*0 be the closest codeword to the local view *x**v* (breaking ties arbitrarily), and let *e**v* : = *x**v* + *c**v* be the local corresponding error (of minimum weight). We slightly abuse notation here and write *c**v* or *e**v* for both the vectors in ${\\mathbb F}\\_2^{Q(v)}$ and the vectors in ${\\mathbb F}\\_2^{Q}$ coinciding with *e**v*, *c**v* on *Q*(*v*) and equal to zero elsewhere.\n#### The decoder.\nThe decoder starts by computing the list (*c**v*)*v* ∈ *V*0 from the decompositions *x**v* = *c**v* + *e**v*. Note that this local decoding can be achieved by brute-force if need be since the local code has constant length Δ2. Note also that the list of these local views is not necessarily equal to the list of local views of a codeword $c\\in{\\EuScript C}$ (if it does then the decoder outputs *c*). The decoder then computes what we may call the *mismatch* of the list (*c**v*), and which is defined as *z* = ∑*v**c**v*. If the weight ∣*z*∣ is too large, namely ${\\geqslant}\\delta\nn/4\\Delta^{3/2+\\eps}$, then the decoder will refuse to continue and output “far from the code”. Otherwise the decoder proceeds by looking for a vertex *v* ∈ *V*0 on which it will update the value of *c**v*, replacing it by *c**v*ʹ = *c**v* + *y**v* for some non-zero *y**v* ∈ *C*0, so as to decrease the weight of the new value *z* + *y**v* of the mismatch. Among all possible vertices *v* and small codewords *y**v* ∈ *C*0, let it choose the one that maximizes ∣*z*∣ − ∣*z* + *y**v*∣.\nThe decoder proceeds iteratively in this way until it has a list of local views with a zero mismatch, corresponding therefore to the list of local views of a global codeword $c'\\in{\\EuScript C}$ which it outputs.\nIf $|z| < \\delta n/4 \\Delta^{3/2+\\eps}$, then the decoder always converges to the zero mismatch and a codeword *c*ʹ of ${\\EuScript C}$.\nThe crucial observation is that *z* is a codeword of ${\\EuScript C}\\_1 = T({\\EuScript G}\\_1^\\square, C\\_A \\otimes {\\mathbb F}\\_2^B + {\\mathbb F}\\_2^A \\otimes C\\_B)$, the Tanner code defined on the graph ${\\EuScript G}\\_1^\\square$ (with vertices in *V*1), and local code given by the *dual* tensor code. In the language of the quantum codes of the previous section, this is simply because ∑*v* ∈ *V*0*c**v* is a sum of generators. Theorem [thm:main] asserts that there exists a generator *y**v* ∈ *C*0 such that ∣*z*∣ − ∣*z* + *y**v*∣ > *a*Δ2, for some constant *a* > 0, independent of *n* (and mentioned in Remark [rem:a]).\nIf $|z| < \\delta n/4 \\Delta^{3/2+\\eps}$, then the distance from *x* to ${\\EuScript C}$ is bounded by\n \n$$\\begin{aligned}\n\\label{eqn:ltc1}\nd(x,{\\EuScript C}) {\\leqslant}n\\Big(1+\\frac{1}{a}\\Big) \\zeta(x). \\end{aligned}$$ \nLet (*c**v*ʹ)*v* ∈ *V*0 be the list of local views of the output codeword *c*ʹ. Writing *x* = *c*ʹ + *e*ʹ, we have that the local views *e*ʹ*v* of *e*ʹ satisfy *x**v*ʹ = *c**v*ʹ + *e**v*ʹ and \n$$d(x,c') =|e'|= \\frac 12\\sum\\_{v\\in V\\_0}|e\\_v'|.$$ \n Let *S* be the list of vertices of *V*0 whose output value *c**v*ʹ differs from the original value *c**v*. Writing *x* = *c*ʹ + *e*ʹ, and remembering that *T* is the set of vertices *v* for which the original *e**v* is non-zero, we may bound from above ∣*e*ʹ∣ by \n∑*v* ∈ *T*∣*e**v*∣ + ∑*v* ∈ *S*∣*e*ʹ*v*∣ ≤ Δ2(∣*T*∣ + ∣*S*∣). \nWe must have ∣*S*∣ ≤ ∣*z*∣/(*a*Δ2). Furthermore, since each bit of the word *x* appears twice in the sum ∑*v* ∈ *V*0*x**v* = ∑*v* ∈ *V*0(*c**v* + *e**v*), we have that ∑*v* ∈ *V*0*c**v* = ∑*v* ∈ *V*0*e**v* = *z*, from which we infer that ∣*z*∣ ≤ ∣*T*∣Δ2. Putting this together, we obtain\n \n$$ |e'|{\\leqslant}\\Delta^2 ( |T| + |T|/a) = \\Delta^2 |V\\_0| \\Big(1+\\frac{1}{a}\\Big) \\zeta(x)$$ \n whence \n$$ d(x,{\\EuScript C}) {\\leqslant}n\\Big(1+\\frac{1}{a}\\Big) \\zeta(x). \\qedhere$$ \nLet us now consider a word $x \\in {\\mathbb F}\\_2^Q$ such that $|z| {\\geqslant}\\delta n/4\n\\Delta^{3/2+\\eps}$, in which case the decoder gives us nothing. We nevertheless have that, again using ∣*z*∣ ≤ Δ2∣*T*∣,\n \n$$\\begin{aligned}\n\\label{eqn:ltc2}\n\\frac 1nd(x, {\\EuScript C}) {\\leqslant}1 {\\leqslant}\\frac{4 \\Delta^{3/2+\\eps} |z|}{\\delta n} {\\leqslant}\\frac{4 \\Delta^{7/2+\\eps} |T|}{\\delta n} = \\frac{4 \\Delta^{7/2+\\eps}\n|V\\_0|}{\\delta n} \\zeta(x) = \\frac{8 \\Delta^{3/2+\\eps}}{\\delta} \\zeta(x). \\end{aligned}$$ \nFrom and, we obtain that the Tanner code ${\\EuScript C}= T({\\EuScript G}\\_0^\\square, C\\_0)$ is *κ*-locally testable with Δ2 queries and \n$$ \\kappa = \\min \\Big( \\frac{a}{a+1}, \\frac{\\delta}{8 \\Delta^{3/2+\\eps}} \\Big).$$ \nWe conclude this section with a word of comment on the choice of the small component code *C**A* = *C**B*. To obtain an LTC we need it to satisfy the hypotheses of Theorem [thm:main]. One way is to obtain this by random choice, namely by applying Theorem [thm:P-robust], as we did in Section [sec:goodLDPC]. But since this time we have no need for the robustness of *C**A*⊥ ⊗ *C**B*⊥, there are alternatives: in, the component codes that are used have better robustness than what is guaranteed by Theorem [thm:P-robust].\nAppendix: proof of Theorem [thm:P-robust]\n=========================================\nThe proof follows the blueprint of, though puncturing is handled a little differently.\nLet $C\\_A\\subset{\\mathbb F}\\_2^A$ and $C\\_B\\subset{\\mathbb F}\\_2^B$ and let *C* be the dual tensor code $C= C\\_A\\otimes\n{\\mathbb F}\\_2^B+{\\mathbb F}\\_2^A\\otimes C\\_B$. Let $X\\subset{\\mathbb F}\\_2^{A\\times B}$ be a Δ × Δ matrix. Let *H**A* be an *r**A* × Δ parity-check matrix for the column code *C**A*. The *r**A* × Δ matrix *H**A**X* is such that, for every *b* ∈ *B*, its *b*-th column is the *H**A*-syndrome of the *b*-th column of *X*. We note that *X* ∈ *C* iff *H**A**X* is such that everyone of its *r**A* rows is a codeword of *C**B*.\n[lem:injective] Let *V* be a set of vectors of ${\\mathbb F}\\_2^{A}$. Suppose that the union of the supports in *A* of the vectors of *V* has size *t* < *d**A*/2, where *d**A* is the minimum distance of *C**A*. Then $\\rk{\\{H\\_Av : v\\in V\\}}=\\rk{V}$, where $\\rk{}$ denotes the rank function.\nThe syndrome map $x\\mapsto H\\_Ax^\\intercal$ is injective on the Hamming ball of radius *t*, therefore for a subset *W* of vectors *x* of *V*, ∑*W**H**A**x* = 0 implies ∑*W**x* = 0. So if *W* ⊂ *V* is a set of linearly independent vectors, $\\rk{\\{H\\_Ax : x\\in W\\}}=|W|$.\nLet *d* > 2*α*Δ*A* be a lower bound on the minimum distance *d**A* of *C**A*, for some constant *α*.\n[lem:rank] Let *X* be a Δ × *n* matrix such that all of its columns are of weight ${\\leqslant}\\sqrt{\\Delta}/\\log\\_2\\Delta$ and such that the number of its non-zero rows is at least *d*/2. Then, for Δ large enough, *X* has rank at least $\\alpha\\sqrt{\\Delta}\\log\\Delta$.\nLet *x*1, *x*2, …, *x**m* be a maximum sequence of non-zero columns of *X* with the property that supp(*x**i*) ⊄ ∪ 1 ≤ *j* < *i* supp(*x**j*). We clearly have that *x*1…, *x**m* are linearly independent, and we have *m* ≥ *d*/2*w* where *w* is the maximum weight of a vector *x**i*, otherwise the union of the supports of all columns of *X* would be less than *d*/2, contrary to our hypothesis. From $w{\\leqslant}\\sqrt{\\Delta}/\\log\\_2\\Delta$ we have the result.\n[cor:rank] Let *X* be a Δ × *n* matrix such that all of its columns are of weight ${\\leqslant}\\sqrt{\\Delta}/\\log\\_2\\Delta$ and that has at least *d*/2 non-zero rows. Then *H**A**X* has rank at least $\\alpha\\sqrt{\\Delta}\\log\\Delta$.\nChoose $\\alpha\\sqrt{\\Delta}\\log\\Delta$ linearly independent columns of *X*, which is possible by Lemma [lem:rank]. Since the weight of these vectors is not more than ${\\leqslant}\\sqrt{\\Delta}/\\log\\_2\\Delta$, the sum of all the weights of these vectors is less than *d*/2, therefore Lemma [lem:injective] applies and the claim is proved.\n[lem:randomC] Let ${\\mathbf C}$ be a random linear code of length *n*, defined by an *r* × *n* uniform random matrix ${\\mathbf H}$. Let *V* be a set of linearly independent vectors of ${\\mathbb F}\\_2^n$. The probability that all vectors of *V* fall into the random code ${\\mathbf C}$ is equal to 1/2*r*∣*V*∣.\nThe ${\\mathbf H}$-syndrome of a fixed non-zero vector *x* is uniformly distributed in ${\\mathbb F}\\_2^r$ and the probability that it equals zero is therefore 1/2*r*. When the fixed vectors *x* ∈ *V* are independent in the sense of linear algebra, their ${\\mathbf H}$-syndromes are independent in the sense of probability, hence the lemma.\n[lem:randomCp] Let ${\\mathbf C}$ be a random linear code of length *n*, defined by an *r* × *n* uniform random matrix ${\\mathbf H}$. Let ${\\mathbf C}\\_p$ be the punctured code obtained from ${\\mathbf C}$ by throwing away the first *p* coordinates. Let *V* be a set of linearly independent vectors of ${\\mathbb F}\\_2^{n-p}$ defined on the set of coordinates {*p* + 1, …*n*}. Then the probability that all vectors of *V* fall into the random code ${\\mathbf C}\\_p$ is at most 1/2(*r* − *p*)∣*V*∣.\nLet ${\\mathbf W}$ be the random linear subspace of ${\\mathbb F}\\_2^r$ generated by the first *p* columns of ${\\mathbf H}$. Let ${\\mathbf H}\\_p$ be the *r* × (*n* − *p*) matrix deduced from ${\\mathbf H}$ by throwing away its first *p* columns. A vector $x\\in{\\mathbb F}\\_2^{n-p}$ is in the punctured code ${\\mathbf C}\\_p$ iff its ${\\mathbf H}\\_p$-syndrome ${\\mathbf H}\\_px^\\intercal$ belongs to ${\\mathbf W}$. Denote by *E**V* the event whereby all vectors of *V* fall into the punctured code ${\\mathbf C}\\_p$. Let *W* be a fixed subspace of ${\\mathbb F}\\_2^r$ of dimension *w* ≤ *p*. Denote by *E**W* the event whereby the first *p* columns of ${\\mathbf H}$ generate *W*. We have that the projections onto ${\\mathbb F}\\_2^r/W$ of the columns of ${\\mathbf H}\\_p$ are uniformly distributed and independent in the sense of probability. Therefore, if the first *p* columns of ${\\mathbf H}$ generate the fixed subspace *W*, Lemma [lem:randomC] applies and we have *P*(*E**V*∣*E**W*) = 1/2(*r* − *w*)∣*V*∣. Now we have \n*P*(*E**V*) = ∑*W**P*(*E**W*)*P*(*E**V*∣*E**W*) \n where the sum is over all subspaces *W* of ${\\mathbb F}\\_2^r$ of dimension at most *p*. Hence, \n$$P(E\\_V){\\leqslant}\\sum\\_WP(E\\_W)\\frac{1}{2^{(r-p)|V|}} = \\frac{1}{2^{(r-p)|V|}}. \\qedhere$$ \n[lem:XP] Let *p*, *r**B* be integers such that 0 < *p* < *r**B* < Δ. Let *X**p* be a fixed Δ × (Δ − *p*) binary matrix such that all its columns are of weight ${\\leqslant}\\sqrt{\\Delta}/\\log\\_2\\Delta$ and that has at least *d*/2 non-zero rows. Let *P* be a subset of *B* of cardinality"},"data_id":{"kind":"string","value":"arxiv_0182398"}}},{"rowIdx":45596,"cells":{"content":{"kind":"string","value":", we outline the background for this survey in Section 2. Preliminaries of DFL are reviewed in Section 3. Section 4 studies possible threats to DFL. Section 5 discusses the potential defense mechanisms in DFL. Section 6 elaborates on the verifiability of DFL. Future research directions are discussed in Section 7. Finally, the survey is concluded in Section 8. This survey contains several abbreviations to make discussions more concise, which are listed in Table [tab:abrv].\nBackground\n==========\nDFL is a paradigm that often encompasses both FL and blockchain technologies, although it is important to note that DFL can exist independently of blockchain. To better comprehend the concept of DFL, it is crucial to have a clear understanding of the fundamentals of FL and blockchain. Thus, the following section provides a concise overview of these concepts.\nFederated Learning\n------------------\nFL was proposed to reduce the risks of data ownership in collaborative training of DL models. Prior to the introduction of FL, collaborative training required immense loads of data exchange between participants of a distributed machine learning framework in which the clients constantly send local training data to a central server. The distributed model is then trained on the accumulated training data gathered from across the network. Nevertheless, communicating user data and overburdening the network with excessive communication load raised new concerns that motivated the invention of FL. In simple words, rather than sharing the training of the local data with the server, the client in FL locally trains a model of the same structure on the user data and sends the obtained model parameters to the server once convergence is achieved. The server aggregates these parameters (e.g., neural network weights) and updates all the client models with the aggregated parameters. As this cycle continues, in each round, clients initialize their local model parameters with the received update from the server before starting the local training and issuing an update to the server. It is worth mentioning that the collaboration schemes between clients at the edge of the network have been a topic of interest in the domain of multi-agent systems even before the birth of FL. For instance, client-server collaboration in FL somehow resembles the umbrella system in which all agents communicate with a server.\nThe aforementioned process in FL, however, assumes the homogeneity of data among all clients, which may not always be the case. This leads to the evolution of FL into a more advanced structure for dealing with heterogeneous data and comes with additional security perks. Specifically, categorizes techniques in this field into three groups: 1) Horizontal FL (HFL), 2) Vertical FL (VFL), and 3) Federated Transfer Learning (FTL). The difference between these groups is in the nature of the training data in the FL systems. HFL which is basically defined based on the initial FL model in, assumes a diverse range of samples across all local training sets that all fall under a similar feature space. VFL on the other hand, considers a common sample space among participants who differ in the feature space. FTL addresses the minimal overlap between both sample and feature space by resorting to transfer learning for transferring knowledge between participants.\nIt is worth mentioning that this categorization has primarily focused on centralized FL. Therefore, the focus of this survey is primarily on DFL which differs from the centralized FL. The central server in the FL systems is in charge of verifying the network clients, aggregating the local parameters, and broadcasting an update. With the absence of servers in the DFL structure, these tasks must be performed using the client networks. The current literature usually devises the concept of blockchain to carry out the mentioned tasks.\nEven though the initial FL model was proposed on the basis of DL, decision tree models can also be employed and adapted to FL frameworks. Threats and defense mechanisms that will be later reviewed in this work generally apply to both DL and tree-based models.\nBlockchain\n----------\nBlockchain is designed primarily with the objective of making the data exchange immutable and traceable over a P2P network. Similar to FL, in a blockchain network data is decentralized, and each user will transfer data using a block. This consists of the data, a hash acting as a unique identifier for the block, and a previous hash pointing to the previous block in the chain. While these hashes make the data traceable in the chain, they also complicate tampering with data, as changing the data in an existing block will also alter the associated hash which creates a discrepancy in the chain.\n[t] [fig:blockchain]\nA key idea behind the blockchain is to prevent a certain participant from controlling the network and provide all network members with an equal chance to verify and control the transactions. Nonetheless, finding the optimal solution to ensure fairness in assigning the evaluators is still an open problem and as a result, many consensus mechanisms have been proposed over time. Perhaps, the most common consensus mechanisms are Proof of Work (PoW) and Proof of State (PoS), albeit they are not necessarily the best. In PoW, the participants (also called miners) need to solve a time-consuming problem and anyone who achieves the answer quicker will get a chance to contribute to the chain. As an alternative to PoW, PoS randomly selects the evaluators with the aim of improving the scalability.\nBased on the current literature, Blockchain approaches can fall into four major categories, namely 1) public, 2) private, 3) consortium, and 4) hybrid blockchain (i.e., we refer to this as DFL). These variations are briefly explained in the following.\n1. **Public:** Blockchain is built in a permissionless network that allows anyone to join the network and participate in the consensus process.\n2. **Private:** Participants can join the network through a received authorization. While permissions are granted in a centralized manner, the consensus mechanisms are decentralized.\n3. **Consortium:** Blockchain is semi-decentralized and the network access is managed by more than one entity.\n4. **Hybrid:** Block construction is carried out within a private network and the storage is done through a public network.\nDecentralized Federated Learning\n================================\n[t]![Advances in decentralized federated learning in time. The order of the methods is set based on the first version that became available online (e.g., pre-prints).](FIG3.png \"fig:\") [fig:diagram]\n[sec:DFL] As mentioned, the main flaw with the centralized FL is the dependency of the entire federation on a central server whose privacy and performance directly affect all the clients in the system. In other words, a compromised server will jeopardize the whole federated learning system. To eliminate the dependency of the whole network on a single node and also enhance the communication efficiency, DFL resorts to P2P communications to bypass the need for a central server. To do so, model aggregation and participant verification should be carried out in a serverless fashion.\nAs explained in Section [sec:back], DFL often uses blockchain to facilitate inter-node communications. While this combination is non-absolute, blockchain can effectively facilitate communication by treating model updates as data within a block and sharing the block with respect to a consensus mechanism. In this process, local model construction is similar to that of FL, and prepared updates will be in the form of a data block. The prepared block will be then shared with the rest of the nodes for consensus and addition to the blockchain. The aggregation mechanisms of DFL work on the basis of decentralized Stochastic Gradient Decent (SGD), which makes use of gossip averaging.\nDFL models that use blockchain have distinct architectures. For instance, the block structure, information included in the headers, aggregation, and consensus mechanisms can all be different in each architecture. However, a generic scheme is shown in Fig. [fig:blockchain] that resembles the most common practices in designing a blockchain-based DFL. In this example, the blockchain is initialized using a single block containing the starting global model. Each update is added as a new block that is linked to the previous block in the chain using a header. On the client’s side, the local model is initialized using the global model *u**t* and trained on the *i*-th client data *D**i*. Once the trained model *M**i* is obtained, a score *S**i* is estimated w.r.t. the difference between the global model and the estimated local model. Nevertheless, not all blockchain-based DFL models necessarily use a scoring system. The client then uploads data to the blockchain by creating a block containing a header, trained model, estimated score, and the uploader identifier (ID). In some architectures, clients are directly communicating with a set of miners to obtain the Merkle root of the data before uploading the block. It is also worthwhile to mention that the block size is often limited to a fixed size. If the model size is larger than the block size, the uploader has to upload a set of serialized blocks to the blockchain. The SC on this blockchain uses a set of rules to determine when the global model should be updated. For instance, a minimum number of unique clients *k* are required to participate in updating *u**t*. Furthermore, considering that each client is allowed to serialize its updates into multiple blocks, a limit is set to define the maximum number of contributions by each client in each round. The aggregation mechanism varies in different DFL architectures. While the majority of DFL models use an averaging function ∑( ⋅ ) similar to that of federated averaging (FedAVG), other frameworks use different approaches such as selecting the update with the highest score.\nWhile the initial DFL model was proposed with the aim of security enhancement and failure robustness of FL using Bayesian beliefs of one-hop neighbors, several research studies extend their work by improving the overall efficiency and security. Fig. [fig:diagram] illustrates the historical evolution of DFL since it was first proposed in 2018. BAFFLE makes use of Smart Contracts (SC) to facilitate storing the FL model and clients’ states. Model updates and aggregation is also carried out by resorting to SC. BAFFLE takes the computational states into account in order to preserve fairness among clients. Another approach, Blade-FL, uses clients for both mining and training via gossip learning. A novel consensus mechanism was proposed in to enhance the efficiency and privacy of the system. The DFL framework proposed based on this committee consensus mechanism is called BFLC. The Biscotti technique integrates DFL with Differential Privacy (DP) to safeguard DFL against a range of attacks. Focused on healthcare applications, introduces the BindaaS method, which integrates DL as a service and lattices-based cryptography. This enables BinDaaS to employ an authentication phase that enhances DFL attack resilience. Another DP-based DFL approach called LearningChain is proposed with the aim of combating Byzantine attacks. To do this, LearningChain utilizes a new aggregation mechanism for decentralized SGD. Reference designs the PIRATE framework based on the sharding approach in blockchain which helps in securing the aggregation process. BEAS employs DP, gradient pruning, and anomaly detection to protect DFL against poisoning attacks. BEAS is also adapted for heterogeneous data to ensure proper convergence of the global model. BrainTorrent is a DFL method that employs version control and arbitrarily assigns a participant to act as the server. A reputation management scheme is used in to select updates from reliable workers in the blockchain which reduces the probability of successful poisoning attacks in the DFL system. BlockFLA, uses smart contracts to detect and penalize malicious nodes by means of monetary penalty. BlockFLA requires uploading hashed updates into separate private and public blockchains. Smart contracts is used for aggregating gradients whereas hashed updates are used for evaluating the updates through recovering the parameters to detect any mismatch. DFedForest is a tree-based DFL framework that uses bagging for tree construction. This method devises a private test dataset for evaluating the uploaded parameters as malicious updates are expected to result in anomalous outputs on this data. Bift,, designs a Proof of FL by combining PoW with FederatedAVG and FederatedSGD aggregation schemes to reach secure DFL with low communication overhead. A verifiable version of DFL incorporates private key sharing and gradient masking into the consensus mechanism to defend against malicious miners and dropouts. Swarm Learning (SL) also integrates FL and blockchain to distributively train a model for diagnosing a number of diseases while safeguarding the patient’s data privacy and security. Security and privacy features of SL are limited to those of blockchain and the P2P DFL design. However, there is an extension of SL that couples it with HE is also presented in to secure SL communications against privacy attacks. Various aspects of SL such as fault tolerance, scalability, and fairness are studied in, and the results indicate an overall improvement over centralized FL.\nDespite the security advantages of blockchain integration with DFL, communication delay and resource consumption issues often become problematic in this domain. As mentioned in Section [sec:back], the process of model aggregation and achieving a consensus often requires sufficient computational power at the edge of the network which may not always be practical (e.g., mobile networks). This becomes more critical for consensus mechanisms such as PoW. As a result, the model aggregation process may be usually delayed due to the potential computational constraints.\nThreats to Decentralized Federated Learning\n===========================================\nAs shown in Fig. [fig:taxonomy], the security and performance of DFL are directly related to those of FL and blockchain. Generally, threats to DFL target privacy or the performance of the global model. In centralized FL, all the possible threats involve the server at some point as the server is the core of the system for aggregation, communication, and validation. As a result, the trustworthiness of the server is of great importance in FL as a malicious server can easily attack all clients. On the other hand, attacking the server and causing a malfunction can also disable the entire system. Table [tab:tab1] lists sources of threats for common attacks on FL. However, in DFL, all possible threats will be traced back to the clients or miners. Table [tab:analysis] indicates attack models that are considered in the design of the aforementioned DFL techniques.\n[t]\nll Attacks & Source of Attack \nData poisoning & Malicious client \nGradient manipulation & Malicious client \nBackdoor attack & Malicious client and malicious server \nEvasion attack & Malicious client and model deployment \nNon-robust aggregation & Aggregation algorithm \nTraining rule manipulation & Malicious client \nInference attacks & Malicious server and communication \nGAN reconstruction & Malicious server and communication \nFree-riding attack & Malicious client \nMan-in-the-middle attack & Communication \n[tab:tab1]\n[t] [fig:taxonomy]\nMalicious clients can follow different attack models to jeopardize the performance or privacy of the system. For instance, an adversary can be either semi-honest or aggressive. In the semi-honest model, the objective is to infer confidential information while complying with the DFL protocol. This phase can be also used as a reconnaissance step prior to launching an aggressive attack where the aim is to degrade the system’s performance. Using this combination, one can first learn the global model parameters using a semi-honest approach and then actively forge malicious updates in order to insert a backdoor in the global model. Due to the immutability of blockchain, hackers cannot directly manipulate the training data in a block. Nevertheless, they can reach their objective by sending the gradients that lead to incorrect predictions. To do so, gradients can be forged targeted or untargeted. As the name implies, the latter aims at degrading the performance of all classes in general whereas the targeted scheme only attacks a certain class without tampering with the rest of the model.\nAttacks on Performance\n----------------------\nThe performance of the DFL model is mostly targeted by means of poisoning attacks. In addition, there are certain blockchain-based attacks that also apply to DFL. An overview of these attacks for DFL is given in the following.\n### Data Poisoning\nData poisoning techniques in FL and DFL could be somewhat different. In centralized FL, data poisoning is referred to as the process of corrupting the local training data via backdoor injection or introducing label noise which in turn produces deviating gradients and misleads the FL model. In this setting, usually, the intruder disguises as a member of the federation and uses its artificially made data. As an example, reference studies different variations of label poisoning in federated learning systems and mitigates these attacks by reformulating the problem as a label noise classification task.\nAside from the aforementioned mechanisms for data poisoning, the following attacks can be planned to take advantage of the blockchain backbone that most DFL methods rely on:\n**Disrupting Communications:** Blockchain is robust by nature against Denial of Service (DoS) attacks as there is no possible SPF that can be targeted. Nevertheless, Distributed DoS (DDoS) attacks can still slow down inter-node communications in the DFL system. DDoS agents continuously generate fake transactions and send them to the chain to fill up the blockchain buffer with spam data once the blockchain capacity is reached. This significantly postpones the inclusion of valid data into the upcoming blocks which puts the functionality of the blockchain under question. This is besides the permanent effect DDoS leaves on the DFL due to the data immutability of the blockchain.\n**Degrading Detection Performance:** Using a targeted attack model, an intruder can inject certain triggers into the local model and obtain gradients in a way that the rest of the model remains intact. This backdoor attack results in sharing the poisoned parameters with the rest of the DFL participants and introducing the backdoor into their model as well. For instance, backdoors are injected into the DFL network in with two different mechanisms, namely label flipping and pixel patch backdoor. The number of malicious nodes is restricted to prevent intruders from controlling the consensus (i.e., 2*N**c**o**r**r**u**p**t**e**d* + 2 < *N*, where *N* and *N**c**o**r**r**u**p**t**e**d* indicate the number of all nodes and corrupted ones, respectively). A pixel pattern backdoor is also simulated in for DFL with smart contract. The untargeted scheme of this attack which is also known as the Crashing DoS attack, follows the same process with the exception of degrading the performance of the entire model rather than for a specific class. While the untargeted version is easier to implement, it is also easier to detect. Note that the literature usually defines DoS differently for blockchain and FL. DoS in blockchain renders the global model inaccessible, whereas in FL, it makes the global model unusable. Reference presents an example of a data poisoning attack in DFL which is simulated with the aim of corrupting the global model performance. In this scenario, 30 percent of nodes are considered malicious, with the assumption that the number of malicious nodes does not change in time. Another case is tested in by generating forged gradients drawn from a Gaussian distribution by several Byzantine data holders (i.e., trusted nodes went rouge in the DFL network). Reference simulates this attack by filliping the class labels randomly on the training data. In contrast to poisoning attacks that primarily corrupt the training process, evasion or exploratory attacks occur during the inference phase after the model has been trained. Their goal is typically not to change the trained model, but rather to generate incorrect predictions or gather information about the model’s characteristics.\n**Causing Legal Problems:** By law, the inclusion of personal data is prohibited in some regions of the world. European general data protection regulation can be mentioned as a good example of this legal constraint. A malicious user, however, can poison the blockchain by inserting personal data into a data block and adding it to the chain which makes the DFL system non-compliant with the existing legal bounds. We call these adversaries privacy poisoning attacks. Despite its importance, not many research studies considering this type of attack in a DFL paradigm have been reported.\n[t]\nlcccc Attacks & Target & Implementation & Effectiveness & Defensibility \nBackdoor & Performance & Challenging & High & Challenging \nDoS/DDoS & Performance & Simple & Moderate & Simple \nGradient manipulation & Performance & Challenging & High & Challenging \nTraining objective manipulation & Performance & Challenging & High & Challenging \nEvasion attacks & Performance & Simple & Moderate & Moderate \nModel inversion & Privacy & Simple & Low & Moderate \nFeature inference & Privacy & Simple & Moderate & Moderate \nMembership inference & Privacy & Challenging & High & Challenging \nHijacking private key & Blockchain & Challenging & High & Challenging \n51% attack & Blockchain & Challenging & High & Challenging \nSybil attacks & Blockchain & Simple & High & Challenging \nDouble spending & Blockchain & Challenging & High & Challenging \nRouting attacks & Blockchain & Simple & Moderate & Challenging \nPrivacy poisoning & Blockchain/Privacy & Moderate & Moderate & Moderate \nAttacks on SC & Blockchain & Challenging & High & Moderate \n[tab:attacks]\n### Model Poisoning\nModel poisoning is the process of maliciously controlling global model training. While data poisoning can be also used as a tool to cause model poisoning, model poisoning without data manipulation is also possible. As an example, one can simply change the objective of the local model to obtain poisoned gradients using valid data on the same model structure as the rest of the network. For instance, an additional term can be added to the objective to penalize sensitivity to malicious data. The same approach can be followed to deteriorate the overall performance in an untargeted manner (e.g., gradient manipulation). This case is also studied in where a set of Byzantine nodes are considered to generate and inject deviating gradients into the blockchain in order to degrade the shared DFL model via an outsider attack. A similar approach is used in to corrupt gradients using pointwise Gaussian noise.\n### Routing Attacks\nReliable network infrastructure is of paramount importance for DFL due to its integration with blockchain. This is while some of the utilized network protocols used by current internet service providers, such as the border gateway protocol, come with security flaws. Blockchain has no control over the network layer as it mostly works in the application layer. Thus, if the service provider network is breached, the routing of packets can be tampered with to either discard the transferring data blocks or change the blockchain structure. An example of a routing attack is a hack that took place in 2014 in which the intruder used the hijacked blocks to provide PoW and steal the associated rewards.\n### Consensus Attacks\nThe objective of these attacks is to obtain the majority of approvals in order to control the consensus mechanism in the DFL system. Depending on the number of agents or nodes involved in these attacks, they will be called 51% or Sybil attacks. Sybil attacks indicate the scenario in which an intruder creates multiple fake nodes and claims to be more than one entity to impose a greater influence on the consensus. If intruders manage to take control of 51 percent of the nodes, the consensus mechanism is fully controllable. At this scale, the attack is known as 51% attack in the literature. It is worth mentioning that consensus attacks are most effective when the blockchain associated with the DFL is at its early stage when the number of participants is still limited. PoW and PoS can drastically complicate consensus attacks for hackers.\nAttacks on Privacy\n------------------\nDespite the decentralization of data in DFL, gradient and parameter information can be visible to system clients. Previously, we studied the effect of such information leakage on the performance of the DFL model. Here, we elaborate on the privacy risks associated with DFL networks.\n### Model Inversion Attacks\nModel inversion can be used to approximate and reconstruct the private data of clients merely based on the classification model. An example of this attack is reported in where gradient leakage of a global DFL model is used for reconstructing the user’s local data. This process is often carried out using a DL-based generator. For instance, Generative Adversarial Networks (GAN) can be employed to perform model inversion by using the target model as the discriminator and training a generator network to minimize a cost that eventually converges and produces data samples similar to that of the targeted client.\n### Membership Inference Attacks\nMembership inference attacks are focused on revealing sensitive information associated with certain samples. Membership inference can disclose the membership of a sample to a certain class, or elicit attribute information. An example of this attack can be found in, where an inference attack by observing the clients’ updates in a DFL network and using the leaked gradients to infer record-level information of local datasets.\n### Hijacking Private Key\nSecuring private keys and public key using cryptography is one of the most critical aspects of blockchain which is the backbone of DFL. If there is any imperfection in the key signing mechanism, hackers can hijack a client’s private key using their public key. Taking over a client’s private key enables the hacker to gain full access to the corresponding data in the blockchain.\n### Vulnerability of Smart Contracts\nSmart contracts, which are coded agreements utilizing blockchain technology for record-keeping, can be susceptible to security flaws. While they eliminate the need for intermediaries and provide immutable contracts, there is a risk associated with poorly coded smart contracts. These coding vulnerabilities create opportunities for attackers to identify flaws in the code and exploit them. By exploiting these weaknesses, attackers can potentially manipulate or extract unauthorized access to the contract’s contents or associated assets. It is crucial to ensure thorough code review and rigorous testing to identify and address these security flaws to safeguard the integrity and trustworthiness of smart contracts.\nAnalyzing the Viability of Attacks\n----------------------------------\nTable [tab:attacks] provides an overview of the analyzed attacks, highlighting their key characteristics. When evaluating these attacks, it is crucial to assess their feasibility and effectiveness to design effective defense mechanisms and allocate appropriate resources to secure DFL systems.\n### Ease of implementation\nImplementing a backdoor attack is considered to be difficult because it requires extensive knowledge of the target DFL system, access to training data, and the ability to modify model parameters. Backdoor attacks often involve sophisticated techniques and may require compromising multiple clients or insiders with privileged access. Conversely, carrying out a DoS or DDoS attack is relatively less complicated. These attacks overwhelm the system with a high volume of requests or malicious traffic, and readily available tools and techniques can be used to execute them.\nGradient manipulation necessitates a deep understanding of the DFL system’s architecture, algorithms, and access to the communication channels. Tampering with gradients exchanged between clients and the server is a challenging task that requires careful execution to avoid detection. Similarly, manipulating the training objective, such as modifying the loss function or optimization process, is also categorized as hard. It demands an in-depth understanding of the system’s algorithms, access to the training process, and the ability to modify the objective without disrupting the overall learning process.\nOn the other hand, attacks such as evasion attacks and model inversion are easier to implement. These attacks can be implemented using existing knowledge and techniques. Other attacks, including feature inference and Sybil attacks, are also categorized as easy to implement. In contrast, attacks such as membership inference, hijacking private keys, 51% attacks, consensus attacks, and double spending are challenging attacks. These attacks require advanced knowledge, sophisticated attacks, and access to specific components or mechanisms of the DFL system. Implementing them successfully is a complex and resource-intensive task. Furthermore, privacy poisoning attacks are not overly complex or technically challenging, leading to a medium level of ease of implementation. The process may involve tampering with the data blocks and ensuring the inclusion of personal information, but it does not require advanced technical skills or extensive resources. Moreover, finding security holes in SC is a complex task as these protocols are usually well-tested before release, and sophisticated technical skills are required to hack SC under this condition.\n### Effectiveness of Attacks\nIn terms of attack effectiveness, the backdoor attack is classified as high. This attack has the potential to significantly impact the model’s predictions, compromising the integrity and reliability of the DFL system. On the other hand, DoS/DDoS attacks are considered to have a moderate level of effectiveness. While they can disrupt the normal functioning of the system and cause temporary delays, their impact on the actual model parameters or the accuracy of the aggregated model may vary.\nGradient manipulation and training objective manipulation attacks are both considered to have a moderate level of effectiveness. Tampering with gradients or modifying the training objective can potentially influence the learning process and impact the accuracy of the aggregated model. However, their success depends on the robustness of the DFL system and the defense mechanisms in place.\nEvasion attacks and model inversion attacks are also considered to have a moderate level of effectiveness. Their impact on the overall model performance may be limited depending on the effectiveness of the defense mechanisms. Membership inference attacks, on the other hand, are classified as having a high level of effectiveness. Successfully inferring membership information about specific individuals participating in the DFL system can have severe privacy implications.\nAttacks targeting the blockchain aspect of DFL, such as hijacking private keys, 51% attacks, consensus attacks, Sybil attacks, double spending, and routing attacks, are generally considered to have a high level of effectiveness. These attacks have the potential to compromise the security, integrity, and trust of the DFL system, depending on the specific vulnerabilities and defense mechanisms in place. Moreover, privacy poisoning attacks can have a moderate impact on compromising the privacy of participants in the DFL system. While the effectiveness of privacy poisoning attacks is not as high as some other attacks, it still poses a significant risk to the privacy of participants and the overall integrity of the system. In addition, exploiting the vulnerabilities of SC can lead to losing digital trust and massive financial loss. For instance, reports indicate that crypto investors nearly lost four billion dollars to hackers in 2022.\n### Defensibility\nBackdoor attacks involve inserting malicious behavior into the training data or model, making it challenging to detect and defend against. Detecting and mitigating backdoor attacks often require advanced techniques and rigorous model verification.\nDoS/DDoS attacks are classified as moderately defensible. While it may be challenging to completely prevent DoS/DDoS attacks, there are various mitigation strategies available, including anomaly detection. Such methods can reduce the impact of DoS/DDoS attacks and keep the system operational.\nDefending against gradient manipulation and training objective manipulation attacks is also categorized as moderately defensible. Implementing techniques like DP, robust aggregation algorithms, and secure communication protocols can enhance the system’s resilience against these attacks. However, the effectiveness of the defense mechanisms may vary based on the sophistication of the attack and the quality of the defense techniques.\nEvasion attacks and model inversion attacks are considered moderately defensible. Privacy-preserving techniques, such as DP or SMC, can help protect against these attacks. However, achieving strong defenses requires careful design and implementation, considering factors like attack vectors and data sensitivity.\nDefending against membership inference attacks is classified as challenging due to the difficulty of protecting individual privacy in the FL setting. Advanced privacy protection techniques, such as privacy amplification and enhanced model aggregation protocols, are needed to effectively mitigate the risk of membership inference attacks.\nRegarding attacks on the blockchain aspect of DFL, the defensibility levels vary depending on the specific attack. Hijacking private keys, 51% attacks, and double spending attacks are challenging to defend against due to the vulnerabilities they exploit in the blockchain infrastructure. Implementing robust consensus mechanisms, multi-factor authentication, and encryption can enhance the defensibility against these attacks. Defending against consensus attacks, Sybil attacks, and routing attacks requires strong identity management systems, network monitoring, and reputation-based mechanisms. Defending against privacy poisoning attacks requires a combination of technical, legal, and regulatory measures, making it of medium difficulty in terms of defensibility. In addition, ensuring the security and integrity of SC has a moderate level of difficulty since it requires sophisticated code review and rigorous testing to identify and eliminate potential security holes and flaws.\n[t] [tab:defs]\nllll Defense mechanism & Description & Weakness & Used against \nHomomorphic Encryption & Encrypts parameters & Computational overhead & Attacks on privacy \nSMC & Protects data with multiparty computation & Communication overhead & Attacks on privacy \nDifferential Privacy & Perturbs parameters with noise & Accuracy loss & Attacks on privacy \nAnomaly Detection & Monitors updates to detect abnormalities & Backdoors can bypass it & Untargeted poisoning \nRobust Aggregation & Safeguards global aggregation against adversaries & Limiting assumptions & Poisoning attacks \n & & Computational overhead & \n& & Performance loss \nTEE & A secure ecosystem for maintaining digital trust & Limited memory size & White box attacks \n & & Computational overhead & \n& & Communication overhead & \n& & Complex implementation & \n & & Computational overhead & Privacy attacks \n& & Not a stand-alone defense & Poisoning attacks \n & & Sensitivity to hyperparameters& Privacy attacks \n& & Computational overhead & Poisoning attacks \n & & Computational overhead & \n& & Blockchain security issues & \nlccccccccc Methods & References & Blockchain & Encryption & Clients & Data poisoning & Model poisoning & Inference \nBindaaS & & & × & Honest & & & \nPIRATE & & & × & Semi-honest& & & \nBAFFLE & & & × & Dishonest && & \nBFLC & & & × & Semi-honest & & & \nLearningChain & & & DP & Dishonest & & & \nBiscotti & & & DP & Dishonest & & & \nBlade-FL & & & DP & Honest & & & \nBEAS & & & DP & Dishonest & & & \nSwarm Learning & && × & Dishonest & & & \nSL+HE & && HE & Dishonest & & & \nBrainTorrent & & × & × & Honest & & & \nP2P-FL & & × & × & Honest & & & \nReputationDFL & & & × & Dishonest & & & \nBlockFLA & & & × & Dishonest & & & \nDFedForest & & & × & Dishonest & & & \nBift & & & × & Dishonest & & & \nVerifiable DFL & & & × & Semi-honest & & & \nDFL for Healthcare & & & DP, SMC & Semi-honest & & & \nFLChain & & & × & Honest & & & \nVFChain & & & × & Semi-honest & & & \n[tab:analysis]\nDefense Mechanisms\n==================\nVarious defense mechanisms are proposed to fortify DFL against privacy and performance-related threats. While this section reviews the literature on DFL defense mechanisms, their characteristics are summarized in Table [tab:defs]. In addition, Table [tab:analysis] specifies mechanisms that each reviewed DFL method employs.\nPrivacy Preserving\n------------------\nDespite the wide diversity of previous efforts on safeguarding FL and blockchain privacy, suggested methods typically fall into one of these three categories: 1) Homomorphic Encryption (HE), 2) Secure Multiparty Computation (SMC), and 3) DP. The following discussion goes through each of these groups.\n### Homomorphic Encryption\nBy processing on cyphertext, HE is commonly used to secure the learning process. Clients can use HE to perform arithmetic operations on encrypted data (i.e., ciphertext) without having to decode it. HE has three major variations that differ in arithmetic complexity and flexibility, namely full, partial, and substantial HE.\nFully HE is capable of doing arbitrary calculations on the encrypted data. This is while partially HE can only execute one operation (e.g., addition or multiplication), and substantially HE can do several operations. The latter, on the other hand, has a restricted amount of additions and multiplications. While full HE offers greater flexibility, it is inefficient when compared to other forms of HE. As an example, shows that swarm learning updates are encrypted using partial HE to defend DFL against inference attacks.\nDespite the benefits of HE, executing arithmetic on the encrypted integers increases the memory and processing time costs. Moreover, non-linear estimations in statistical models demand approximating polynomials which make it important to find a balance between utility and privacy. In, for instance, additively HE is used to secure distributed learning by securing model changes and maintaining gradient privacy. Another example is, which uses an additively homomorphic architecture to defeat honest-but-curious adversaries using federated logistic regression on the encrypted vertical FL data. However, the overburdening of the system with additional computational costs is a typical downside of such systems.\n### Secure Multiparty Computation\nSecure Multiparty Computation (SMC), is a distributed cryptography technique in which several entities participate in the estimation of a function. The distinct feature of SMC is securing each participant’s data by creating a set of random values that are not equal to the participant’s data, and sending them to the rest of the parties for local calculation of the function. The outputs of these functions then will be averaged to obtain the desired estimation. In this scheme, the data on the participant’s side is meaningless, and the function estimations are only usable once they are averaged. For instance, SMC was used in for private model training. It is worth mentioning that SMC is followed by excessive communication and computational cost. It has been also mentioned that SMC is best to be coupled with DP to secure the communications.\nIn conclusion, SMC usage in large-scale DFL could be inefficient due to the dramatic rise in communication and processing costs. Secondly, encryption-based solutions corresponding to these functions must be properly defined and implemented. Finally, all cryptography-based protocols preclude an audition phase of the received updates by the shared model, hence, leaving holes for rogue users to exploit.\n### Differential Privacy\nThe idea of DP is to inject random noise into the generating updates so that the data interpretation becomes infeasible for malicious entities. DP is primarily used to safeguard DFL communications against privacy attacks (e.g., inference attacks); however, the literature also shows that DP can be also beneficial against data poisoning attacks as these attacks are usually designed based on the communicated gradients. Biscotti couples DP with a secure aggregation technique to safeguard DFL systems against poisoning and inference attacks. Another report shows the effectiveness of pruning-based DP in safeguarding horizontal DFL systems. DP in is implemented through perturbing local gradients using the exponential mechanism and a predefined probability density function before uploading them into the blockchain and broadcasting them across the DFL network. Similarly, Blade-FL adds noise to the gradients prior to the encapsulation phase, albeit using a Gaussian distribution.\nIn contrast to HE and SMC whose main disadvantage was communication overhead, DP has a lower computational cost and does not overburden the system in this sense. Instead, DP comes at the cost of deteriorating the model quality. This is mainly because the injected noise can potentially add up to the noise within the constructed model. Another issue of concern in conventional DP is the cumulative privacy loss resulting from iterative training processes that utilize local data from multiple individuals or sources. These iterations are crucial for enhancing the accuracy and performance of trained models. However, with each iteration, a certain degree of privacy loss is introduced, and this loss accumulates over time, potentially reaching a significant level. Researchers have dedicated efforts to address this problem by exploring various approaches, including subsampling and privacy amplification by iteration. These techniques aim to mitigate cumulative privacy loss and improve the overall privacy guarantees of DP schemes. Since these methods derive a tight upper bound of cumulative privacy loss, they can also be applied to preserve model utility even in cases where the gradients are perturbed by noise. Moreover, DP provides resistance to poisoning attempts due to its group privacy trait. As a result, as the number of attackers increases, this defense will reduce significantly. As mentioned before, all privacy-preserving methods have a set of advantages and disadvantages, and thus, there is no perfect alternative for DP. In other words, DP, HE, and SMC each result in a different privacy-utility trade-off. As a solution, hybrid approaches can be implemented to build privacy protocols that are more robust than using only one of these methods. As an example, SMC is combined with DP to balance the trade-off between them. This combination offsets excessive noise injection when the number of clients is growing while preserving the desired rate of trust.\nGaussian DP is a specific case within the functional DP (*f*-DP) framework, which characterizes privacy through hypothesis testing. In *f*-DP, a randomized algorithm is considered to satisfy privacy if the difficulty of distinguishing between two neighboring datasets (quantified by a trade-off function) is element-wise larger than a convex and non-increasing function *f*. When this function f is constructed using two Gaussian distributions, the resulting form of *f*-DP is termed Gaussian Differential Privacy. In other words, Gaussian DP defines privacy guarantees by examining the distinguishability of neighboring datasets through a hypothesis testing approach, with the specific choice of a trade-off function derived from Gaussian distributions. This provides a mathematical and analytical framework for evaluating privacy in the context of private data analysis. For instance, applies Gaussian DP in an FL system to provide record-level privacy guarantees for each client. Building upon the ability of this approach to handle composition and subsampling, extends the work to apply DP to SGD and Adam optimizers.\nIn the context of applying DP to FL, two privacy notations, namely user level and instance level privacy, can be found in the literature. User-level privacy in DP ensures that clients’ data remains private and the privacy of the global model remains intact, even if an adversary removes a client or its data from the aggregation process. Additionally, adversaries are unable to determine whether a client has participated in the training. On the other hand, employing DP at the record level offers privacy guarantees for each individual record within a client’s data, protecting them against potential adversaries. Instance-level DP is often perceived as offering a relatively weaker privacy guarantee, as adversaries may potentially determine whether a user participated in the training or not. However, there are cases where discerning user participation is not crucial, and the primary concern is maintaining data privacy. Considering the trade-off between privacy and utility in differential privacy, it is prudent to adopt instance-level privacy in such scenarios to achieve improved performance while avoiding overly stringent privacy constraints.\nDP can be centralized, local, or distributed. In centralized DP, the noise addition is performed via a server, which makes it impractical in DFL. On the other hand, local and distributed DP both assume that the aggregator is not trusted which perfectly complies with the DFL paradigm. In the local variant, participants inject noise in their estimated gradients before sharing them over the blockchain. However, research on local DP indicates its inability to provide a privacy guarantee on large-scale and heterogeneous models with numerous parameters. In DFL, the injected noise should be calibrated to ensure successful DP. Despite the appealing security qualities of local DP, its practicality becomes questionable when dealing with an immense number of users.\nDistributed DP combines cryptographic techniques to provide the benefits of both local and centralized DP without compromising the clients’ privacy. As a result, it avoids putting faith in any server and is more effective in that sense. Decentralized DP, in theory, has the same benefit as the centralized variant because the overall quantity of noise is similar for them. The concept of distributed DP alludes to the notion that the required amount of noise is derived from several individuals.\nModel Robustness\n----------------\nDefenses are classified into two types: proactive and reactive. Proactive defense is a low-cost method of anticipating attacks and associated consequences. The reactive defense operates by detecting an invasion and taking preventative steps. In the production environment, reactive defense is often deployed as a patch-up. DFL presents multiple additional attack surfaces throughout training, resulting in complicated and unique countermeasures. In this part, we will look at some of the most common types of DFL defensive tactics and investigate their usefulness and limits.\n### Anomaly Detection\nAnomaly detection methods have a long history in real-time identification of data attacks in intelligent systems. In centralized FL, anomaly detection is often performed on the server to identify malicious updates that can lead to model or data poisoning. It is worthwhile to mention that this approach usually works best against untargeted attacks. In the DFL paradigm, however, anomalies can take place either due to fraudulent transactions or changes in blockchain networks such as network division and blockchain fork. Each of these issues is separately studied in the literature and there is no anomaly detection framework designed to handle both. For instance, proposes a fast-paced detection scheme for monitoring transactions using a subgraph-based approach. The designed detection method is optimized for parallel processing GPU acceleration. On the other hand, another work,, designs a distributed anomaly detection scheme, called BAD, to combat eclipse attacks in blockchain networks by re-connecting unauthorized forks to the P2P network.\nThe majority of the anomaly detection methods for FL security are proposed with respect to the centralized architecture of the FL, and anomaly detection in DFL is only studied in a limited number of works. As an example, BEAS employs two anomaly detection protocols, namely Multi-KRUM and FoolsGold to detect poisoning attacks and identifying Sybil groups (i.e., group of malicious nodes coordinating a cyber-attack), respectively. The former monitors change in variance or performance of generated gradients with respect to the majority of updates. The latter, on the other hand, detects Sybil groups based on the correlation between the generated updates, as it is expected for malicious nodes to generate highly similar gradients. Anomaly detection in high-dimensional space poses challenges, leading some detectors to suggest using dimensionality reduction to make monitoring easier. However, reducing the dimensionality size may result in information loss, even with advanced techniques preserving data characteristics.\n### Robust Aggregation\nThe aggregation algorithm used in an FL system should tolerate communications disturbances, client dropout, and incorrect model updates on top of hostile participants. Extensive research has been dedicated to advancing robust aggregation in centralized FL. Similarly, in DFL networks, the security of the aggregation phase is of paramount importance. Under a DFL paradigm, the aggregation phase is mainly secured by employing the same techniques used in FL in combination with the consensus mechanism of the utilized blockchain scheme. These protocols (e.g., PoW, PoS, SC) control which nodes can participate in the model aggregation phase for both generating updates and aggregating parameters. On the downside, many of the advanced Byzantine-robust aggregations rely on assumptions that are either unrealistic or incompatible within the context of FL.\n### Pruning\nEven though pruning is not specific to DFL and can be applied to any neural network structure, it can help in eliminating backdoors. The idea of pruning is to drop some neurons in order to enhance the efficiency and accuracy of the network. However, this feature makes the parameter usage somewhat unpredictable for attackers which complicates injecting backdoors into a neural network since inactive neurons will be removed eventually in the network. While pruning has been mainly studied for FL, current literature on pruning under a DFL paradigm is very limited. In particular, BEAS makes use of gradient pruning in DFL to facilitate DP and complicating model poisoning for attackers in the system. Nevertheless, since these techniques are mainly used at the edge of the network, most pruning approaches should be adaptable to DFL as well.\nPruning comes with a set of drawbacks. One drawback is the potential loss of model capacity, resulting from the removal of connections or components. Another concern is the sensitivity of pruning to initialization and training, necessitating careful fine-tuning and experimentation for optimal outcomes. Moreover, the pruned model may struggle to generalize effectively to new data, limiting its real-world applicability. Lastly, the computational overhead increases due to the additional resources and time required during the pruning and retraining stages. Considering these factors is vital when assessing the viability of pruning as a defense mechanism against backdoors.\n### Trusted Execution Environment\nThe blockchain backbone of DFL requires the data to be replicated on each node which makes it challenging to keep smart contracts confidential. TEE (Trusted Execution Environment), on the other hand, is a tamper-resistant ecosystem that can be used to maintain digital trust between nodes of a distributed network. TEE is often referred to as a secure and isolated part of the processor that requires all codes and data signatures to be verified with respect to the designer’s expectations. The validity of a participating device in a TEE Authentication should be checked by the connected service with which it is attempting to enroll. Until the matching party provides a message, the status of code execution stays hidden. The execution route of the code cannot be changed until it takes explicit input or a validated interruption. Data stored on and processed by participants is safe, and interactions between various parties are carried out in a secure manner. The TEE is in charge of all data access privileges. Cryptographic technologies are used to secure TEE communications. Only the TEE secure environment stores, maintains, and uses private and public encryption keys. The TEE can show a remote client what code is presently being executed as well as the starting state. TEE can aid in resolving a key challenge for FL security since it is becoming progressively important in securing the central server and clients against hackers and preventing data theft.\nTEE is coupled with FL to safeguard against algorithmic attacks. TEE can hide model parameters on local devices so that the model is not accessible to the attacker. Under this condition, the attacker is only able to launch black box attacks on the system. Nevertheless, TEE often suffers from a limited memory size, that is only a limited part of the model can be secured with this approach. It has been also suggested that TEE can secure smart contract data in a blockchain which is the main vulnerability of this architecture. Theoretically, the same concept can be applied to DFL. However, current literature lacks an experimental study on the integration of DFL and TEE.\n### Zero-Knowledge Proofs\nThe origin of zero-knowledge proofs goes back to the mid-1980s. This cryptographic approach allows a verification process that does not involve data exchange between parties. This process often involves using unconnected bits of information to keep data private during the verification.\nAs an example, zero-knowledge proofs have the potential to be utilized in DFL to verify the authenticity of the features used by clients for training and generating updates. While zero-knowledge proofs offer a promising avenue for enhancing secure update monitoring, additional research is necessary to identify challenges in constructing and implementing their modules. Notably, zero-knowledge proof protocols generally maintain their performance regardless of the data volume.\nZero-knowledge proofs offer advantages for enhancing secure update monitoring in DFL. However, their adoption also entails certain drawbacks. These include computational overhead due to resource-intensive operations, the complexity of implementation requiring meticulous attention to cryptographic techniques, potential scalability issues in large-scale DFL systems due to communication complexity, and the trust assumption associated with the setup phase.\n### Knowledge Distillation\nKnowledge distillation is a fundamental algorithm in federated distillation. The goal of knowledge distillation is to perform TL from a large teacher model (T) to a compact student model (S) without sacrificing performance significantly. Smaller models have fewer parameters and are less susceptible to overfitting, making them more resilient against attacks. This increased resistance makes it more challenging for attackers to reverse-engineer or manipulate the model. Nonetheless, for certain threats such as backdoor attacks, knowledge distillation alone may not be directly effective, as these attacks typically involve modifying the training data or model parameters to embed a hidden trigger. However, knowledge distillation can be used as part of a broader defense strategy to mitigate the impact of backdoor attacks. For instance, reference mitigates backdoor attacks by utilizing knowledge distillation in deep neural networks. This approach independently fine-tunes T on a clean subset and uses it to clean backdoored S. To do so, this method tries to align intermediate-layer attention in S with that of T. Another example trains a T ensemble on disjoint training subsets and trains S based on aggregated noisy voting among T models. The goal of this approach is to provide a privacy guarantee for training data. In DFL, this idea translates into sharing the knowledge of a model rather than the parameters which improves FL’s robustness against both poisoning and privacy attacks. In addition to the mentioned security advantages, knowledge distillation also results in communication and computation efficiency in DFL. Exchanging model parameters becomes burdensome when communication resources are limited, especially for contemporary big deep neural networks. In this sense, federated distillation is an appealing FL option since it only transmits model outputs which are often considerably less in size than the model sizes.\n### Regularization\nIn DFL, the computational model that is being distributively updated is most likely a DL structure. One approach to mislead this model is to make it overfit by introducing malicious samples which may in turn lead to membership inference. Regularization techniques such as *L*2 regularizer and dropout can eliminate the effect of such malicious samples to a great extent. Reference designs a semidefinite relaxation method that generates a differentiable certificate for network robustness and optimizes it alongside network parameters to encourage robustness against all attacks. Furthermore, determining suitable hyperparameters, such as regularization strength, for regularization techniques can be a difficult task, and incorrect choices may yield undesirable results. Moreover, the use of certain regularization methods with intricate terms or penalties can introduce considerable computational complexity, especially in scenarios involving extensive datasets or intricate models, resulting in prolonged training durations.\nVerifiable DFL\n==============\nDefense mechanisms reviewed in the previous section mainly combat an adversary after a malicious attack is launched by an intruder. Nevertheless, another approach to combat these threats is to prevent malicious parties to take part in the training process. A DFL system that can verify the trustworthiness of its clients is called verifiable.\nA DFL system is considered verifiable if clients of a network can prove to each other that the given task has been carried out without compromising privacy. Security, transparency, and automation of blockchain can enhance the verifiability of DFL which in turn prevent malicious activities. In this structure, some nodes are in charge of generating updates and training the model (i.e., trainers) and groups of participants verify the generated updates (i.e., workers) with respect to consensus protocol. If the update is verified, the worker who completes this task first will create a new block. Then, if the majority of the workers approve the content of the created block, it will be appended to the chain. At this point, DFL clients can update their models using this new block of information. This process is repeated periodically to keep the DFL model up to date.\nTrustable DFL Trainers\n----------------------\nA vulnerability of DFL systems is that dishonest trainers can upload maliciously crafted updates to the blockchain. For instance, BlockFL verifies the submitted updates through a random selection of workers before letting the update take effect. In BlockFL, each client is associated with a random worker who is rewarded when evaluating the updates prior to the aggregation. The evaluation criteria are designed based on the relationship between the data size and the elapsed time for generating the update.\nAnother approach for verifying trainers is to use proof of correctness. Examples of this are presented in, where produced parameters are paired with their proof of correctness which is used by registered workers for determining the approved updates. This approach uses verifiable random functions or reliability ranking to select a worker whose aggregation results will be appended to the blockchain. The trainer’s reliability, which is an indicator of its performance, will affect its chance to take part in the training process thereafter.\nTrustable DFL Workers\n---------------------\nAssuming that all trainers are trustworthy, the aggregated model could still be compromised if the responsible worker for the model aggregation is unreliable, that is, it does not follow the defined protocol. VFChain addresses this issue by using trainer signatures that are appended to the uploaded block by the trainer. To verify the aggregated model by the responsible worker, a committee is randomly selected among other workers to evaluate the aggregated model with respect to the verifying contract. The members of this committee are continuously updated in each training round. The blockchain preserves signatures and their corresponding aggregated models of each training round which makes all records tamper-resistant and transparent to all network members. It is worth mentioning that this structure is only compatible with semi-honest trainers.\nFuture Research Directions\n==========================\nDespite the advancements of DFL in the past five years, this paradigm is fairly new and still has a lot of potential to be improved. From a security standpoint, many of the available countermeasures in the literature are either studied for FL or blockchain, and their effectiveness in DFL remains unclear. Here, we outline future development requirements that we believe will be promising for DFL in this sense.\nWeakness of Privacy-Preserving Techniques\n-----------------------------------------\nAs mentioned before, privacy-preserving techniques often trade excessive computational burden for enhancing privacy (e.g., HE, SMC). Furthermore, the implementation of cryptographic techniques such as SMC is sensitive and can imperil privacy if not defined properly. DP, on the other hand, can affect the accuracy of the aggregated model since the utilized noise can leak into the model. These issues are beside the fact that the audition phase in these techniques can lead to security holes. Hence, despite the history of privacy-preserving techniques, they all have imperfections that require further research and adaptation to the DFL paradigm.\nAddressing these challenges requires ongoing research and adaptation of privacy-preserving techniques to suit the specific requirements of DFL. Developing optimized implementations of cryptographic methods, such as SMC, can help alleviate the computational overhead and address sensitivity concerns. Paying attention to secure parameter configurations and robust cryptographic protocols is essential to minimize privacy risks. Exploring advanced noise reduction techniques within DP can strike a balance between preserving privacy and maintaining the accuracy of the model. Robust auditing mechanisms need to be established to ensure the security of privacy-preserving techniques and safeguard against potential vulnerabilities. Moreover, investigating hybrid approaches that integrate multiple privacy-preserving techniques can offer a compromise between privacy and computational efficiency.\nContinued exploration and refinement of privacy-preserving techniques in the DFL domain will enhance privacy and security while minimizing the impact on computational performance. This ongoing research and innovation will pave the way for more effective and privacy-aware DFL systems.\nOptimizing Defense Mechanism Deployment\n---------------------------------------\nDevising defense mechanisms in DFL presents specific challenges that need to be addressed. One key challenge is the requirement for additional computational power, which may not always be feasible, particularly in resource-constrained environments like mobile networks. The limited computational resources available can pose limitations on the implementation of defense mechanisms.\nAnother challenge is the diverse nature of threats in DFL. Different threats often require different countermeasures. Designing a comprehensive defense mechanism that addresses a wide range of threats can be complex and challenging. It requires analyzing various threat models, understanding their unique characteristics, and developing tailored defense strategies accordingly.\nWhen considering DFL deployed on blockchain, resource and computational constraints specific to blockchain systems must also be taken into account. The scalability of defense mechanisms becomes a critical aspect to consider. Strategies that optimize smart contract execution or leverage off-chain computations can help ensure efficient operation within the resource constraints of blockchain platforms.\nContinuous evaluation, benchmarking, and improvement of existing defense mechanisms are vital. The field of defense mechanisms in DFL is dynamic, with emerging threats and advancements in technology. Regular evaluation and improvement are necessary to keep up with evolving challenges and maintain effective protection.\nOverall, addressing the challenges of additional computational power, diverse threats, resource constraints in blockchain, and the optimal deployment of defense mechanisms are critical areas for further research and development in the field of DFL. By tackling these challenges and exploring the proposed solutions, researchers can advance the field and enhance the security and resilience of DFL systems.\nBlockchain-Related Security Issues\n----------------------------------\nThe decentralization and traceability offered by DFL introduce unique challenges related to blockchain-related threats. These threats specifically exploit the structure of the blockchain, posing risks to the overall security of DFL systems. Consequently, there is a need for research and development aiming at creating more secure blockchain structures that can enhance the resilience of DFL ecosystems against these attacks.\nOne key challenge is the susceptibility of DFL to cyber-attacks that target the underlying blockchain infrastructure. Adversaries may attempt to manipulate or tamper with the blockchain, compromising the integrity and reliability of the system. This poses a significant risk to the confidentiality and privacy of participants’ data in DFL. To address these challenges, research efforts should focus on the development of more secure blockchain structures tailored to the specific requirements of DFL. This involves exploring techniques such as improved consensus mechanisms, advanced cryptographic protocols, and enhanced smart contract design. These solutions aim to bolster the security of the blockchain layer, mitigating the vulnerabilities that can be exploited by malicious actors. Additionally, advancements in blockchain technology, such as the integration of privacy-preserving techniques and zero-knowledge proofs, can contribute to building a more attack-resilient DFL ecosystem. By leveraging techniques that enhance data privacy and confidentiality, DFL systems can better withstand attacks and protect sensitive information. Furthermore, the development of robust monitoring and auditing mechanisms is crucial for detecting and mitigating blockchain-related threats in DFL. Implementing effective monitoring systems can help identify suspicious activities and ensure the integrity of the blockchain. Auditing mechanisms can verify the correctness and security of the system, providing transparency and accountability in DFL deployments.\nHeterogeneity of Decentralized Federated Learning\n-------------------------------------------------\nDFL convergence in a distributed ecosystem (e.g., blockchain) is not guaranteed when various clients have diverse computation power (i.e., heterogeneous users). Limited processing power and poor connectivity can delay the training and communication of some nodes, delaying the global aggregation. Disregarding this heterogeneity results deteriorates the overall efficiency of DFL training.\nThe heterogeneity of clients in DFL not only poses challenges to convergence and efficiency but also gives rise to security issues. In an asynchronous DFL setting, where clients operate with different computation power and connectivity, ensuring secure aggregation becomes challenging. Asynchronous DFL is not compatible with secure aggregation protocols because all clients need to be incorporated in the aggregation step when using secure aggregation. This also limits the ability to perform centralized or distributed DP. Consequently, the use of local DP techniques becomes the viable option in asynchronous DFL scenarios, potentially compromising the level of privacy protection and increasing the risk of information leakage during the aggregation process. These security concerns highlight the need for innovative approaches that strike a balance between achieving privacy-preserving aggregation and accommodating the heterogeneity of clients in DFL. By exploring novel cryptographic techniques, adaptive privacy mechanisms, and secure communication protocols, it is possible to address the security challenges that arise from the heterogeneity of DFL and ensure the confidentiality, integrity, and privacy of the FL process.\nChallenges in Verifiable Decentralized Federated Learning\n---------------------------------------------------------\nJoining a DFL network requires a set of verifiers to evaluate the trustworthiness of the joining party. Since the joining party will not disclose their information for privacy reasons, the evaluation process often involves monitoring the performance of the joining party in the training process. Nevertheless, this evaluation scheme is not efficient as it exhausts computational and communication resources. Another challenge in this domain is to gain insight into the misbehavior of clients based on the history of their behavior. It is important to discover how the verification results can be employed to facilitate this task.\nOne possible solution is to employ privacy-preserving techniques during the trust evaluation process. Techniques such as SMC or HE can enable verifiers to evaluate the performance and trustworthiness of the joining party without directly accessing their sensitive information. This way, privacy is maintained while still obtaining the necessary insights for trust evaluation. Furthermore, optimizing the evaluation process can enhance its efficiency. Implementing techniques that reduce the computational and communication overhead, such as lightweight monitoring mechanisms or selective data aggregation, can help alleviate the resource exhaustion issues associated with the evaluation scheme. By carefully designing evaluation protocols and employing efficient algorithms, the overall efficiency of the process can be improved without compromising the evaluation’s effectiveness.\nIn terms of addressing misbehavior based on historical behavior, leveraging machine learning techniques and anomaly detection algorithms can be beneficial. By analyzing the behavior patterns and performance metrics of clients over time, it is possible to identify potential anomalies or deviations that may indicate misbehavior. Establishing a comprehensive system for tracking and monitoring client behavior, coupled with intelligent analysis and detection mechanisms, can aid in detecting and addressing misbehavior in a timely manner. Moreover, incorporating reputation systems within the DFL network can provide additional insights into the trustworthiness of participating clients. Verifiers can leverage the verification results and feedback from previous collaborations to build reputation scores for each client. These scores can serve as indicators of past behavior and can be used as a basis for decision-making during the trust evaluation process.\nConclusion\n==========\nThe integration of FL and blockchain technologies has alleviated the need for a server in the network. This is followed by a number of advantages such as efficient communication and the elimination of a single point of failure in the federation. While there are a limited number of surveys on the security analysis of FL, since the previous studies were all based on centralized architectures, further analysis is required to study the new paradigm of DFL from a security perspective. This work first reviewed common trends and preliminaries of FL and blockchain. It then performed a security analysis on DFL by identifying possible threats and defense mechanisms in such systems.\n[]Ehsan Hallaji (Graduate Student Member, IEEE) received the B.Sc. degree in software engineering from Shahid Rajaee University, Tehran, Iran, in 2015, and the M.A.Sc. degree in electrical and computer engineering from the University of Windsor, Windsor, ON, Canada, in 2018, where he is currently pursuing a Ph.D. degree with the Department of Electrical and Computer Engineering. His current research interests include machine learning, data mining, federated learning, and cybersecurity. He is a reviewer for several journals and conferences in his area of research. He also served as the Vice-Chair of the IEEE SMC Society, Windsor Section, from 2019 to 2022. \n[]Roozbeh Razavi-Far (Senior Member, IEEE) is an Assistant Professor at the Faculty of Computer Science and Canadian Institute for Cybersecurity, at the University of New Brunswick. His research focuses on machine learning, big data analytics, computational intelligence, and cybersecurity of cyber-physical systems. He has authored or co-authored more than 150 papers in scholarly journals and international conferences. Stanford lists his name among the top two percent most cited researchers for 2022. He is the recipient of several awards and grants including NSERC-DG, NSERC-ECR, NBIF, USRG and NSERC-PDF. He is an Associate Editor at several journals, including *Neurocomputing*, *Machine Learning with Applications*, *Discover Artificial Intelligence*, and IEEE Transactions on Industrial Cyber-Physical Systems. He served as a Guest Editor and Chair for several journals and peer-reviewed conferences, and the Chapter Chair of IEEE Computational Intelligence, and Systems, Man and Cybernetics Societies at Windsor Section. \n[]Mehrdad Saif(Fellow, IEEE) is a distinguished figure in the field of systems and control, with a career spanning nearly four decades. He received the B.S., M.S., and D.Eng. degrees in electrical engineering from Cleveland State University, OH, USA, in 1982, 1984, and 1987, respectively. Throughout his graduate studies, he was involved in research sponsored by NASA Lewis (now Glenn) Research Center and the Cleveland Advanced Manufacturing Program (CAMP). In 1987, Dr. Saif joined Simon Fraser University’s School of Engineering Science as an Assistant Professor. He rose through the ranks, becoming a Full Professor in 1997. From 2002 to 2011, he took on the role of Director of the School, spearheading its significant expansion during his tenure. Subsequently, he served as the Dean of the Faculty of Engineering at the University of Windsor from July 2011 until September 2021. There, he initiated major developments, including substantial enrollment growth and the addition of new programs in aerospace engineering, engineering management, B.Eng. technology, mechatronics, among others. Under his leadership, the Faculty of Engineering at University of Windsor also saw an increase in both faculty/staff numbers and research output. Dr. Saif’s research contributions are both vast and impactful, specializing in systems and control, estimation, observer theory, and AI/ML-based approaches to fault diagnostics and condition monitoring. His work has applications in a range of sectors, including automotive, power, and autonomous systems. To date, he has published over 400 refereed journal and conference papers and edited a book in these subject areas. He has garnered over 10,000 citations and holds a Google h-index of 50. Moreover, Dr. Saif is listed in Stanford University’s database of the top 100,000 career scientists from 1965 to the present. In his own area of expertise, the same database ranks him in the top 0.7%. Research.com also places him at 2681 worldwide and 139Th nationally among all electrical and electronics engineers. Dr. Saif expertise has been sought after by notable organizations such as GM, NASA, B.C. Hydro, and Canadian Space Agency (CSA). He has also significantly contributed to the IEEE Control Systems Society as the Chair of its Vancouver Section and serves on the Editorial Board of several esteemed IEEE journals such as the IEEE Access, IEEE SMC Magazine, IEEE Industrial Informatics, IEEE Transactions on Industrial and Cyber-physical Systems, among others. Dr. Saif is a Registered Professional Engineer in Ontario, Canada, and has been honored with Fellowships in several prestigious organizations, including IEEE, the Canadian Academy of Engineering (CAE), the Engineering Institute of Canada (EIC), the Institution of Engineering and Technology (IET), and Asia-Pacific Artificial Intelligence Association (AAIA). His IEEE fellowship is particularly noteworthy, earned *“for contributions to monitoring, diagnosis and prognosis in cyber-physical health systems”*.\n[]Boyu Wang (Member, IEEE) received his B.Eng. degree in Electronic Information Engineering from Tianjin University, Tianjin, China, M.Sc. degree in Electrical and Computer Engineering from University of Macau, Macau, China, and Ph.D. in Computer Science from McGill University, Montreal, QC, Canada. He is currently an Assistant Professor with the Department of Computer Science, University of Western Ontario, London, ON, Canada. He is also affiliated with the Brain and Mind Institute and the Vector Institute. He was a Post-Doctoral Research Fellow at the University of Pennsylvania and Princeton University. His research interests include machine learning theory, algorithms, and applications. \n[]Qiang Yang (Fellow, IEEE) received the B.Sc. degree in astrophysics from Peking University, Beijing, China, in 1982, and the M.Sc. degree in astrophysics and the Ph.D. degree in computer science from the University of Maryland, College Park, MD, USA, in 1985 and 1989, respectively. He was a Faculty Member with the University of Waterloo, Waterloo, ON, Canada, from 1989 to 1995, and Simon Fraser University, Burnaby, BC, Canada, from 1995 to 2001. He was the Founding Director of Huawei’s Noah’s Ark Lab, Hong Kong, from 2012 to 2014 and a Co-Founder of 4Paradigm Corporation, Beijing, an artificial intelligence (AI) platform company. He is currently the Head of the AI Department and the Chief AI Officer of WeBank, Shenzhen, China. Dr. Yang has been a Professor Emeritus with the Computer Science and Engineering Department, Hong Kong University of Science and Technology (HKUST), Hong Kong, since 2023, where he was previously a Chair Professor, and a former Head of the CSE Department and the Founding Director of the Big Data Institute from 2015 to 2018. He is the author of several books, including *Intelligent Planning* (Springer), *Crafting Your Research Future* (Morgan & Claypool), and *Constraint-Based Design Recovery for Software Engineering* (Springer). His research interests include AI, machine learning, and data mining, especially in transfer learning, automated planning, federated learning, and case-based reasoning. Dr. Yang has served as an Executive Council Member of the Advancement of AI (AAAI) from 2016 to 2020. He is a fellow of several international societies, including ACM, AAAI, IEEE, IAPR, and AAAS. He was a recipient of several awards, including the 2004/2005 ACM KDDCUP Championship, the ACM SIGKDD Distinguished Service Award in 2017, and the AAAI Innovative AI Applications Awards in 2018 and 2020. He was the Founding Editorin-Chief of the *ACM Transactions on Intelligent Systems and Technology* and IEEE Transactions on Big Data. He has served as the President of the International Joint Conference on AI (IJCAI) from 2017 to 2019. \n---\n1. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).[↩](#fnref1)\n2. Ehsan Hallaji and Mehrdad Saif are with the Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada. E-mail: hallaji@uwindsor.ca, msaif@uwindsor.ca.[↩](#fnref2)\n3. Roozbeh Razavi-Far is with the Faculty of Computer Science, University of New Brunswick, Fredericton, NB E3B 5A3, Canada, and also with the Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada. E-mail: roozbeh.razavi-far@unb.ca.[↩](#fnref3)\n4. Boyu Wang is with the Department of Computer Science and the Brain Mind Institute, University of Western Ontario, London, ON N6A 5B7, Canada. E-mail: bwang@csd.uwo.ca.[↩](#fnref4)\n5. Qiang Yang is with the Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, and also with the Department of AI, WeBank, Gaungdong Province, China. E-mail: qyang@cse.ust.hk.[↩](#fnref5)\n\nDecentralized Federated Learning: A Survey on Security and Privacy\n==================================================================\nFederated learning has been rapidly evolving and gaining popularity in recent years due to its privacy-preserving features, among other advantages. Nevertheless, the exchange of model updates and gradients in this architecture provides new attack surfaces for malicious users of the network which may jeopardize the model performance and user and data privacy. For this reason, one of the main motivations for decentralized federated learning is to eliminate server-related threats by removing the server from the network and compensating for it through technologies such as blockchain. However, this advantage comes at the cost of challenging the system with new privacy threats. Thus, performing a thorough security analysis in this new paradigm is necessary. This survey studies possible variations of threats and adversaries in decentralized federated learning and overviews the potential defense mechanisms. Trustability and verifiability of decentralized federated learning are also considered in this study.\nFederated learning, privacy-preserving, security, blockchain, adversarial attacks, decentralized federated learning, verifiable federated learning.\nIntroduction\n============\nthe widespread use of machine learning over the recent years, new concerns have been raised regarding user and data privacy. The data-driven nature of these intelligent models necessitates gathering users’ data to constantly improve and maintain the operating statistical model. This issue becomes more problematic in large-scale distributed systems with millions of users such as mobile networks.\nAside from privacy issues, in large-scale systems, communicating user data may pose an overhead to the network. This is while information technology and intelligent devices are evolving at a rapid pace, and in the wake of it, there is an explosion of data at the edge of the network. Given the potential benefits of this data collection process for improving their model, organizations tend to make the best use of it for knowledge extraction with minimal waste of data. Thus, modern intelligent systems struggle to find the optimal trade-off between user privacy and service quality.\nInspired by recent breakthroughs in distributed optimization, Federated Learning (FL) has proposed as a potential solution to resolve the aforementioned challenges. In contrast to distributed machine learning which is shown in Fig. [fig:workflow](a), FL proposes training local models at the edges of the network and then sharing the model parameters to a central server that aggregates the received information and then updates all the client models.\nAside from its use in distributed training of Deep Learning (DL) models, FL offers additional advantages. Firstly, by communicating the local model parameters rather than the user’s data, the central model is trained on decentralized data, which no longer jeopardizes data privacy. Secondly, the communication is extensively reduced for large-scale systems, as the volume of model parameters is often smaller than the training data itself. The general scheme of FL workflow is illustrated in Fig. [fig:workflow](b). As shown in this figure, for client *i*, a model *M**i* is constructed locally based on the local user data *D**i*. The model parameters *M**i* are then sent to a server that aggregates all models in the form of ∑(*M*1, *M*2, *M*3) = *u*, where ∑( ⋅ ) and *u* are the aggregation function and the generated update, respectively. Finally, *u* is sent back to the clients to update local models.\n[t] [fig:workflow]\nDespite the breakthrough made by FL, the proposed architecture was not flawless and demanded further research endeavors on the topic. To begin with, although the user data is not being shared in the network, communicating local parameters is still vulnerable to sniffing attacks which will lead to stealing model parameters for launching an inference attack to extract sensitive information from the local training data at the edges of the networks. In addition, the trustability of the central server is sometimes questionable in environments such as wireless networks. Moreover, having a Single Point of Failure (SPF) in the system is not ideal since a compromised server can affect the whole network. While various works have been dedicated to designing defense mechanisms for FL, it was proposed that decentralizing the FL architecture will eliminate part of these security challenges.\nRelying on Peer-to-Peer (P2P) communications, Decentralized Federated Learning (DFL) improves both the reliability and scalability of FL by eliminating SPF and enhancing the communication efficiency. The process of local update preparation is similar to that of FL; however, exchanging model parameters and model aggregation is mostly undertaken through P2P communication or blockchain technology. For instance, a generic diagram of a DFL workflow is shown in Fig. [fig:workflow](c), where a client is assigned with the aggregator role in each round (i.e., shown with dotted lines) to apply the aggregating function ∑(*M*1, *M*2, *M*3) = *u* on the received parameters and update all nodes with *u*. The protocol for selecting the aggregating node and the ∑( ⋅ ) function varies among different DFL architectures. Integration with blockchain brings about additional advantages including traceability and immutability. Despite the security and efficiency features of DFL, this architecture is not flawless. For instance, incorporating blockchain into FL may come at the cost of making the system defenseless against blockchain-related security adversaries. The trustability of the DFL participants is another issue of concern that requires further research.\nll Abbreviation & Definition \nBAFFLE & Blockchain-based aggregator free FL \nBEAS & Blockchain enabled asynchronous and secure FL \nBFLC & Blockchain-based FL with committee consensus \nBindaaS & Blockchain-based deep learning as-a-service \nDDoS & Distributed denial of service \nDFL & Decentralized federated learning \nDL & Deep learning \nDoS & Denial of service \nDP & Differential privacy \n*f*-DP & Functional differential privacy \nFL & Federated learning \nFTL & Federated transfer learning \nGAN & Generative adversarial network \nHE & Homomorphic encryption \nHFL & Horizontal federated learning \nIoT & Internet of things \nP2P & Peer-to-peer \nPoS & Proof of state \nPoW & Proof of work \nTEE & Trusted Executive Environment \nTL & Transfer learning \nVFL & Vertical federated learning \nSC & Smart contract \nSGD & Stochastic gradient decent \nSL & Swarm learning \nSMC & Secure multiparty computation \nSPF & Single point of failure \n[tab:abrv]\nCurrently, the literature lacks a survey that primarily studies the security and privacy of DFL. Available reviews on DFL generally study trending approaches in DFL with an emphasis on the application in the Internet of Things (IoT). On the other hand, surveys on security and privacy of centralized FL do not discuss the applicability of their analysis and findings to DFL. Among the limited number of surveys on DFL, studies security issues in centralized FL that can be addressed through blockchain integration. However, the literature still lacks a thorough security analysis of threats in DFL and its potential defense mechanisms. Hence, it is worthwhile to perform a comprehensive survey on the new attack surface created on DFL, and defense mechanisms that can be directly used or adapted for DFL. Toward this goal, this survey makes the following contributions:\n* State-of-the-art DFL methods are reviewed in terms of security robustness and employed technologies.\n* Potential threats to DFL systems are identified and explained.\n* Defense mechanisms that can safeguard DFL systems against attacks are studied and analyzed.\n* The effect of blockchain integration on the security and privacy of DFL is studied.\n* The connection between verifiable FL and DFL is studied from a security standpoint.\n* Undiscovered domains and demanding research directions in enhancing DFL security and privacy are identified and introduced.\nReviewed works in this survey are collected from Scopus and Google Scholar searches. The keywords used for the search are FL, DFL, security, privacy, attack, defense, and blockchain. After the initial search, irrelevant papers to the topic of this survey were excluded. Furthermore, the search was limited to works after 2015, and we consider published or in-press research works, as well as arXiv preprints.\nTowards this, we outline the background for this survey in Section 2. Preliminaries of DFL are reviewed in Section 3. Section 4 studies possible threats to DFL. Section 5 discusses the potential defense mechanisms in DFL. Section 6 elaborates on the verifiability of DFL. Future research directions are discussed in Section 7. Finally, the survey is concluded in Section 8. This survey contains several abbreviations to make discussions more concise, which are listed in Table [tab:abrv].\nBackground\n==========\nDFL is a paradigm that often encompasses both FL and blockchain technologies, although it is important to note that DFL can exist independently of blockchain. To better comprehend the concept of DFL, it is crucial to have a clear understanding of the fundamentals of FL and blockchain. Thus, the following section provides a concise overview of these concepts.\nFederated Learning\n------------------\nFL was proposed to reduce the risks of data ownership in collaborative training of DL models. Prior to the introduction of FL, collaborative training required immense loads of data exchange between participants of a distributed machine learning framework in which the clients constantly send local training data to a central server. The distributed model is then trained on the accumulated training data gathered from across the network. Nevertheless, communicating user data and overburdening the network with excessive communication load raised new concerns that motivated the invention of FL. In simple words, rather than sharing the training of the local data with the server, the client in FL locally trains a model of the same structure on the user data and sends the obtained model parameters to the server once convergence is achieved. The server aggregates these parameters (e.g., neural network weights) and updates all the client models with the aggregated parameters. As this cycle continues, in each round, clients initialize their local model parameters with the received update from the server before starting the local training and issuing an update to the server. It is worth mentioning that the collaboration schemes between clients at the edge of the network have been a topic of interest in the domain of multi-agent systems even before the birth of FL. For instance, client-server collaboration in FL somehow resembles the umbrella system in which all agents communicate with a server.\nThe aforementioned process in FL, however, assumes the homogeneity of data among all clients, which may not always be the case. This leads to the evolution of FL into a more advanced structure for dealing with heterogeneous data and comes with additional security perks. Specifically, categorizes techniques in this field into three groups: 1) Horizontal FL (HFL), 2) Vertical FL (VFL), and 3) Federated Transfer Learning (FTL). The difference between these groups is in the nature of the training data in the FL systems. HFL which is basically defined based on the initial FL model in, assumes a diverse range of samples across all local training sets that all fall under a similar feature space. VFL on the other hand, considers a common sample space among participants who differ in the feature space. FTL addresses the minimal overlap between both sample and feature space by resorting to transfer learning for transferring knowledge between participants.\nIt is worth mentioning that this categorization has primarily focused on centralized FL. Therefore, the focus of this survey is primarily on DFL which differs from the centralized FL. The central server in the FL systems is in charge of verifying the network clients, aggregating the local parameters, and broadcasting an update. With the absence of servers in the DFL structure, these tasks must be performed using the client networks. The current literature usually devises the concept of blockchain to carry out the mentioned tasks.\nEven though the initial FL model was proposed on the basis of DL, decision tree models can also be employed and adapted to FL frameworks. Threats and defense mechanisms that will be later reviewed in this work generally apply to both DL and tree-based models.\nBlockchain\n----------\nBlockchain is designed primarily with the objective of making the data exchange immutable and traceable over a P2P network. Similar to FL, in a blockchain network data is decentralized, and each user will transfer data using a block. This consists of the data, a hash acting as a unique identifier for the block, and a previous hash pointing to the previous block in the chain. While these hashes make the data traceable in the chain, they also complicate tampering with data, as changing the data in an existing block will also alter the associated hash which creates a discrepancy in the chain.\n[t] [fig:blockchain]\nA key idea behind the blockchain is to prevent a certain participant from controlling the network and provide all network members with an equal chance to verify and control the transactions. Nonetheless, finding the optimal solution to ensure fairness in assigning the evaluators is still an open problem and as a result, many consensus mechanisms have been proposed over time. Perhaps, the most common consensus mechanisms are Proof of Work (PoW) and Proof of State (PoS), albeit they are not necessarily the best. In PoW, the participants (also called miners) need to solve a time-consuming problem and anyone who achieves the answer quicker will get a chance to contribute to the chain. As an alternative to PoW, PoS randomly selects the evaluators with the aim of improving the scalability.\nBased on the current literature, Blockchain approaches can fall into four major categories, namely 1) public, 2) private, 3) consortium, and 4) hybrid blockchain (i.e., we refer to this as DFL). These variations are briefly explained in the following.\n1. **Public:** Blockchain is built in a permissionless network that allows anyone to join the network and participate in the consensus process.\n2. **Private:** Participants can join the network through a received authorization. While permissions are granted in a centralized manner, the consensus mechanisms are decentralized.\n3. **Consortium:** Blockchain is semi-decentralized and the network access is managed by more than one entity.\n4. **Hybrid:** Block construction is carried out within a private network and the storage is done through a public network.\nDecentralized Federated Learning\n================================\n[t]![Advances in decentralized federated learning in time. The order of the methods is set based on the first version that became available online (e.g., pre-prints).](FIG3.png \"fig:\") [fig:diagram]\n[sec:DFL] As mentioned, the main flaw with the centralized FL is the dependency of the entire federation on a central server whose privacy and performance directly affect all the clients in the system. In other words, a compromised server will jeopardize the whole federated learning system. To eliminate the dependency of the whole network on a single node and also enhance the communication efficiency, DFL resorts to P2P communications to bypass the need for a central server. To do so, model aggregation and participant verification should be carried out in a serverless fashion.\nAs explained in Section [sec:back], DFL often uses blockchain to facilitate inter-node communications. While this combination is non-absolute, blockchain can effectively facilitate communication by treating model updates as data within a block and sharing the block with respect to a consensus mechanism. In this process, local model construction is similar to that of FL, and prepared updates will be in the form of a data block. The prepared block will be then shared with the rest of the nodes for consensus and addition to the blockchain. The aggregation mechanisms of DFL work on the basis of decentralized Stochastic Gradient Decent (SGD), which makes use of gossip averaging.\nDFL models that use blockchain have distinct architectures. For instance, the block structure, information included in the headers, aggregation, and consensus mechanisms can all be different in each architecture. However, a generic scheme is shown in Fig. [fig:blockchain] that resembles the most common practices in designing a blockchain-based DFL. In this example, the blockchain is initialized using a single block containing the starting global model. Each update is added as a new block that is linked to the previous block in the chain using a header. On the client’s side, the local model is initialized using the global model *u**t* and trained on the *i*-th client data *D**i*. Once the trained model *M**i* is obtained, a score *S**i* is estimated w.r.t. the difference between the global model and the estimated local model. Nevertheless, not all blockchain-based DFL models necessarily use a scoring system. The client then uploads data to the blockchain by creating a block containing a header, trained model, estimated score, and the uploader identifier (ID). In some architectures, clients are directly communicating with a set of miners to obtain the Merkle root of the data before uploading the block. It is also worthwhile to mention that the block size is often limited to a fixed size. If the model size is larger than the block size, the uploader has to upload a set of serialized blocks to the blockchain. The SC on this blockchain uses a set of rules to determine when the global model should be updated. For instance, a minimum number of unique clients *k* are required to participate in updating *u**t*. Furthermore, considering that each client is allowed to serialize its updates into multiple blocks, a limit is set to define the maximum number of contributions by each client in each round. The aggregation mechanism varies in different DFL architectures. While the majority of DFL models use an averaging function ∑( ⋅ ) similar to that of federated averaging (FedAVG), other frameworks use different approaches such as selecting the update with the highest score.\nWhile the initial DFL model was proposed with the aim of security enhancement and failure robustness of FL using Bayesian beliefs of one-hop neighbors, several research studies extend their work by improving the overall efficiency and security. Fig. [fig:diagram] illustrates the historical evolution of DFL since it was first proposed in 2018. BAFFLE makes use of Smart Contracts (SC) to facilitate storing the FL model and clients’ states. Model updates and aggregation is also carried out by resorting to SC. BAFFLE takes the computational states into account in order to preserve fairness among clients. Another approach, Blade-FL, uses clients for both mining and training via gossip learning. A novel consensus mechanism was proposed in to enhance the efficiency and privacy of the system. The DFL framework proposed based on this committee consensus mechanism is called BFLC. The Biscotti technique integrates DFL with Differential Privacy (DP) to safeguard DFL against a range of attacks. Focused on healthcare applications, introduces the BindaaS method, which integrates DL as a service and lattices-based cryptography. This enables BinDaaS to employ an authentication phase that enhances DFL attack resilience. Another DP-based DFL approach called LearningChain is proposed with the aim of combating Byzantine attacks. To do this, LearningChain utilizes a new aggregation mechanism for decentralized SGD. Reference designs the PIRATE framework based on the sharding approach in blockchain which helps in securing the aggregation process. BEAS employs DP, gradient pruning, and anomaly detection to protect DFL against poisoning attacks. BEAS is also adapted for heterogeneous data to ensure proper convergence of the global model. BrainTorrent is a DFL method that employs version control and arbitrarily assigns a participant to act as the server. A reputation management scheme is used in to select updates from reliable workers in the blockchain which reduces the probability of successful poisoning attacks in the DFL system. BlockFLA, uses smart contracts to detect and penalize malicious nodes by means of monetary penalty. BlockFLA requires uploading hashed updates into separate private and public blockchains. Smart contracts is used for aggregating gradients whereas hashed updates are used for evaluating the updates through recovering the parameters to detect any mismatch. DFedForest is a tree-based DFL framework that uses bagging for tree construction. This method devises a private test dataset for evaluating the uploaded parameters as malicious updates are expected to result in anomalous outputs on this data. Bift,, designs a Proof of FL by combining PoW with FederatedAVG and FederatedSGD aggregation schemes to reach secure DFL with low communication overhead. A verifiable version of DFL incorporates private key sharing and gradient masking into the consensus mechanism to defend against malicious miners and dropouts. Swarm Learning (SL) also integrates FL and blockchain to distributively train a model for diagnosing a number of diseases while safeguarding the patient’s data privacy and security. Security and privacy features of SL are limited to those of blockchain and the P2P DFL design. However, there is an extension of SL that couples it with HE is also presented in to secure SL communications against privacy attacks. Various aspects of SL such as fault tolerance, scalability, and fairness are studied in, and the results indicate an overall improvement over centralized FL.\nDespite the security advantages of blockchain integration with DFL, communication delay and resource consumption issues often become problematic in this domain. As mentioned in Section [sec:back], the process of model aggregation and achieving a consensus often requires sufficient computational power at the edge of the network which may not always be practical (e.g., mobile networks). This becomes more critical for consensus mechanisms such as PoW. As a result, the model aggregation process may be usually delayed due to the potential computational constraints.\nThreats to Decentralized Federated Learning\n===========================================\nAs shown in Fig. [fig:taxonomy], the security and performance of DFL are directly related to those of FL and blockchain. Generally, threats to DFL target privacy or the performance of the global model. In centralized FL, all the possible threats involve the server at some point as the server is the core of the system for aggregation, communication, and validation. As a result, the trustworthiness of the server is of great importance in FL as a malicious server can easily attack all clients. On the other hand, attacking the server and causing a malfunction can also disable the entire system. Table [tab:tab1] lists sources of threats for common attacks on FL. However, in DFL, all possible threats will be traced back to the clients or miners. Table [tab:analysis] indicates attack models that are considered in the design of the aforementioned DFL techniques.\n[t]\nll Attacks & Source of Attack \nData poisoning & Malicious client \nGradient manipulation & Malicious client \nBackdoor attack & Malicious client and malicious server \nEvasion attack & Malicious client and model deployment \nNon-robust aggregation & Aggregation algorithm \nTraining rule manipulation & Malicious client \nInference attacks & Malicious server and communication \nGAN reconstruction & Malicious server and communication \nFree-riding attack & Malicious client \nMan-in-the-middle attack & Communication \n[tab:tab1]\n[t] [fig:taxonomy]\nMalicious clients can follow different attack models to jeopardize the performance or privacy of the system. For instance, an adversary can be either semi-honest or aggressive. In the semi-honest model, the objective is to infer confidential information while complying with the DFL protocol. This phase can be also used as a reconnaissance step prior to launching an aggressive attack where the aim is to degrade the system’s performance. Using this combination, one can first learn the global model parameters using a semi-honest approach and then actively forge malicious updates in order to insert a backdoor in the global model. Due to the immutability of blockchain, hackers cannot directly manipulate the training data in a block. Nevertheless, they can reach their objective by sending the gradients that lead to incorrect predictions. To do so, gradients can be forged targeted or untargeted. As the name implies, the latter aims at degrading the performance of all classes in general whereas the targeted scheme only attacks a certain class without tampering with the rest of the model.\nAttacks on Performance\n----------------------\nThe performance of the DFL model is mostly targeted by means of poisoning attacks. In addition, there are certain blockchain-based attacks that also apply to DFL. An overview of these attacks for DFL is given in the following.\n### Data Poisoning\nData poisoning techniques in FL and DFL could be somewhat different. In centralized FL, data poisoning is referred to as the process of corrupting the local training data via backdoor injection or introducing label noise which in turn produces deviating gradients and misleads the FL model. In this setting, usually, the intruder disguises as a member of the federation and uses its artificially made data. As an example, reference studies different variations of label poisoning in federated learning systems and mitigates these attacks by reformulating the problem as a label noise classification task.\nAside from the aforementioned mechanisms for data poisoning, the following attacks can be planned to take advantage of the blockchain backbone that most DFL methods rely on:\n**Disrupting Communications:** Blockchain is robust by nature against Denial of Service (DoS) attacks as there is no possible SPF that can be targeted. Nevertheless, Distributed DoS (DDoS) attacks can still slow down inter-node communications in the DFL system. DDoS agents continuously generate fake transactions and send them to the chain to fill up the blockchain buffer with spam data once the blockchain capacity is reached. This significantly postpones the inclusion of valid data into the upcoming blocks which puts the functionality of the blockchain under question. This is besides the permanent effect DDoS leaves on the DFL due to the data immutability of the blockchain.\n**Degrading Detection Performance:** Using a targeted attack model, an intruder can inject certain triggers into the local model and obtain gradients in a way that the rest of the model remains intact. This backdoor attack results in sharing the poisoned parameters with the rest of the DFL participants and introducing the backdoor into their model as well. For instance, backdoors are injected into the DFL network in with two different mechanisms, namely label flipping and pixel patch backdoor. The number of malicious nodes is restricted to prevent intruders from controlling the consensus (i.e., 2*N**c**o**r**r**u**p**t**e**d* + 2 < *N*, where *N* and *N**c**o**r**r**u**p**t**e**d* indicate the number of all nodes and corrupted ones, respectively). A pixel pattern backdoor is also simulated in for DFL with smart contract. The untargeted scheme of this attack which is also known as the Crashing DoS attack, follows the same process with the exception of degrading the performance of the entire model rather than for a specific class. While the untargeted version is easier to implement, it is also easier to detect. Note that the literature usually defines DoS differently for blockchain and FL. DoS in blockchain renders the global model inaccessible, whereas in FL, it makes the global model unusable. Reference presents an example of a data poisoning attack in DFL which is simulated with the aim of corrupting the global model performance. In this scenario, 30 percent of nodes are considered malicious, with the assumption that the number of malicious nodes does not change in time. Another case is tested in by generating forged gradients drawn from a Gaussian distribution by several Byzantine data holders (i.e., trusted nodes went rouge in the DFL network). Reference simulates this attack by filliping the class labels randomly on the training data. In contrast to poisoning attacks that primarily corrupt the training process, evasion or exploratory attacks occur during the inference phase after the model has been trained. Their goal is typically not to change the trained model, but rather to generate incorrect predictions or gather information about the model’s characteristics.\n**Causing Legal Problems:** By law, the inclusion of personal data is prohibited in some regions of the world. European general data protection regulation can be mentioned as a good example of this legal constraint. A malicious user, however, can poison the blockchain by inserting personal data into a data block and adding it to the chain which makes the DFL system non-compliant with the existing legal bounds. We call these adversaries privacy poisoning attacks. Despite its importance, not many research studies considering this type of attack in a DFL paradigm have been reported.\n[t]\nlcccc Attacks & Target & Implementation & Effectiveness & Defensibility \nBackdoor & Performance & Challenging & High & Challenging \nDoS/DDoS & Performance & Simple & Moderate & Simple \nGradient manipulation & Performance & Challenging & High & Challenging \nTraining objective manipulation & Performance & Challenging & High & Challenging \nEvasion attacks & Performance & Simple & Moderate & Moderate \nModel inversion & Privacy & Simple & Low & Moderate \nFeature inference & Privacy & Simple & Moderate & Moderate \nMembership inference & Privacy & Challenging & High & Challenging \nHijacking private key & Blockchain & Challenging & High & Challenging \n51% attack & Blockchain & Challenging & High & Challenging \nSybil attacks & Blockchain & Simple & High & Challenging \nDouble spending & Blockchain & Challenging & High & Challenging \nRouting attacks & Blockchain & Simple & Moderate & Challenging \nPrivacy poisoning & Blockchain/Privacy & Moderate & Moderate & Moderate \nAttacks on SC & Blockchain & Challenging & High & Moderate \n[tab:attacks]\n### Model Poisoning\nModel poisoning is the process of maliciously controlling global model training. While data poisoning can be also used as a tool to cause model poisoning, model poisoning without data manipulation is also possible. As an example, one can simply change the objective of the local model to obtain poisoned gradients using valid data on the same model structure as the rest of the network. For instance, an additional term can be added to the objective to penalize sensitivity to malicious data. The same approach can be followed to deteriorate the overall performance in an untargeted manner (e.g., gradient manipulation). This case is also studied in where a set of Byzantine nodes are considered to generate and inject deviating gradients into the blockchain in order to degrade the shared DFL model via an outsider attack. A similar approach is used in to corrupt gradients using pointwise Gaussian noise.\n### Routing Attacks\nReliable network infrastructure is of paramount importance for DFL due to its integration with blockchain. This is while some of the utilized network protocols used by current internet service providers, such as the border gateway protocol, come with security flaws. Blockchain has no control over the network layer as it mostly works in the application layer. Thus, if the service provider network is breached, the routing of packets can be tampered with to either discard the transferring data blocks or change the blockchain structure. An example of a routing attack is a hack that took place in 2014 in which the intruder used the hijacked blocks to provide PoW and steal the associated rewards.\n### Consensus Attacks\nThe objective of these attacks is to obtain the majority of approvals in order to control the consensus mechanism in the DFL system. Depending on the number of agents or nodes involved in these attacks, they will be called 51% or Sybil attacks. Sybil attacks indicate the scenario in which an intruder creates multiple fake nodes and claims to be more than one entity to impose a greater influence on the consensus. If intruders manage to take control of 51 percent of the nodes, the consensus mechanism is fully controllable. At this scale, the attack is known as 51% attack in the literature. It is worth mentioning that consensus attacks are most effective when the blockchain associated with the DFL is at its early stage when the number of participants is still limited. PoW and PoS can drastically complicate consensus attacks for hackers.\nAttacks on Privacy\n------------------\nDespite the decentralization of data in DFL, gradient and parameter information can be visible to system clients. Previously, we studied the effect of such information leakage on the performance of the DFL model. Here, we elaborate on the privacy risks associated with DFL networks.\n### Model Inversion Attacks\nModel inversion can be used to approximate and reconstruct the private data of clients merely based on the classification model. An example of this attack is reported in where gradient leakage of a global DFL model is used for reconstructing the user’s local data. This process is often carried out using a DL-based generator. For instance, Generative Adversarial Networks (GAN) can be employed to perform model inversion by using the target model as the discriminator and training a generator network to minimize a cost that eventually converges and produces data samples similar to that of the targeted client.\n### Membership Inference Attacks\nMembership inference attacks are focused on revealing sensitive information associated with certain samples. Membership inference can disclose the membership of a sample to a certain class, or elicit attribute information. An example of this attack can be found in, where an inference attack by observing the clients’ updates in a DFL network and using the leaked gradients to infer record-level information of local datasets.\n### Hijacking Private Key\nSecuring private keys and public key using cryptography is one of the most critical aspects of blockchain which is the backbone of DFL. If there is any imperfection in the key signing mechanism, hackers can hijack a client’s private key using their public key. Taking over a client’s private key enables the hacker to gain full access to the corresponding data in the blockchain.\n### Vulnerability of Smart Contracts\nSmart contracts, which are coded agreements utilizing blockchain technology for record-keeping, can be susceptible to security flaws. While they eliminate the need for intermediaries and provide immutable contracts, there is a risk associated with poorly coded smart contracts. These coding vulnerabilities create opportunities for attackers to identify flaws in the code and exploit them. By exploiting these weaknesses, attackers can potentially manipulate or extract unauthorized access to the contract’s contents or associated assets. It is crucial to ensure thorough code review and rigorous testing to identify and address these security flaws to safeguard the integrity and trustworthiness of smart contracts.\nAnalyzing the Viability of Attacks\n----------------------------------\nTable [tab:attacks] provides an overview of the analyzed attacks, highlighting their key characteristics. When evaluating these attacks, it is crucial to assess their feasibility and effectiveness to design effective defense mechanisms and allocate appropriate resources to secure DFL systems.\n### Ease of implementation\nImplementing a backdoor attack is considered to be difficult because it requires extensive knowledge of the target DFL system, access to training data, and the ability to modify model parameters. Backdoor attacks often involve sophisticated techniques and may require compromising multiple clients or insiders with privileged access. Conversely, carrying out a DoS or DDoS attack is relatively less complicated. These attacks overwhelm the system with a high volume of requests or malicious traffic, and readily available tools and techniques can be used to execute them.\nGradient manipulation necessitates a deep understanding of the DFL system’s architecture, algorithms, and access to the communication channels. Tampering with gradients exchanged between clients and the server is a challenging task that requires careful execution to avoid detection. Similarly, manipulating the training objective, such as modifying the loss function or optimization process, is also categorized as hard. It demands an in-depth understanding of the system’s algorithms, access to the training process, and the ability to modify the objective without disrupting the overall learning process.\nOn the other hand, attacks such as evasion attacks and model inversion are easier to implement. These attacks can be implemented using existing knowledge and techniques. Other attacks, including feature inference and Sybil attacks, are also categorized as easy to implement. In contrast, attacks such as membership inference, hijacking private keys, 51% attacks, consensus attacks, and double spending are challenging attacks. These attacks require advanced knowledge, sophisticated attacks, and access to specific components or mechanisms of the DFL system. Implementing them successfully is a complex and resource-intensive task. Furthermore, privacy poisoning attacks are not overly complex or technically challenging, leading to a medium level of ease of implementation. The process may involve tampering with the data blocks and ensuring the inclusion of personal information, but it does not require advanced technical skills or extensive resources. Moreover, finding security holes in SC is a complex task as these protocols are usually well-tested before release, and sophisticated technical skills are required to hack SC under this condition.\n### Effectiveness of Attacks\nIn terms of attack effectiveness, the backdoor attack is classified as high. This attack has the potential to significantly impact the model’s predictions, compromising the integrity and reliability of the DFL system. On the other hand, DoS/DDoS attacks are considered to have a moderate level of effectiveness. While they can disrupt the normal functioning of the system and cause temporary delays, their impact on the actual model parameters or the accuracy of the aggregated model may vary.\nGradient manipulation and training objective manipulation attacks are both considered to have a moderate level of effectiveness. Tampering with gradients or modifying the training objective can potentially influence the learning process and impact the accuracy of the aggregated model. However, their success depends on the robustness of the DFL system and the defense mechanisms in place.\nEvasion attacks and model inversion attacks are also considered to have a moderate level of effectiveness. Their impact on the overall model performance may be limited depending on the effectiveness of the defense mechanisms. Membership inference attacks, on the other hand, are classified as having a high level of effectiveness. Successfully inferring membership information about specific individuals participating in the DFL system can have severe privacy implications.\nAttacks targeting the blockchain aspect of DFL, such as hijacking private keys, 51% attacks, consensus attacks, Sybil attacks, double spending, and routing attacks, are generally considered to have a high level of effectiveness. These attacks have the potential to compromise the security, integrity, and trust of the DFL system, depending on the specific vulnerabilities and defense mechanisms in place. Moreover, privacy poisoning attacks can have a moderate impact on compromising the privacy of participants in the DFL system. While the effectiveness of privacy poisoning attacks is not as high as some other attacks, it still poses a significant risk to the privacy of participants and the overall integrity of the system. In addition, exploiting the vulnerabilities of SC can lead to losing digital trust and massive financial loss. For instance, reports indicate that crypto investors nearly lost four billion dollars to hackers in 2022.\n### Defensibility\nBackdoor attacks involve inserting malicious behavior into the training data or model, making it challenging to detect and defend against. Detecting and mitigating backdoor attacks often require advanced techniques and rigorous model verification.\nDoS/DDoS attacks are classified as moderately defensible. While it may be challenging to completely prevent DoS/DDoS attacks, there are various mitigation strategies available, including anomaly detection. Such methods can reduce the impact of DoS/DDoS attacks and keep the system operational.\nDefending against gradient manipulation and training objective manipulation attacks is also categorized as moderately defensible. Implementing techniques like DP, robust aggregation algorithms, and secure communication protocols can enhance the system’s resilience against these attacks. However, the effectiveness of the defense mechanisms may vary based on the sophistication of the attack and the quality of the defense techniques.\nEvasion attacks and model inversion attacks are considered moderately defensible. Privacy-preserving techniques, such as DP or SMC, can help protect against these attacks. However, achieving strong defenses requires careful design and implementation, considering factors like attack vectors and data sensitivity.\nDefending against membership inference attacks is classified as challenging due to the difficulty of protecting individual privacy in the FL setting. Advanced privacy protection techniques, such as privacy amplification and enhanced model aggregation protocols, are needed to effectively mitigate the risk of membership inference attacks.\nRegarding attacks on the blockchain aspect of DFL, the defensibility levels vary depending on the specific attack. Hijacking private keys, 51% attacks, and double spending attacks are challenging to defend against due to the vulnerabilities they exploit in the blockchain infrastructure. Implementing robust consensus mechanisms, multi-factor authentication, and encryption can enhance the defensibility against these attacks. Defending against consensus attacks, Sybil attacks, and routing attacks requires strong identity management systems, network monitoring, and reputation-based mechanisms. Defending against privacy poisoning attacks requires a combination of technical, legal, and regulatory measures, making it of medium difficulty in terms of defensibility. In addition, ensuring the security and integrity of SC has a moderate level of difficulty since it requires sophisticated code review and rigorous testing to identify and eliminate potential security holes and flaws.\n[t] [tab:defs]\nllll Defense mechanism & Description & Weakness & Used against \nHomomorphic Encryption & Encrypts parameters & Computational overhead & Attacks on privacy \nSMC & Protects data with multiparty computation & Communication overhead & Attacks on privacy \nDifferential Privacy & Perturbs parameters with noise & Accuracy loss & Attacks on privacy \nAnomaly Detection & Monitors updates to detect abnormalities & Backdoors can bypass it & Untargeted poisoning \nRobust Aggregation & Safeguards global aggregation against adversaries & Limiting assumptions & Poisoning attacks \n & & Computational overhead & \n& & Performance loss \nTEE & A secure ecosystem for maintaining digital trust & Limited memory size & White box attacks \n & & Computational overhead & \n& & Communication overhead & \n& & Complex implementation & \n & & Computational overhead & Privacy attacks \n& & Not a stand-alone defense & Poisoning attacks \n & & Sensitivity to hyperparameters& Privacy attacks \n& & Computational overhead & Poisoning attacks \n & & Computational overhead & \n& & Blockchain security issues & \nlccccccccc Methods & References & Blockchain & Encryption & Clients & Data poisoning & Model poisoning & Inference \nBindaaS & & & × & Honest & & & \nPIRATE & & & × & Semi-honest& & & \nBAFFLE & & & × & Dishonest && & \nBFLC & & & × & Semi-honest & & & \nLearningChain & & & DP & Dishonest & & & \nBiscotti & & & DP & Dishonest & & & \nBlade-FL & & & DP & Honest & & & \nBEAS & & & DP & Dishonest & & & \nSwarm Learning & && × & Dishonest & & & \nSL+HE & && HE & Dishonest & & & \nBrainTorrent & & × & × & Honest & & & \nP2P-FL & & × & × & Honest & & & \nReputationDFL & & & × & Dishonest & & & \nBlockFLA & & & × & Dishonest & & & \nDFedForest & & & × & Dishonest & & & \nBift & & & × & Dishonest & & & \nVerifiable DFL & & & × & Semi-honest & & & \nDFL for Healthcare & & & DP, SMC & Semi-honest & & & \nFLChain & & & × & Honest & & & \nVFChain & & & × & Semi-honest & & & \n[tab:analysis]\nDefense Mechanisms\n==================\nVarious defense mechanisms are proposed to fortify DFL against privacy and performance-related threats. While this section reviews the literature on DFL defense mechanisms, their characteristics are summarized in Table [tab:defs]. In addition, Table [tab:analysis] specifies mechanisms that each reviewed DFL method employs.\nPrivacy Preserving\n------------------\nDespite the wide diversity of previous efforts on safeguarding FL and blockchain privacy, suggested methods typically fall into one of these three categories: 1) Homomorphic Encryption (HE), 2) Secure Multiparty Computation (SMC), and 3) DP. The following discussion goes through each of these groups.\n### Homomorphic Encryption\nBy processing on cyphertext, HE is commonly used to secure the learning process. Clients can use HE to perform arithmetic operations on encrypted data (i.e., ciphertext) without having to decode it. HE has three major variations that differ in arithmetic complexity and flexibility, namely full, partial, and substantial HE.\nFully HE is capable of doing arbitrary calculations on the encrypted data. This is while partially HE can only execute one operation (e.g., addition or multiplication), and substantially HE can do several operations. The latter, on the other hand, has a restricted amount of additions and multiplications. While full HE offers greater flexibility, it is inefficient when compared to other forms of HE. As an example, shows that swarm learning updates are encrypted using partial HE to defend DFL against inference attacks.\nDespite the benefits of HE, executing arithmetic on the encrypted integers increases the memory and processing time costs. Moreover, non-linear estimations in statistical models demand approximating polynomials which make it important to find a balance between utility and privacy. In, for instance, additively HE is used to secure distributed learning by securing model changes and maintaining gradient privacy. Another example is, which uses an additively homomorphic architecture to defeat honest-but-curious adversaries using federated logistic regression on the encrypted vertical FL data. However, the overburdening of the system with additional computational costs is a typical downside of such systems.\n### Secure Multiparty Computation\nSecure Multiparty Computation (SMC), is a distributed cryptography technique in which several entities participate in the estimation of a function. The distinct feature of SMC is securing each participant’s data by creating a set of random values that are not equal to the participant’s data, and sending them to the rest of the parties for local calculation of the function. The outputs of these functions then will be averaged to obtain the desired estimation. In this scheme, the data on the participant’s side is meaningless, and the function estimations are only usable once they are averaged. For instance, SMC was used in for private model training. It is worth mentioning that SMC is followed by excessive communication and computational cost. It has been also mentioned that SMC is best to be coupled with DP to secure the communications.\nIn conclusion, SMC usage in large-scale DFL could be inefficient due to the dramatic rise in communication and processing costs. Secondly, encryption-based solutions corresponding to these functions must be properly defined and implemented. Finally, all cryptography-based protocols preclude an audition phase of the received updates by the shared model, hence, leaving holes for rogue users to exploit.\n### Differential Privacy\nThe idea of DP is to inject random noise into the generating updates so that the data interpretation becomes infeasible for malicious entities. DP is primarily used to safeguard DFL communications against privacy attacks (e.g., inference attacks); however, the literature also shows that DP can be also beneficial against data poisoning attacks as these attacks are usually designed based on the communicated gradients. Biscotti couples DP with a secure aggregation technique to safeguard DFL systems against poisoning and inference attacks. Another report shows the effectiveness of pruning-based DP in safeguarding horizontal DFL systems. DP in is implemented through perturbing local gradients using the exponential mechanism and a predefined probability density function before uploading them into the blockchain and broadcasting them across the DFL network. Similarly, Blade-FL adds noise to the gradients prior to the encapsulation phase, albeit using a Gaussian distribution.\nIn contrast to HE and SMC whose main disadvantage was communication overhead, DP has a lower computational cost and does not overburden the system in this sense. Instead, DP comes at the cost of deteriorating the model quality. This is mainly because the injected noise can potentially add up to the noise within the constructed model. Another issue of concern in conventional DP is the cumulative privacy loss resulting from iterative training processes that utilize local data from multiple individuals or sources. These iterations are crucial for enhancing the accuracy and performance of trained models. However, with each iteration, a certain degree of privacy loss is introduced, and this loss accumulates over time, potentially reaching a significant level. Researchers have dedicated efforts to address this problem by exploring various approaches, including subsampling and privacy amplification by iteration. These techniques aim to mitigate cumulative privacy loss and improve the overall privacy guarantees of DP schemes. Since these methods derive a tight upper bound of cumulative privacy loss, they can also be applied to preserve model utility even in cases where the gradients are perturbed by noise. Moreover, DP provides resistance to poisoning attempts due to its group privacy trait. As a result, as the number of attackers increases, this defense will reduce significantly. As mentioned before, all privacy-preserving methods have a set of advantages and disadvantages, and thus, there is no perfect alternative for DP. In other words, DP, HE, and SMC each result in a different privacy-utility trade-off. As a solution, hybrid approaches can be implemented to build privacy protocols that are more robust than using only one of these methods. As an example, SMC is combined with DP to balance the trade-off between them. This combination offsets excessive noise injection when the number of clients is growing while preserving the desired rate of trust.\nGaussian DP is a specific case within the functional DP (*f*-DP) framework, which characterizes privacy through hypothesis testing. In *f*-DP, a randomized algorithm is considered to satisfy privacy if the difficulty of distinguishing between two neighboring datasets (quantified by a trade-off function) is element-wise larger than a convex and non-increasing function *f*. When this function f is constructed using two Gaussian distributions, the resulting form of *f*-DP is termed Gaussian Differential Privacy. In other words, Gaussian DP defines privacy guarantees by examining the distinguishability of neighboring datasets through a hypothesis testing approach, with the specific choice of a trade-off function derived from Gaussian distributions. This provides a mathematical and analytical framework for evaluating privacy in the context of private data analysis. For instance, applies Gaussian DP in an FL system to provide record-level privacy guarantees for each client. Building upon the ability of this approach to handle composition and subsampling, extends the work to apply DP to SGD and Adam optimizers.\nIn the context of applying DP to FL, two privacy notations, namely user level and instance level privacy, can be found in the literature. User-level privacy in DP ensures that clients’ data remains private and the privacy of the global model remains intact, even if an adversary removes a client or its data from the aggregation process. Additionally, adversaries are unable to determine whether a client has participated in the training. On the other hand, employing DP at the record level offers privacy guarantees for each individual record within a client’s data, protecting them against potential adversaries. Instance-level DP is often perceived as offering a relatively weaker privacy guarantee, as adversaries may potentially determine whether a user participated in the training or not. However, there are cases where discerning user participation is not crucial, and the primary concern is maintaining data privacy. Considering the trade-off between privacy and utility in differential privacy, it is prudent to adopt instance-level privacy in such scenarios to achieve improved performance while avoiding overly stringent privacy constraints.\nDP can be centralized, local, or distributed. In centralized DP, the noise addition is performed via a server, which makes it impractical in DFL. On the other hand, local and distributed DP both assume that the aggregator is not trusted which perfectly complies with the DFL paradigm. In the local variant, participants inject noise in their estimated gradients before sharing them over the blockchain. However, research on local DP indicates its inability to provide a privacy guarantee on large-scale and heterogeneous models with numerous parameters. In DFL, the injected noise should be calibrated to ensure successful DP. Despite the appealing security qualities of local DP, its practicality becomes questionable when dealing with an immense number of users.\nDistributed DP combines cryptographic techniques to provide the benefits of both local and centralized DP without compromising the clients’ privacy. As a result, it avoids putting faith in any server and is more effective in that sense. Decentralized DP, in theory, has the same benefit as the centralized variant because the overall quantity of noise is similar for them. The concept of distributed DP alludes to the notion that the required amount of noise is derived from several individuals.\nModel Robustness\n----------------\nDefenses are classified into two types: proactive and reactive. Proactive defense is a low-cost method of anticipating attacks and associated consequences. The reactive defense operates by detecting an invasion and taking preventative steps. In the production environment, reactive defense is often deployed as a patch-up. DFL presents multiple additional attack surfaces throughout training, resulting in complicated and unique countermeasures. In this part, we will look at some of the most common types of DFL defensive tactics and investigate their usefulness and limits.\n### Anomaly Detection\nAnomaly detection methods have a long history in real-time identification of data attacks in intelligent systems. In centralized FL, anomaly detection is often performed on the server to identify malicious updates that can lead to model or data poisoning. It is worthwhile to mention that this approach usually works best against untargeted attacks. In the DFL paradigm, however, anomalies can take place either due to fraudulent transactions or changes in blockchain networks such as network division and blockchain fork. Each of these issues is separately studied in the literature and there is no anomaly detection framework designed to handle both. For instance, proposes a fast-paced detection scheme for monitoring transactions using a subgraph-based approach. The designed detection method is optimized for parallel processing GPU acceleration. On the other hand, another work,, designs a distributed anomaly detection scheme, called BAD, to combat eclipse attacks in blockchain networks by re-connecting unauthorized forks to the P2P network.\nThe majority of the anomaly detection methods for FL security are proposed with respect to the centralized architecture of the FL, and anomaly detection in DFL is only studied in a limited number of works. As an example, BEAS employs two anomaly detection protocols, namely Multi-KRUM and FoolsGold to detect poisoning attacks and identifying Sybil groups (i.e., group of malicious nodes coordinating a cyber-attack), respectively. The former monitors change in variance or performance of generated gradients with respect to the majority of updates. The latter, on the other hand, detects Sybil groups based on the correlation between the generated updates, as it is expected for malicious nodes to generate highly similar gradients. Anomaly detection in high-dimensional space poses challenges, leading some detectors to suggest using dimensionality reduction to make monitoring easier. However, reducing the dimensionality size may result in information loss, even with advanced techniques preserving data characteristics.\n### Robust Aggregation\nThe aggregation algorithm used in an FL system should tolerate communications disturbances, client dropout, and incorrect model updates on top of hostile participants. Extensive research has been dedicated to advancing robust aggregation in centralized FL. Similarly, in DFL networks, the security of the aggregation phase is of paramount importance. Under a DFL paradigm, the aggregation phase is mainly secured by employing the same techniques used in FL in combination with the consensus mechanism of the utilized blockchain scheme. These protocols (e.g., PoW, PoS, SC) control which nodes can participate in the model aggregation phase for both generating updates and aggregating parameters. On the downside, many of the advanced Byzantine-robust aggregations rely on assumptions that are either unrealistic or incompatible within the context of FL.\n### Pruning\nEven though pruning is not specific to DFL and can be applied to any neural network structure, it can help in eliminating backdoors. The idea of pruning is to drop some neurons in order to enhance the efficiency and accuracy of the network. However, this feature makes the parameter usage somewhat unpredictable for attackers which complicates injecting backdoors into a neural network since inactive neurons will be removed eventually in the network. While pruning has been mainly studied for FL, current literature on pruning under a DFL paradigm is very limited. In particular, BEAS makes use of gradient pruning in DFL to facilitate DP and complicating model poisoning for attackers in the system. Nevertheless, since these techniques are mainly used at the edge of the network, most pruning approaches should be adaptable to DFL as well.\nPruning comes with a set of drawbacks. One drawback is the potential loss of model capacity, resulting from the removal of connections or components. Another concern is the sensitivity of pruning to initialization and training, necessitating careful fine-tuning and experimentation for optimal outcomes. Moreover, the pruned model may struggle to generalize effectively to new data, limiting its real-world applicability. Lastly, the computational overhead increases due to the additional resources and time required during the pruning and retraining stages. Considering these factors is vital when assessing the viability of pruning as a defense mechanism against backdoors.\n### Trusted Execution Environment\nThe blockchain backbone of DFL requires the data to be replicated on each node which makes it challenging to keep smart contracts confidential. TEE (Trusted Execution Environment), on the other hand, is a tamper-resistant ecosystem that can be used to maintain digital trust between nodes of a distributed network. TEE is often referred to as a secure and isolated part of the processor that requires all codes and data signatures to be verified with respect to the designer’s expectations. The validity of a participating device in a TEE Authentication should be checked by the connected service with which it is attempting to enroll. Until the matching party provides a message, the status of code execution stays hidden. The execution route of the code cannot be changed until it takes explicit input or a validated interruption. Data stored on and processed by participants is safe, and interactions between various parties are carried out in a secure manner. The TEE is in charge of all data access privileges. Cryptographic technologies are used to secure TEE communications. Only the TEE secure environment stores, maintains, and uses private and public encryption keys. The TEE can show a remote client what code is presently being executed as well as the starting state. TEE can aid in resolving a key challenge for FL security since it is becoming progressively important in securing the central server and clients against hackers and preventing data theft.\nTEE is coupled with FL to safeguard against algorithmic attacks. TEE can hide model parameters on local devices so that the model is not accessible to the attacker. Under this condition, the attacker is only able to launch black box attacks on the system. Nevertheless, TEE often suffers from a limited memory size, that is only a limited part of the model can be secured with this approach. It has been also suggested that TEE can secure smart contract data in a blockchain which is the main vulnerability of this architecture. Theoretically, the same concept can be applied to DFL. However, current literature lacks an experimental study on the integration of DFL and TEE.\n### Zero-Knowledge Proofs\nThe origin of zero-knowledge proofs goes back to the mid-1980s. This cryptographic approach allows a verification process that does not involve data exchange between parties. This process often involves using unconnected bits of information to keep data private during the verification.\nAs an example, zero-knowledge proofs have the potential to be utilized in DFL to verify the authenticity of the features used by clients for training and generating updates. While zero-knowledge proofs offer a promising avenue for enhancing secure update monitoring, additional research is necessary to identify challenges in constructing and implementing their modules. Notably, zero-knowledge proof protocols generally maintain their performance regardless of the data volume.\nZero-knowledge proofs offer advantages for enhancing secure update monitoring in DFL. However, their adoption also entails certain drawbacks. These include computational overhead due to resource-intensive operations, the complexity of implementation requiring meticulous attention to cryptographic techniques, potential scalability issues in large-scale DFL systems due to communication complexity, and the trust assumption associated with the setup phase.\n### Knowledge Distillation\nKnowledge distillation is a fundamental algorithm in federated distillation. The goal of knowledge distillation is to perform TL from a large teacher model (T) to a compact student model (S) without sacrificing performance significantly. Smaller models have fewer parameters and are less susceptible to overfitting, making them more resilient against attacks. This increased resistance makes it more challenging for attackers to reverse-engineer or manipulate the model. Nonetheless, for certain threats such as backdoor attacks, knowledge distillation alone may not be directly effective, as these attacks typically involve modifying the training data or model parameters to embed a hidden trigger. However, knowledge distillation can be used as part of a broader defense strategy to mitigate the impact of backdoor attacks. For instance, reference mitigates backdoor attacks by utilizing knowledge distillation in deep neural networks. This approach independently fine-tunes T on a clean subset and uses it to clean backdoored S. To do so, this method tries to align intermediate-layer attention in S with that of T. Another example trains a T ensemble on disjoint training subsets and trains S based on aggregated noisy voting among T models. The goal of this approach is to provide a privacy guarantee for training data. In DFL, this idea translates into sharing the knowledge of a model rather than the parameters which improves FL’s robustness against both poisoning and privacy attacks. In addition to the mentioned security advantages, knowledge distillation also results in communication and computation efficiency in DFL. Exchanging model parameters becomes burdensome when communication resources are limited, especially for contemporary big deep neural networks. In this sense, federated distillation is an appealing FL option since it only transmits model outputs which are often considerably less in size than the model sizes.\n### Regularization\nIn DFL, the computational model that is being distributively updated is most likely a DL structure. One approach to mislead this model is to make it overfit by introducing malicious samples which may in turn lead to membership inference. Regularization techniques such as *L*2 regularizer and dropout can eliminate the effect of such malicious samples to a great extent. Reference designs a semidefinite relaxation method that generates a differentiable certificate for network robustness and optimizes it alongside network parameters to encourage robustness against all attacks. Furthermore, determining suitable hyperparameters, such as regularization strength, for regularization techniques can be a difficult task, and incorrect choices may yield undesirable results. Moreover, the use of certain regularization methods with intricate terms or penalties can introduce considerable computational complexity, especially in scenarios involving extensive datasets or intricate models, resulting in prolonged training durations.\nVerifiable DFL\n==============\nDefense mechanisms reviewed in the previous section mainly combat an adversary after a malicious attack is launched by an intruder. Nevertheless, another approach to combat these threats is to prevent malicious parties to take part in the training process. A DFL system that can verify the trustworthiness of its clients is called verifiable.\nA DFL system is considered verifiable if clients of a network can prove to each other that the given task has been carried out without compromising privacy. Security, transparency, and automation of blockchain can enhance the verifiability of DFL which in turn prevent malicious activities. In this structure, some nodes are in charge of generating updates and training the model (i.e., trainers) and groups of participants verify the generated updates (i.e., workers) with respect to consensus protocol. If the update is verified, the worker who completes this task first will create a new block. Then, if the majority of the workers approve the content of the created block, it will be appended to the chain. At this point, DFL clients can update their models using this new block of information. This process is repeated periodically to keep the DFL model up to date.\nTrustable DFL Trainers\n----------------------\nA vulnerability of DFL systems is that dishonest trainers can upload maliciously crafted updates to the blockchain. For instance, BlockFL verifies the submitted updates through a random selection of workers before letting the update take effect. In BlockFL, each client is associated with a random worker who is rewarded when evaluating the updates prior to the aggregation. The evaluation criteria are designed based on the relationship between the data size and the elapsed time for generating the update.\nAnother approach for verifying trainers is to use proof of correctness. Examples of this are presented in, where produced parameters are paired with their proof of correctness which is used by registered workers for determining the approved updates. This approach uses verifiable random functions or reliability ranking to select a worker whose aggregation results will be appended to the blockchain. The trainer’s reliability, which is an indicator of its performance, will affect its chance to take part in the training process thereafter.\nTrustable DFL Workers\n---------------------\nAssuming that all trainers are trustworthy, the aggregated model could still be compromised if the responsible worker for the model aggregation is unreliable, that is, it does not follow the defined protocol. VFChain addresses this issue by using trainer signatures that are appended to the uploaded block by the trainer. To verify the aggregated model by the responsible worker, a committee is randomly selected among other workers to evaluate the aggregated model with respect to the verifying contract. The members of this committee are continuously updated in each training round. The blockchain preserves signatures and their corresponding aggregated models of each training round which makes all records tamper-resistant and transparent to all network members. It is worth mentioning that this structure is only compatible with semi-honest trainers.\nFuture Research Directions\n==========================\nDespite the advancements of DFL in the past five years, this paradigm is fairly new and still has a lot of potential to be improved. From a security standpoint, many of the available countermeasures in the literature are either studied for FL or blockchain, and their effectiveness in DFL remains unclear. Here, we outline future development requirements that we believe will be promising for DFL in this sense.\nWeakness of Privacy-Preserving Techniques\n-----------------------------------------\nAs mentioned before, privacy-preserving techniques often trade excessive computational burden for enhancing privacy (e.g., HE, SMC). Furthermore, the implementation of cryptographic techniques such as SMC is sensitive and can imperil privacy if not defined properly. DP, on the other hand, can affect the accuracy of the aggregated model since the utilized noise can leak into the model. These issues are beside the fact that the audition phase in these techniques can lead to security holes. Hence, despite the history of privacy-preserving techniques, they all have imperfections that require further research and adaptation to the DFL paradigm.\nAddressing these challenges requires ongoing research and adaptation of privacy-preserving techniques to suit the specific requirements of DFL. Developing optimized implementations of cryptographic methods, such as SMC, can help alleviate the computational overhead and address sensitivity concerns. Paying attention to secure parameter configurations and robust cryptographic protocols is essential to minimize privacy risks. Exploring advanced noise reduction techniques within DP can strike a balance between preserving privacy and maintaining the accuracy of the model. Robust auditing mechanisms need to be established to ensure the security of privacy-preserving techniques and safeguard against potential vulnerabilities. Moreover, investigating hybrid approaches that integrate multiple privacy-preserving techniques can offer a compromise between privacy and computational efficiency.\nContinued exploration and refinement of privacy-preserving techniques in the DFL domain will enhance privacy and security while minimizing the impact on computational performance. This ongoing research and innovation will pave the way for more effective and privacy-aware DFL systems.\nOptimizing Defense Mechanism Deployment\n---------------------------------------\nDevising defense mechanisms in DFL presents specific challenges that need to be addressed. One key challenge is the requirement for additional computational power, which may not always be feasible, particularly in resource-constrained environments like mobile networks. The limited computational resources available can pose limitations on the implementation of defense mechanisms.\nAnother challenge is the diverse nature of threats in DFL. Different threats often require different countermeasures. Designing a comprehensive defense mechanism that addresses a wide range of threats can be complex and challenging. It requires analyzing various threat models, understanding their unique characteristics, and developing tailored defense strategies accordingly.\nWhen considering DFL deployed on blockchain, resource and computational constraints specific to blockchain systems must also be taken into account. The scalability of defense mechanisms becomes a critical aspect to consider. Strategies that optimize smart contract execution or leverage off-chain computations can help ensure efficient operation within the resource constraints of blockchain platforms.\nContinuous evaluation, benchmarking, and improvement of existing defense mechanisms are vital. The field of defense mechanisms in DFL is dynamic, with emerging threats and advancements in technology. Regular evaluation and improvement are necessary to keep up with evolving challenges and maintain effective protection.\nOverall, addressing the challenges of additional computational power, diverse threats, resource constraints in blockchain, and the optimal deployment of defense mechanisms are critical areas for further research and development in the field of DFL. By tackling these challenges and exploring the proposed solutions, researchers can advance the field and enhance the security and resilience of DFL systems.\nBlockchain-Related Security Issues\n----------------------------------\nThe decentralization and traceability offered by DFL introduce unique challenges related to blockchain-related threats. These threats specifically exploit the structure of the blockchain, posing risks to the overall security of DFL systems. Consequently, there is a need for research and development aiming at creating more secure blockchain structures that can enhance the resilience of DFL ecosystems against these attacks.\nOne key challenge is the susceptibility of DFL to cyber-attacks that target the underlying blockchain infrastructure. Adversaries may attempt to manipulate or tamper with the blockchain, compromising the integrity and reliability of the system. This poses a significant risk to the confidentiality and privacy of participants’ data in DFL. To address these challenges, research efforts should focus on the development of more secure blockchain structures tailored to the specific requirements of DFL. This involves exploring techniques such as improved consensus mechanisms, advanced cryptographic protocols, and enhanced smart contract design. These solutions aim to bolster the security of the blockchain layer, mitigating the vulnerabilities that can be exploited by malicious actors. Additionally, advancements in blockchain technology, such as the integration of privacy-preserving techniques and zero-knowledge proofs, can contribute to building a more attack-resilient DFL ecosystem. By leveraging techniques that enhance data privacy and confidentiality, DFL systems can better withstand attacks and protect sensitive information. Furthermore, the development of robust monitoring and auditing mechanisms is crucial for detecting and mitigating blockchain-related threats in DFL. Implementing effective monitoring systems can help identify suspicious activities and ensure the integrity of the blockchain. Auditing mechanisms can verify the correctness and security of the system, providing transparency and accountability in DFL deployments.\nHeterogeneity of Decentralized Federated Learning\n-------------------------------------------------\nDFL convergence in a distributed ecosystem (e.g., blockchain) is not guaranteed when various clients have diverse computation power (i.e., heterogeneous users). Limited processing power and poor connectivity can delay the training and communication of some nodes, delaying the global aggregation. Disregarding this heterogeneity results deteriorates the overall efficiency of DFL training.\nThe heterogeneity of clients in DFL not only poses challenges to convergence and efficiency but also gives rise to security issues. In an asynchronous DFL setting, where clients operate with different computation power and connectivity, ensuring secure aggregation becomes challenging. Asynchronous DFL is not compatible with secure aggregation protocols because all clients need to be incorporated in the aggregation step when using secure aggregation. This also limits the ability to perform centralized or distributed DP. Consequently, the use of local DP techniques becomes the viable option in asynchronous DFL scenarios, potentially compromising the level of privacy protection and increasing the risk of information leakage during the aggregation process. These security concerns highlight the need for innovative approaches that strike a balance between achieving privacy-preserving aggregation and accommodating the heterogeneity of clients in DFL. By exploring novel cryptographic techniques, adaptive privacy mechanisms, and secure communication protocols, it is possible to address the security challenges that arise from the heterogeneity of DFL and ensure the confidentiality, integrity, and privacy of the FL process.\nChallenges in Verifiable Decentralized Federated Learning\n---------------------------------------------------------\nJoining a DFL network requires a set of verifiers to evaluate the trustworthiness of the joining party. Since the joining party will not disclose their information for privacy reasons, the evaluation process often involves monitoring the performance of the joining party in the training process. Nevertheless, this evaluation scheme is not efficient as it exhausts computational and communication resources. Another challenge in this domain is to gain insight into the misbehavior of clients based on the history of their behavior. It is important to discover how the verification results can be employed to facilitate this task.\nOne possible solution is to employ privacy-preserving techniques during the trust evaluation process. Techniques such as SMC or HE can enable verifiers to evaluate the performance and trustworthiness of the joining party without directly accessing their sensitive information. This way, privacy is maintained while still obtaining the necessary insights for trust evaluation. Furthermore, optimizing the evaluation process can enhance its efficiency. Implementing techniques that reduce the computational and communication overhead, such as lightweight monitoring mechanisms or selective data aggregation, can help alleviate the resource exhaustion issues associated with the evaluation scheme. By carefully designing evaluation protocols and employing efficient algorithms, the overall efficiency of the process can be improved without compromising the evaluation’s effectiveness.\nIn terms of addressing misbehavior based on historical behavior, leveraging machine learning techniques and anomaly detection algorithms can be beneficial. By analyzing the behavior patterns and performance metrics of clients over time, it is possible to identify potential anomalies or deviations that may indicate misbehavior. Establishing a comprehensive system for tracking and monitoring client behavior, coupled with intelligent analysis and detection mechanisms, can aid in detecting and addressing misbehavior in a timely manner. Moreover, incorporating reputation systems within the DFL network can provide additional insights into the trustworthiness of participating clients. Verifiers can leverage the verification results and feedback from previous collaborations to build reputation scores for each client. These scores can serve as indicators of past behavior and can be used as a basis for decision-making during the trust evaluation process.\nConclusion\n==========\nThe integration of FL and blockchain technologies has alleviated the need for a server in the network. This is followed by a number of advantages such as efficient communication and the elimination of a single point of failure in the federation. While there are a limited number of surveys on the security analysis of FL, since the previous studies were all based on centralized architectures, further analysis is required to study the new paradigm of DFL from a security perspective. This work first reviewed common trends and preliminaries of FL and blockchain. It then performed a security analysis on DFL by identifying possible threats and defense mechanisms in such systems.\n[]Ehsan Hallaji (Graduate Student Member, IEEE) received the B.Sc. degree in software engineering from Shahid Rajaee University, Tehran, Iran, in 2015, and the M.A.Sc. degree in electrical and computer engineering from the University of Windsor, Windsor, ON, Canada, in 2018, where he is currently pursuing a Ph.D. degree with the Department of Electrical and Computer Engineering. His current research interests include machine learning, data mining, federated learning, and cybersecurity. He is a reviewer for several journals and conferences in his area of research. He also served as the Vice-Chair of the IEEE SMC Society, Windsor Section, from 2019 to 2022. \n[]Roozbeh Razavi-Far (Senior Member, IEEE) is an Assistant Professor at the Faculty of Computer Science and Canadian Institute for Cybersecurity, at the University of New Brunswick. His research focuses on machine learning, big data analytics, computational intelligence, and cybersecurity of cyber-physical systems. He has authored or co-authored more than 150 papers in scholarly journals and international conferences. Stanford lists his name among the top two percent most cited researchers for 2022. He is the recipient of several awards and grants including NSERC-DG, NSERC-ECR, NBIF, USRG and NSERC-PDF. He is an Associate Editor at several journals, including *Neurocomputing*, *Machine Learning with Applications*, *Discover Artificial Intelligence*, and IEEE Transactions on Industrial Cyber-Physical Systems. He served as a Guest Editor and Chair for several journals and peer-reviewed conferences, and the Chapter Chair of IEEE Computational Intelligence, and Systems, Man and Cybernetics Societies at Windsor Section. \n[]Mehrdad Saif(Fellow, IEEE) is a distinguished figure in the field of systems and control, with a career spanning nearly four decades. He received the B.S., M.S., and D.Eng. degrees in electrical engineering from Cleveland State University, OH, USA, in 1982, 1984, and 1987, respectively. Throughout his graduate studies, he was involved in research sponsored by NASA Lewis (now Glenn) Research Center and the Cleveland Advanced Manufacturing Program (CAMP). In 1987, Dr. Saif joined Simon Fraser University’s School of Engineering Science as an Assistant Professor. He rose through the ranks, becoming a Full Professor in 1997. From 2002 to 2011, he took on the role of Director of the School, spearheading its significant expansion during his tenure. Subsequently, he served as the Dean of the Faculty of Engineering at the University of Windsor from July 2011 until September 2021. There, he initiated major developments, including substantial enrollment growth and the addition of new programs in aerospace engineering, engineering management, B.Eng. technology, mechatronics, among others. Under his leadership, the Faculty of Engineering at University of Windsor also saw an increase in both faculty/staff numbers and research output. Dr. Saif’s research contributions are both vast and impactful, specializing in systems and control, estimation, observer theory, and AI/ML-based approaches to fault diagnostics and condition monitoring. His work has applications in a range of sectors, including automotive, power, and autonomous systems. To date, he has published over 400 refereed journal and conference papers and edited a book in these subject areas. He has garnered over 10,000 citations and holds a Google h-index of 50. Moreover, Dr. Saif is listed in Stanford University’s database of the top 100,000 career scientists from 1965 to the present. In his own area of expertise, the same database ranks him in the top 0.7%. Research.com also places him at 2681 worldwide and 139Th nationally among all electrical and electronics engineers. Dr. Saif expertise has been sought after by notable organizations such as GM, NASA, B.C. Hydro, and Canadian Space Agency (CSA). He has also significantly contributed to the IEEE Control Systems Society as the Chair of its Vancouver Section and serves on the Editorial Board of several esteemed IEEE journals such as the IEEE Access, IEEE SMC Magazine, IEEE Industrial Informatics, IEEE Transactions on Industrial and Cyber-physical Systems, among others. Dr. Saif is a Registered Professional Engineer in Ontario, Canada, and has been honored with Fellowships in several prestigious organizations, including IEEE, the Canadian Academy of Engineering (CAE), the Engineering Institute of Canada (EIC), the Institution of Engineering and Technology (IET), and Asia-Pacific Artificial Intelligence Association (AAIA). His IEEE fellowship is particularly noteworthy, earned *“for contributions to monitoring, diagnosis and prognosis in cyber-physical health systems”*.\n[]Boyu Wang (Member, IEEE) received his B.Eng. degree in Electronic Information Engineering from Tianjin University, Tianjin, China, M.Sc. degree in Electrical and Computer Engineering from University of Macau, Macau, China, and Ph.D. in Computer Science from McGill University, Montreal, QC, Canada. He is currently an Assistant Professor with the Department of Computer Science, University of Western Ontario, London, ON, Canada. He is also affiliated with the Brain and Mind Institute and the Vector Institute. He was a Post-Doctoral Research Fellow at the University of Pennsylvania and Princeton University. His research interests include machine learning theory, algorithms, and applications. \n[]Qiang Yang (Fellow, IEEE) received the B.Sc. degree in astrophysics from Peking University, Beijing, China, in 1982, and the M.Sc. degree in astrophysics and the Ph.D. degree in computer science from the University of Maryland, College Park, MD, USA, in 1985 and 1989, respectively. He was a Faculty Member with the University of Waterloo, Waterloo, ON, Canada, from 1989 to 1995, and Simon Fraser University, Burnaby, BC, Canada, from 1995 to 2001. He was the Founding Director of Huawei’s Noah’s Ark Lab, Hong Kong, from 2012 to 2014 and a Co-Founder of 4Paradigm Corporation, Beijing, an artificial intelligence (AI) platform company. He is currently the Head of the AI Department and the Chief AI Officer of WeBank, Shenzhen, China. Dr. Yang has been a Professor Emeritus with the Computer Science and Engineering Department, Hong Kong University of Science and Technology (HKUST), Hong Kong, since 2023, where he was previously a Chair Professor, and a former Head of the CSE Department and the Founding Director of the Big Data Institute from 2015 to 2018. He is the author of several books, including *Intelligent Planning* (Springer), *Crafting Your Research Future* (Morgan & Claypool), and *Constraint-Based Design Recovery for Software Engineering* (Springer). His research interests include AI, machine learning, and data mining, especially in transfer learning, automated planning, federated learning, and case-based reasoning. Dr. Yang has served as an Executive Council Member of the Advancement of AI (AAAI) from 2016 to 2020. He is a fellow of several international societies, including ACM, AAAI, IEEE, IAPR, and AAAS. He was a recipient of several awards, including the 2004/2005 ACM KDDCUP Championship, the ACM SIGKDD Distinguished Service Award in 2017, and the AAAI Innovative AI Applications Awards in 2018 and 2020. He was the Founding Editorin-Chief of the *ACM Transactions on Intelligent Systems and Technology* and IEEE Transactions on Big Data. He has served as the President of the International Joint Conference on AI (IJCAI) from 2017 to 2019. \n---\n1. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).[↩](#fnref1)\n2. Ehsan Hallaji and Mehrdad Saif are with the Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada. E-mail: hallaji@uwindsor.ca, msaif@uwindsor.ca.[↩](#fnref2)\n3. Roozbeh Razavi-Far is with the Faculty of Computer Science, University of New Brunswick, Fredericton, NB E3B 5A3, Canada, and also with the Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada. E-mail: roozbeh.razavi-far@unb.ca.[↩](#fnref3)\n4. Boyu Wang is with the Department of Computer Science and the Brain Mind Institute, University of Western Ontario, London, ON N6A 5B7, Canada. E-mail: bwang@csd.uwo.ca.[↩](#fnref4)\n5. Qiang Yang is with the Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, and also with the Department of AI, WeBank, Gaungdong Province, China. E-mail: qyang@cse.ust.hk.[↩](#fnref5)\n"},"data_id":{"kind":"string","value":"arxiv_0182401"}}},{"rowIdx":45597,"cells":{"content":{"kind":"string","value":"On Security Research TowardsFuture Mobile Network Generations\n=============================================================\nOver the last decades, numerous security and privacy issues in all three active mobile network generations have been revealed that threaten users as well as network providers. In view of the newest generation (5G) currently under development, we now have the unique opportunity to identify research directions for the next generation based on existing security and privacy issues as well as already proposed defenses. This paper aims to unify security knowledge on mobile phone networks into a comprehensive overview and to derive pressing open research questions. To achieve this systematically, we develop a methodology that categorizes known attacks by their aim, proposed defenses, underlying causes, and root causes. Further, we assess the impact and the efficacy of each attack and defense. We then apply this methodology to existing literature on attacks and defenses in all three network generations. By doing so, we identify ten causes and four root causes for attacks.\nMapping the attacks to proposed defenses and suggestions for the 5G specification enables us to uncover open research questions and challenges for the development of next-generation mobile networks. The problems of unsecured pre-authentication traffic and jamming attacks exist across all three mobile generations. They should be addressed in the future, in particular to wipe out the class of downgrade attacks and, thereby, strengthen the users’ privacy. Further advances are needed in the areas of inter-operator protocols as well as secure baseband implementations. Additionally, mitigations against denial-of-service attacks by smart protocol design represent an open research question.\nSecurity Research, Mobile Networks, GSM, UMTS, LTE, 5G, Systematization of Knowledge\nIntroduction\n============\nthe past decades, mobile communication has become an integral part of our daily life. For instance, in 2016 the mobile network comprised 4.61 billion users and the revenue of all mobile network operators totaled 1,331 billion USD . In many markets, the number of mobile Internet subscribers has outnumbered the stationary ones. A vast and diverse mobile communication and application ecosystem has emerged. These applications include private as well as business communication, and even critical infrastructures. For example, payment services, energy infrastructure, and emergency services (e.g., FirstNet ) highly depend on mobile networks. As a consequence, the reliability and security of mobile networks have become a substantial aspect of our daily lives.\nHowever, over the last years, a large body of literature has revealed numerous security and privacy issues in mobile networks. There is a broad set of attacks that affect the users’ privacy and data secrecy, the mobile network operators’ revenue, and the availability of the infrastructure. Various countermeasures against these attacks have been proposed, some of which have become security features of new mobile generations. Besides the academic community, the non-academic community also substantially contributed to the comprehension of mobile network security. Unfortunately, attacks and countermeasures were mostly considered in an isolated manner and the research efforts have not been systematized or categorized into a big picture. However, these insights are necessary to develop generic countermeasures instead of separate fixes or mitigations. For example, messages being exchanged before the authentication and key agreement is the cause of multiple attacks. Considering the attacks separately, one might not assume that this is a broader problem present in all three mobile generations.\nAs network standards tend to stay in use for decades, structural or backwards-incompatible changes are only possible for new network generations. We would like to use the window of opportunity with regards to 5G for the development of future mobile security specifications in order to eliminate insecure legacies. While considering the next mobile network generation, we systematized the research efforts of the last decades to improve and provide a basis for future security research and specifications. Since the contributions in mobile network security research are fragmented, we develop a methodology to categorize attacks and their countermeasures and thus provide an abstract overview on the topic. We project the design errors and attacks across the network generations to illustrate the specifications’ development. Furthermore, we give an outlook on future developments in mobile communication and map the extracted issues to them. Finally, we identify open research questions regarding mobile network security and point out challenges for future specifications.\nThe scope of our survey is on the technical side of mobile networks (Figure [fig:genericarchitecture]) to fill a blank space between highly researched topics. There are surveys on mobile applications such as secure messaging systems and mobile operating systems, e.g., Android. On the other hand, there are generic Internet security surveys and telephony security contributions focusing on fraud attacks . Recently, Jover pointed out 5G security challenges, but without systematizing prior work.\nIn summary, the main contributions of this article are as follows:\nWe develop a **systematization methodology** for attacks and defenses in mobile networks. Starting from security requirements, we classify the attacks by their aims. We use attack characteristics for estimating the attack impact, e.g., different attacker capabilities. Defenses characteristics help us to describe the advancement for defenses. To gain an abstract overview on the topic, we logically group technical attacks and defenses into *causes* and high-level *root causes*.\nWe **categorize and systematize** attacks and defenses on mobile networks using our systematization methodology to obtain a comprehensive picture of research in this field. To this end, we incorporate publications from the academic as well as non-academic communities to represent the big picture.\nWe derive **open research questions and challenges** building upon our systematization for further studies in both offensive and defensive work. We do this to shape the future research in the field of mobile network security. In order to achieve this, we investigate the shortcomings of existing work, the implications of future technologies, and the concrete challenges of defenses. We underline the challenges of future technologies by mapping implications of 5G technologies to open research questions.\nMobile Network Background\n=========================\nIn the following, we briefly describe the technical background of mobile networks, including an overview of the currently active generations and a generic overview of the network architecture.\nGenerations\n-----------\nOver the years, the requirements for mobile networks have shifted from rather single-purpose networks (voice service) to multi-purpose networks (data). In the following, we introduce the currently active network generations.\n **(2G, )** is the first digital mobile communication system and was designed for voice transmissions. It uses circuit-switched scheduling in which fixed slots are allocated for transmissions over the air and on network components along the transmission path. The is a packet-switched extension on top of the circuit-switched architecture.\n **(3G, Universal Mobile Telecommunications System)**: In order to meet the increasing demand for data transmissions, the next generation was optimized for data transmission on the radio layer. Additionally, UMTS added new security features such as mutual authentication and new encryption algorithms. Although the network is packet-switched in its core, voice and SMS transmissions are still offered as distinct network services.\n **(4G, Long Term Evolution)** uses a completely redesigned radio layer and a strict IP-based packet-switched architecture with guaranteed classes. In contrast to its predecessors, voice and SMS transmissions are no longer network services, but offered as IP-based services (SIP, VoIP) on top of a general-purpose IP data network. However, fallback options exists for phones or operators which do not support.\nNames and abbreviations for equivalent network components and concepts vary between the different network generations. Overall, we try to stay agnostic to the generations and access technologies. If we need to specially differentiate between different terms, we denote them with a 2G for GSM, 3G for UMTS, and 4G for LTE.\nNetwork Architecture\n--------------------\n [fig:genericarchitecture]\nFigure [fig:genericarchitecture] shows a generic network architecture including the scope of this paper. In general, the architecture consists of the following components:\n### User Equipment\n[sec:backgroundTMSI] [sec:architectureTMSI] The 4G (e.g., smartphone) is the device utilized to communicate with the network and consume its services. It comprises different components, such as the application processor that runs the mobile operating system, the graphical user interface, and all its locally installed applications. The baseband processor implements the mobile protocol stacks for multiple network generations and thereby establishes the communication with the network. The SIM2G/USIM3G,4G () directly identifies a customer and stores the authentication information as a pair of the permanent identity (, ) and the secret long-term symmetric key used for encryption and authentication. From outside, a user is identifiable and thus callable via a public phone number (called ). Besides permanent identities, for privacy reasons temporary identities are dynamically allocated to the, such as a that is used for paging and core network communication.\n### Radio Access Network\nThe purpose of the is to transmit data between the and the core network that provides service to the user. Therefore, the mobile phone establishes a radio connection to the base station (BTS2G, nodeB3G, eNodeB4G) that acts as a network access point. For mobility management, base stations are organized into cells which are in turn grouped for circuit-switched services into 2G,3G, and for packet-switched services into 2G,3G and 4G.\n### Core Network\nThe core network’s task is to manage the connection mobility and to deliver the services, e.g., phone calls and Internet connection. For this mobility management, several core network elements are utilized. A central database, the 2G or 3G,4G, stores the authentication, mapping, and other information about the users. Its security functionality is often referred as. Core network elements manage the mobility, connection, and security establishment. is used within and networks for signaling purposes such as mobility management and call setup as well as externally for roaming. was developed in the mid-1970s for landline networks and was later extended for mobile telephony networks. Unfortunately, the protocol only provides limited security mechanisms. Today, SS7 is mostly used as an SS7-over-IP adaptation. introduced new IP-based protocols for the core-network infrastructure, e.g., the SIP-based handles voice, video, and text messages.\n### Inter Network\n[sec:architecturess7] Many services require a connection to other communication networks such as the or the Internet leading to the introduction of subsystems and gateways.\nMobile networks are connected to each other via or its successor, the Diameter protocol for global inter-network operator roaming, text messages, and call forwarding. Diameter inherited most of the semantics, but offers improved authentication and confidentiality through the use of IPsec and.\n### Radio Channel\nThe radio layer shares some common design choices between GSM, UMTS, and LTE, whereas other characteristics like frequencies, modulation, or access technologies are highly individual. All generations incorporate three main types of logical channels into the physical radio channel: (i) Broadcast control channels carry information about the base station, its neighbors, and the network configuration. (ii) Paging channels are used to call out for specific UEs when the network wants to transmit data to them. (iii) Dedicated channels are used for traffic to and from each single device. These are the only channels that can be encrypted and integrity protected, if initiated by the network.\n### Pre-Authentication Traffic and Security Establishment\nUnless initiated by the network, the traffic is unencrypted, not integrity protected and, thus, not authenticated. This means that only dedicated traffic to and from a specific device is secured. Thus paging, other broadcasts, most of the radio resource allocations, and low-level signaling traffic are always unprotected. All traffic that happens before the setup of an authenticated session is defined as *pre-authentication traffic*.\n[sec:architectureenc] The authenticated session is established via an protocol which is a challenge-response protocol, that authenticates the partner(s) and derives a session key for the communication. While only establishes user authentication, newer generations (, ) establish mutual authentication. The session keys are derived from a common long-term shared secret stored in the (U)SIM. The particular used depend on the deployed SIM, the operators, and the access technology. On GSM, the example algorithm COMP128 became the de-facto standard, albeit operators could issue SIMs with a custom algorithm. In later generation key derivations are split between the and the where the publicly reviewed Milenage (and TUAK) algorithms are used. Operators are still able to customize AuC’s and USIM’s algorithms.\n### Mobility Management and Paging\nWhen no active data transmission or phone call is ongoing, the phone goes into the idle state. In this state, the network only knows the coarse Location Area where the subscriber is located. The phone listens to the paging channel as an incoming phone call, message, or data triggers a paging message of the subscriber in the Location Area. Upon receiving a paging message, the phone contacts the network and requests a dedicated (logical) channel for further communication. Thus, only if the phone switches to another Location Area (circuit switched), it has to inform the network using a *Location Update Request*. Additionally, the phone sends *periodic* location updates at a low interval (typically every 24h) to reassure the network of its continued presence. Analogue semantics exist for Routing Areas and Tracking Areas in packet-switched context. Additionally, each cell broadcasts a list of neighbor cells (e.g., their frequencies) to help the phone find these cells faster.\n### RAN Sessions and Data Tunnels\n[sec:architecturebearers] As most data services need stable addresses, tunnels are used between the UE and an IP endpoint. These tunnels hide the mobile network’s mobility management and also allow to offer multiple connections to different IP networks, such as Internet access or private networks. Tunnels terminate at the packet gateways (, 4G). If necessary, middleboxes separate the mobile network from the Internet by translating the private IP address and port to a public IP address and port. Tunnels aim at guaranteeing certain parameters, such as latency or bandwidth.\n[t!] [fig:methodology]\nMethodology of Systematization\n==============================\nIn this section, we introduce our systematization methodology that we apply to categorize attacks on mobile networks and their defenses. In the course of this article, we recognize and formulate research questions and challenges for future research based on this systematization.\nMethodology\n-----------\nWe structure the systematization process according to the flow depicted in Figure [fig:methodology], beginning with the selection of particular security requirements, continuing with the assessment of recent attacks and defenses, and resulting in a set of research questions and challenges to shape future research in mobile network security. This process allows us to incorporate multiple aspects of offensive and defensive previous work resulting in a high-level perspective on essential root causes of security issues.\n**Security requirements** define concrete features to protect mobile networks and their users. Such requirements are challenged by **attack aims**, i.e., the major interests of an attacker (Table [tab:results]). Accordingly, **attacks** are instantiations of these attack aims and exploit existing vulnerabilities in the definitions and implementations of systems or system components. In order to assess the impact of attacks on the mobile network, we use a set of **attack characteristics** that give a precise definition of an adversary’s technical and organizational capabilities, e.g., the preliminaries for an attack. We assess the scope of existing **defenses** in relation to known attack vectors (Table [tab:countermeasures]) and aggregate **defense characteristics** to assess their research and deployment status. However, multiple flaws are often manifestations of a broader problem that we define as a **cause**. Such causes facilitate the differentiation of attacks into distinct classes, thus allowing to derive open research questions and challenges in relation to the current state of the art. All causes are grouped into four fundamental **root causes**, which form the logical structure of our systematization.\nBelow, we give a short example to illustrate the application of this method.\n#### Example\n=2em *Radio Measurement Reports* are used for the maintenance of radio access networks and can be requested by a base station without authentication. This flaw can be exploited via the *Radio Measurement Report Request Attack* that allows an attacker with *active radio capabilities* to pinpoint a victim. Hence, this attacks the user’s privacy (*attack aim*) as it breaks the location confidentiality *requirement*. The attack can be executed in the non-authenticated state of UEs, allowing an active attacker to fake measurement report requests. Therefore, the *flaw* is that requests for Radio Measurement Reports are part of the unsecured pre-authentication traffic (*cause*). As this is a legitimate request according to the specification, the *root cause* lies in the specification. A proposed *mitigation* is to require this specific request to be authenticated. However, this does not fix the underlying general problem of insecure pre-authentication traffic. A *generic solution* would be to eliminate pre-authenticated traffic completely, as it is the source of many other vulnerabilities as well. An *open research question* is how to develop a privacy-preserving specification while keeping the maintainability of mobile networks.\n### Coverage of Literature\nWe focus on academic research, i.e., scientifically peer-reviewed publications, as a foundation for the assessment. In addition, we use non-academic research (including publications, presentations, and demonstrations at hacker venues) and white papers from industry. Even though these publications are not peer-reviewed, they complement the academic body of the systematization with a comprehensive picture of mobile network security. In particular, the hacker community has contributed a lot to the understanding of mobile protocols and has provided tools that academic researchers have built upon.\nWe require that the literature must present security- or privacy-related attacks and defenses and need to be unique to mobile networks and not focus solely on applications. We specifically exclude mobile operating systems security and common challenges of the public phone network. Preferentially, the literature should have a high impact, e.g., it affects many users, is operable from a large distance, or produces considerable damage.\n### Structure\nWe base the structure of our systematization (Sections [sec:rcsi]–[sec:syswc]) on the root causes, and each section is further grouped into causes. Thus, attacks that are evoked by the same cause and root cause are logically grouped together. Mitigations and solutions are discussed directly within the respective subsections. Regarding the structure of this document, we traverse the systematization process (Figure [fig:methodology]) backwards from root causes, to causes, to attacks. Research questions originate from causes, root causes, and the implications of 5G technologies.\nSecurity Requirements\n---------------------\nSecurity requirements describe the demands that need to be met by the system in order to protect the interests of its stakeholders. For our systematization, we aim to establish generic and long-lasting security requirements spanning all three mobile network generations. However, the standardization bodies, e.g.,, have issued diverging requirements over time, which is why, they do not allow us to provide a holistic view and might not fit modern security concepts. In order to define generic and long-lasting security requirements, we therefore base our work on the publication of Avizienis et al. who define a taxonomy of dependable and secure computing and general security requirements that we underpin with some more concrete requirements published by the 3GPP .\n### Confidentiality\nAvizienis et al. define confidentiality as the “*absence of unauthorized disclosure of information*”. This statement is substantiated by the 3GPP with the following requirement: “*the network shall provide several appropriate levels of user privacy including communication confidentiality, location privacy, and identity protection*”. In detail, this means :\nCommunication confidentiality: “[…] *contents, origin, and destination of a particular communication shall be protected from disclosure to unauthorized parties*”.\nIdentity protection: The network shall “*hide the identities of users from unauthorized third parties*”.\nLocation privacy: The network shall “*hide the user location from unauthorized parties*”.\n### Availability\nAvailability denotes the readiness and the continuity of correct services . With the pervasiveness of mobile communications in our everyday lives, availability becomes a crucial factor for customers as well as part of critical infrastructures .\n### System Integrity\nIn contrast to data or transmission integrity, system integrity focuses on the hard- and software of the network components. Integrity is defined as the absence of unauthorized system alterations . System integrity is an essential security requirement as it is crucial for the proper operation and trustworthiness of the system.\n### Unauthorized Service Access and Correct Charging\nThe service should only be accessible to authorized parties . This requirement includes correct recording and offsetting call data records and other chargeable items . In other words, a proper authorization and charging system should only allow subscribed services to be consumed and it should charge the *right* user for the *correct* volume .\nSubsequently, we will use these high-level security requirements to assign an attack aim to each identified attack that challenges one or more of those requirements.\nAttack Aims\n-----------\nEach attack has a clear primary aim that challenges one of the identified security requirements. An attacker might also pursue a secondary attack aim. For example, using side-channels, an attacker can obtain the shared key on the SIM card that undermines primarily the secrecy aim. However, the attacker might also clone the SIM card for free calls and thereby commit fraud attacks (secondary). We define five distinct attack aims:\n**Attacks on Privacy:** This aim covers all attacks that undermine the privacy of the user, including the *identity protection* and the *location privacy*.\n**Attacks on Secrecy:** This category includes attacks on *communication confidentiality*, e.g., the content of the transmission.\n**Denial of Service:** This attack aim contains all the objectives that impact the *availability* of services, or parts of them. Thus, *downgrade attacks*, such as disabling encryption or stepping back to less secure protocols belong here.\n**Attacks on Integrity:** This category comprises all the attempts which undermine the requirements for system integrity.\n**Fraud Attacks:** This aim covers attacks that aim towards directly or indirectly targeting financial benefits for the attacker or financial losses for others. *Direct under-billing* attacks dodge service charges at the expense of the operator, whereas *direct over-billing* produces financial loss to customers. *Indirect fraud* includes scams or spam via telephone.\nAttacks\n-------\nAttacks exploit system flaws under the defined attack aims. We use the following characteristics for an assessment of the attack impact (see Table [tab:results]). In general, as for Table [tab:results], denotes a fully applicable attack for the characteristic, refers to limitations, and characterizes attacks that are not applicable.\n### Attacker Capabilities\n[sec:attackerCapabilities] For mobile radio attacks, an attacker often combines several capabilities to perform an attack (e.g., retrieving session keys over SS7 and passively monitoring and decoding traffic); thus, we describe the attacker model as a set of distinct capabilities (i.e., building blocks). We assume that the attacker is a-priori not in possession of any *private* information (secret keys) of the victim, but might be in possession of *public* identifiers such as the phone number ().\n**Passive Radio**: An attacker with passive radio capabilities is able to capture radio transmissions, decode signals, and read raw messages. The recent developments of and re-purposed hardware render this type of attack quite affordable .\n[sec:attackerCapabilityRadioPassive]\n**Active Radio**: An attacker with this capability has full control over radio transmissions and is therefore able to put arbitrary messages on the radio channel. This enables an attacker to setup a own base station or a fully controllable phone stack using an. [sec:attackerCapabilityRadioTrasnsmit]\n**User Traffic**: The attacker is able to control or initiate traffic on a commodity mobile phone. The phone performs normal radio emissions, but the attacker accesses the higher (user-land controlled) network layers (e.g., IP) or dedicated network services (e.g., SMS). In most cases, this ability does not require a rooted or jail-broken phone.\n[sec:attackerCapabilityUserTraffic]\n**SS7/Diameter Interface**: An attacker with access to SS7/Diameter is able to send and receive or Diameter messages to and from other networks. Some network providers even sell these services. [sec:attackerCapabilitySS7]\n**Nondestructive Physical**: A nondestructive-physical attacker temporarily has physical access to the victim’s device, but neither destroys nor modifies hardware or software. Thus, the attack leaves no visible or measurable trace. We exclude the destructive attacker, because these visible traces would raise doubts by the users. [sec:attackerCapabilityPhysical]\n**PSTN Interface**: An attacker has access to voice or text services of the. [sec:attackerCapabilityPSTN]\n**Internet Traffic**: An attacker with the ability to access the Internet in a way that can specifically contact the victim’s phone. That can be achieved by knowing the phone’s public IP address and the TCP/UDP port mapping on the operator’s packet gateway. Other possibilities include identifiers of chat services, instant messaging apps, or cloud messaging services (such as or Apple’s Push Notifications ), and the ability to transfer such messages. [sec:attackerCapabilityInet]\n### Limitations of Attacker Capabilities\nFor our systematization, we assume that the operator’s authorized personnel is trusted and thus exclude such attacker capabilities from systematization. However, such attacks have occurred in the past and are a threat to the mobile user’s data secrecy and privacy. For instance, in the 2005 Vodafone Greece incident, a staff technician was suspected to have planted a backdoor in mobile switches that allowed copying traffic on government phones. In the Gemalto SIM key material theft , secret key material for the SIM cards was transferred by the use of unprotected means. However, such attacker capabilities are beyond the scope of this paper, as the attackers had the permissions in the first place and deliberately misused them.\n### Target\nThe target category depicts who is harmed by the attack, and if there is a relation to other categories, e.g., privacy attacks predominantly target the user. We focus on the primary goal and disregard secondary effects such as bad publicity due to data breaches.\n### Technology\nThis category maps the applicability of an attack to the three major access technology generations and assesses if there has been a security development, e.g., if defenses have been introduced in later access technologies, or if new technologies open new attack vectors. The former does not necessarily prevent attacks, as multiple downgrade attacks are known.\nFor example, only lacks mutual authentication (*cause*), hence it is prone to attacks. However, UMTS and LTE are open to various downgrade attacks; therefore the problem will not be resolved until phones stop to (unconditionally) support GSM. Downgrade attacks that trick or force a specific party to fall back on older and less secure access technology must be kept in mind when discussing fixes or mitigations for new access technology generations. We filed downgrade attacks as part of attacks, as they deserve a separate review.\n### Range\nThe range of an attack is an indirect indicator of impact and cost. A higher range (e.g., a globally performable attack) increases its impact and versatility and might justify higher costs for an attacker. In contrast, an attack that requires more physical vicinity increases involvement of the attacker and reduces the set of victims. In Table [tab:results], we classify the range by technical boundaries: Physical access (**Phy**), same radio cell including simulated ones (**Cell**), same location area (**LA**), same network (**Net**), and globally executable attacks (**Glo**).\nFlaw\n----\nFor our systematization, we define a flaw as a specific and distinct vulnerability that is exploited by a particular attack. We coalesced attacks that exploit the same technical flaw or are otherwise very similar in their technical or operational principle. This leads to a 1:1 relationship between flaws and attacks.\n[!pht] [tab:results]\nyes, applicable, needed for attack\npartially/supportive/optional\nno, not applicable, or does not apply\n**?** property unknown\nCause\n-----\nWe group flaws that have similar technical or organizational reasons via a common cause. A cause is a broader technical reason summarizing multiple individual flaws and, if dealt with appropriately, would foil an entire class of attacks.\nRoot cause\n----------\nRoot causes are the underlying reason for certain classes of attacks; they are defined on an abstract level and independent from technology. Each root cause summarizes particular flaws and vulnerabilities according to their structural or causal dependency (Table [tab:rootcauseoverview]) and is completely disjoint from other root causes. We use this abstract structure as the foundation for our systematization.\nDefenses\n--------\nIn order to systematize and assess the defenses presented in the literature, we relate them to attacks, causes, and root causes. We show the coverage of the suggested defenses in Table [tab:countermeasures]. If a defense encounters all the attacks of a cause, it can be considered a *generic* defense, otherwise it is a *specialized* defense. We differentiate between two kinds of defenses, namely, **detections** () and **mitigations** (). While the detection of an attack is an important step to impede losses or disadvantages, it does not ultimately prevent the attack. Mitigations foil attacks by fixing the underlying flaw or by making attacks very unlikely to succeed.\nAdditionally, we evaluated the defenses according to a set of quality characteristics that help to assess the realization effort, the sustainability, and the current research status of a defense.\n### Realization Method\nThe realization method (Table [tab:countermeasures]) specifies *how* a defense is achieved. A **specification defense** (S) needs to pass the change-request process, such as protocol changes, to guarantee the interoperability. It has a higher realization effort time-wise, as it needs to pass the specification process. However, it will potentially reach more people in the long run, as new products are likely to implement such measures. In contrast, **implementation defenses** (I) do not depend on a specification and can be implemented directly into network components. While this task can be accomplished by a vendor, only users of an updated product benefit from it.\nThe realization method is an indicator of the *effort*, the *swiftness* with which it can be rolled out, and the *reachability*.\n### Affected Components\nOn the right hand side of Table [tab:countermeasures], we denote which network components need modifications to implement a particular countermeasure. On the one hand, it indicates who needs to take action to roll-out a defense. On the other hand, as a rough estimate, the less components are affected by a defense measure, the easier it can be implemented in practice, and vice-versa. However, particular updates on the UE are hard to roll out due to the high number of units and manufacturers involved.\n### Deployment Status\nThe deployment status has more a practical than a scientific value: It helps to evaluate the feasibility of a defense, if information about its deployment is available. A defense which has been deployed by at least one operator can naturally be considered feasible. The following notation is used:\n**No information** (?). No information about deployment.\n**Not deployed** (). Not deployed by any operator.\n**Partially deployed** (). At least one operator or phone that partially implements the defense.\n**Fully deployed** (). At least one operator or phone that fully implements the countermeasure.\n### Research Status\nIn the literature, defenses are discussed at vastly different levels of detail. Some papers about attacks conclude with rather vague defense proposals, while other works focus on the concrete realization and evaluation of a defense. The research status of a defense helps us to disclose possible shortcomings of the existing work. For this, we estimated the research status by examining the detail level and evaluation degree of a defense proposal and denote three status levels in Table [tab:countermeasures]:\n**Vague Proposal** (). A vague proposal mentions a defense without any precise scheme or architecture.\n**Concrete Proposal** (). A concrete proposal is a detailed scheme for a defense. However, the scheme is not evaluated.\n**Evaluated Proposal** (). The security of the proposal was comprehensively evaluated—preferably by more then one literature source. The evaluation can be done either theoretically and/or practically.\nResearch Questions and Challenges\n---------------------------------\nFrom the assessment of attacks and countermeasures, we derived open questions and challenges that should shape future security research in context of 5G mobile security. Following the systematization structure, we defined three leading questions or challenges for each cause:\n**Shortcomings of Existing Work**. The limitations of prior research lead to individual sets of shortcomings for the different causes that we identified through our systematization. We define concrete starting points to address these shortcomings in future offensive and defensive work.\n**Concrete Challenge of Defenses**. As we systematize the defenses in the context of causes, we emphasize the problems and challenges that all proposed defenses are trying to solve. We evaluate if the defenses are successful and might encounter multiple attacks in the cause. If they are not already solved successfully or just covering a small amount of attacks in a cause, we give starting points using literature of other related disciplines in which similar problems were solved. **Security Implications of 5G**. When possible, we match these research questions onto the current 5G specification development. As the 5G is currently under development, we briefly refer to discussions and suggestions made within technical reports. This matching assists us in finding the difference between current research and the specification process. From today’s perspective, we point out security challenges of 5G technologies that should be addressed by future security research. To do this, we introduce the new technologies as well as the associated 5G challenges.\n[!t] [tab:rootcauseoverview]\n[!pht] [tab:countermeasures]\nSystematization Overview\n========================\nIn the following, we introduce the abstract *root causes* and their individual *causes* as used in this systematization (see Table [tab:rootcauseoverview]). Later, we present each cause in detail in relation to attacks, defenses, and open research questions, following the reverse process of Figure [fig:methodology]. The results regarding attacks are condensed in Table [tab:results] and as graph in Figure [fig:graphvizdiagramall] (see Appendix). Likewise, defenses are aggregated in Table [tab:countermeasures].\n*Specification Issues* originate from incomplete, inaccurate, or faulty definitions of system behavior and comprise five individual causes: *Unsecured Pre-Authentication Traffic* allows to send messages to the phone or network prior to the key agreement and ciphering setup has taken place. *Non-Existing Mutual Authentication* relates to an attack vector exclusive to networks, but is still an issue in recent technologies due to downgrade attacks. The use of *Weak Cryptography* significantly reduces the required effort for attacks on encrypted data, while *Insecure Inter-Network Protocols* undermine the users’ privacy and confidentiality by poor protocol design choices. *Resource Usage Asymmetry* enables an attacker to perform cheap requests that result in intensive operations on the network side and hence can lead to.\n*Implementation Issues* are either caused by *Insecure Implementations* that open attack vectors in components of the mobile network, which can impair the system’s integrity, or by *Leaky Implementations*, which means that sensitive information can be accessed through unintended side channels.\nThe root cause *Protocol Context Discrepancy* summarizes a class of security issues which use a protocol differently or in another environment than it was originally designed for. *Cross-Layer Information Loss* occurs at the interface of different layers in the network stack, e.g., when necessary, trustworthy security information is lost between network layers. *Accounting Policy Inconsistencies* result from different accounting schemes that can be played against each other, e.g., voice data is charged in minutes, whereas other data is charged by volume.\nThe *Wireless Channel* and its characteristics is essential for the transmission of information in mobile communication and comes with several physical limitations that impact the security.\nFollowing the structure of root causes, we discuss offensive and defensive characteristics of specific causes and derive open research questions and challenges for future mobile network technologies.\nRoot Cause: Specification Issue\n===============================\nSpecifications ensure the interoperability between implementations by specifying protocols, state machines, and interfaces. However, there may exists issues in the specification that might lead to flaws that can be exploited by attackers. In the specification-related root cause, we collate all flaws that are based on specification issues. The reasons for these problems range from technical trade-offs to political motivation.\nCause: Unsecured Pre-Authentication Traffic\n-------------------------------------------\nThe signaling traffic prior the security establishment with the protocol is unprotected: it is neither encrypted nor integrity-protected and thus unauthenticated[6](#fn6). This leads to implicit trust between the phone and the network. In this unauthenticated state, the phone fully obeys the network, even if the latter is not genuine. A malicious usage of messages in this unauthenticated state can serve for downgrade, track, or locate a specific user or handset.\n### Attacks\nOne prominent example for attacks based on unsecured pre-authentication is the deployment of *fake base stations*. Fake base stations (also known as *rogue* or *fraudulent base stations*, *IMSI Catchers*, *cell-site simulators*, a *DRT-Box*, or by product names such as *Stingray*) are active devices simulating a genuine base station to the phone by broadcasting genuine network identifiers. These fake base stations exploit the fact that mobile phones cannot verify the authenticity of the network prior to the protocol.\nIn the unauthenticated state, the attacker is allowed to ask for the permanent identity, such as the or, and can thus undermine the user’s identity and location privacy. Besides obvious requests such as the *identity request*, an attacker can also use more subtle ways to determine the vicinity of a victim, e.g., with the linkability attack. Additionally, an attacker can repeatedly page the victim’s IMSI and, thereby, determine if a user is in radio range. Moreover, an attacker can retrieve a more precise location by requesting measurement reports from the victim’s handset enabling an attacker to track a victim or to request the identity of people within radio range.\nFurthermore, unsecured pre-authentication traffic allows *downgrade attacks* to a less secure access technology by denying service using the *tracking area update reject* or a combination of other messages. This serves as a stepping stone for further attacks such as the attack (Section [sec:MITMimsicatcher]). Additionally, a fake base station can disable the location service on some phones from the late 2000s by sending out the country code of Egypt, because GPS receivers used to be forbidden in Egypt at that time ,. Furthermore, Golde et al. showed that unsecured pre-authentication uplink traffic in can be misused for a attack dropping calls in the entire location area by winning the race answering paging requests. This is a problem of the GSM state-machine specification, as it can not recover once it proceeds to the ciphering setup.\nThe missing protection of broadcast and paging messages also enables attacks that retrieve the temporary identity of a victim by triggering the paging process multiple times and statically analyzing the paged s . An attacker can trigger the paging procedure in multiple ways: for example, with targeted Internet traffic, a short phone call and immediate hang-up, e.g., before the ring starts, or with a *Silent SMS* that is a text message which is silently discarded by the phone.\n### Assessment\nAttacks based on unauthenticated uplink traffic or on passively exploitable downlink traffic are vastly outnumbered by active radio attacks based on pre-authentication traffic. While potentially having a very severe impact, an active radio attacker is limited to his/her radio vicinity. Most of these attacks undermine the victim’s data or location privacy. Many commercially available products exploit unsecured pre-authentication traffic, hence making it a high priority to be addressed.\n### Defenses\nThe research community proposed and implemented multiple detection and mitigation techniques against fake base stations. Detection schemes against the client include geographical mapping (e.g., via GPS) of the network structure to detect new—and possible fraudulent—base stations, finding unusual frequency or cell parameter configurations, and behavioral analysis of base stations. Some are implemented as smartphone apps others use dedicated smartphones with baseband firewalls. Furthermore, a sensor network can detect such changes. Recently, operator-based detection schemes were proposed. Some of these approaches also detect large-scale paging race attacks .\nBesides detection schemes, mitigations and fixes based on protocol changes have been proposed. An ephemeral identifier, e.g., dynamic IMSI or pseudonymic IMSI (P-IMSI) instead of the static IMSI has been proposed . All of them protect against an unauthorized IMSI request by providing a new, seemingly unrelated number as the identifier for each request. However, the ephemeral identifier require severe changes to the network structure as the IMSI is used as a primary key to link all the subscriber-related information in a network. Additionally, protocol changes have been proposed against the paging response race attack and the linkability attacks.\nBesides these specialized solutions, securing the paging and other broadcast traffic would eliminate the cause for all these attacks, e.g., using a source origin authentication scheme. Some proposed options are based upon asymmetric cryptography with a public key infrastructure or upon broadcast authentication schemes .\nAttacks that map a public identifier to the temporary identifier (TMSI de-anonymization) are currently countered by frequently changing TMSIs even after unsuccessful calls. Hong et al.pointed out that TMSI reallocation schemes lack randomness in the real world and remain attackable . A proper reallocation scheme must be eimplemented by the operators for encountering the threat of identity mapping attacks. The GSMMap project provides a rough estimation on the deployment of this workaround.\n### Research Questions and Challenges\nThe detection of fake base stations via the handset (e.g., behavioral analysis) or with externally deployed sensors only benefits a small number of people or a certain geographical area.\nCountermeasures based on protocol changes often have a hard time to get through specifications for current network generations, as they introduce non-backward-compatible changes. For example, dynamic identifiers such as PMSI (as a replacement for TMSI and IMSI) require deep changes in many systems at once (VLR, HLR, Packet Gateways). The importance of such measures influenced the 5G design process and some of them might be included in the new specification . However, they introduce the risk of a permanent SIM card lock if the SIM and home network HLR/HSS should get out of sync. New solutions that address this need to be sought. In contrast, moving measurement reports from an unauthenticated to an authenticated protocol state is possible in current network generations, as the functionality of currently deployed handsets is unaffected (but stay vulnerable). However, in recent years, new vulnerabilities based on pre-authentication traffic have been revealed . In conclusion, none of the proposed protocol changes abandons non-protected pre-authenticated traffic in its entirety.\nThus, a more general approach is based on abandoning pre-authentication traffic in particular – or unprotected signaling in general – completely. Two main ideas have been proposed: Schemes based on asymmetric cryptography and broadcast-authentication schemes such as the TESLA protocol. For example, 5G currently explores ways to protect base station identity signatures using asymmetric cryptography. However, both were not extensively researched in the context of mobile networks with its limited bandwidth, power consumption restrictions, and low computational USIM card. Another related question is, if the protection of signaling uplink traffic from the to the network would increase the over-all security of the system. Such a general solutions would be desirable for future network generations as to remove the entire threat class.\nNevertheless, formal analysis of proposed protocols should prove the authenticity, confidentiality, privacy, and availability requirements . This can be either accomplished by manually proving the protocols or the use of automated tools.\n[backgroundcolor=blue!5,nobreak] **Summary:** Attacks based on pre-authentication traffic affect the privacy and aim to downgrade the service. Such attacks are possible across all three mobile generations. Defenses include either attack detection or protocol changes that aim to mitigate specific attacks or abandon the entire pre-authentication traffic. Future research must focus on completely abandoning pre-authentication traffic, e.g., with asymmetric cryptography. Automated and manual inspection aid the goal of provable security by identifying shortcomings.\nCause: Non-Existing Mutual Authentication\n-----------------------------------------\nThe original specification of does not include network authentication and, thus, allows a attack. While 3G can be used in GSM if supported by all parties, no downgrade prevention exists.\nAlthough the fact of non-existing mutual authentication originally exploits a specific vulnerability of networks, they are still a relevant threat in today’s networks as the weakest-link-principle applies. Downgrade attacks via unsecured pre-authentication traffic on UMTS or LTE (Section [sec:UnsecuredPreAuthTraffic]) still allow to exploit this GSM vulnerability on modern phones. The difference to pre-authentication traffic (Section [sec:UnsecuredPreAuthTraffic]) is the lack of mutual authentication. In that sense, the non-existing mutual authentication is an extension of the unsecured pre-authentication traffic issue and has similarities in attacks and defenses with the former cause.\n### Attacks\nIf the phone cannot verify the authenticity of the network, an unconditional trust of the phone to the network and, thus, to a potential attacker is established. In a network-centric architecture, where most decisions are made by the network, an attacker faking a base station gains excessive power over the handset.\nFake base stations often employ additional techniques to keep a victim in the fake cell, such as not supplying information on neighboring cells or manipulating cell reselection thresholds . The phone behaves inconspicuously and is able to make phone calls as well as send text messages and data to the fake network. However, without any further exploit, the attacker can not gain the possession of the cryptographic keys. Still, the attacker can downgrade the communication to the null-cipher or an easily attackable cipher (see Section [sec:GSMciphers]) for passing it to the real network. In this case, the phone remains reachable for the genuine network. Alternatively, calls, SMS, and data could be forwarded with additional modems or SIP, in which case the original caller-ID is lost, and the phone is not reachable from the outside. The impact of the attacks can be increased by an attacker with SS7 capabilities, e.g., she/he can directly inject the traffic into the phone network.\nSimilar to measurement reports on LTE, the GSM radio resource location service protocol enables the network to request GPS coordinates from the phone . Developed for emergency services, most phones will answer the request even though it is not an emergency call . Additionally, the non-existing mutual authentication has been a rich source for location-based SMS spam—mainly in China . Commercially available fake base stations with ready-to-use exploits are a reminder of the urgency with which this threat should be addressed.\n### Defenses\nSimilar to unsecured pre-authentication traffic, fake base stations that exploit the non-existing mutual authentication can be detected with mobile apps, baseband firewalls, sensors, or network-based sensors . Nevertheless, mobile apps do not sufficiently protect against the threat of fake base stations . Besides that, fake base stations that send out SMS spam can be detected based on the content of the SMS and meta-information (e.g., signal strength, duration of cell contact) . Many defenses (and caveats) from unsecured pre-authentication (Section [sec:PATdefenses]) traffic also apply here.\nWhile and later network generations introduced mutual authentication, the literature urges retrofitting mutual authentication to GSM. Some of the proposed schemes need major changes in the specification , while others focus on minimizing changes to ensure a fast roll-out . Different proposals for UMTS authentication schemes over GSM exist, but the current used one does neither protect against downgrades nor integrity-protect the CipherModeCommand. Thus, giving an MitM-attacker attacker the ability to choose a weak or no cipher.\n### Research Questions and Challenges\nEven the phase-out of by some network providers does not protect users against attacks, as the phone continues to “speak”. Research can thus proceed into several directions. Modern authentication protocols could be retrofitted into GSM with a downgrade protection that prohibits legacy GSM operations if both the phone and the network can support newer methods. A downgrade protection scheme will also benefit future network generations with their security problems. For this, we refer to the causes *unsecured pre-authentication traffic* and *wireless channel*, as they are responsible for downgrade attacks.\n[backgroundcolor=blue!5,nobreak] **Summary:** GSM has no network authentication, which leads to privacy and confidentiality issues. Nevertheless, the GSM specification will not be addressed by any improvements. For future generations downgrade attacks to the insecure GSM standard open old attack vectors. Future research must suggest technologies for prevention downgrade attacks, e.g., by securing the pre-authentication traffic. Furthermore, the retrofitting network authentication to GSM provides protection in case of a downgrade attack.\nCause: Weak Cryptography\n------------------------\nCryptography provides the means to achieve data confidentiality. However, weak cryptography can lead to attacks revealing protected information. This can emerge from intentionally weakened algorithms or by evolving attack methods . Cryptographic systems should be designed following Kerckhoffs’ principle , which states that a system should only rely on the secrecy of the key, while everything else might be known to the attacker (or the public). In the following, we describe attacks that aim specifically at weak cryptography.\n[b] [tab:ciphering]\nnot attackable\nattacks with commodity hardware known\nattacks known, but not practicable or not demonstrated\n### Attacks\nCryptography is used for the encryption algorithms on the air interface, for the handover, and initial key derivation. In all these parts, we identify attacks due to the use of weak cryptography. The found attacks undermine the data confidentially requirement, either by breaking the used session key or the shared key on the SIM card.\n#### Encryption Algorithms\nTable [tab:ciphering] depicts the air interface encryption algorithms for all three generations, in particular the type of cipher, the effective key length, and if the cipher is attackable. All cipher suites in GSM except for A5/{3,4} are breakable within minutes on commodity hardware for different reasons. A5/1 is a 64-bit cipher based on three connected Linear Feedback Shift Registers (LFSR) with major cryptographic vulnerabilities that have led to passive decryption attacks. A5/2 was designed as a stripped-down export version of A5/1 with a shorter effective key length, and Goldberg et al. showed how to break this cipher in near real-time. Rainbow table approaches are further eased by the predictable padding of messages. Theoretical attacks exist against KASUMI but they are impractical in terms of space requirements, as they require 226 captured data frames and do not directly translate into A5/3. A5/3 rainbow tables similar to A5/1 were proposed but never published . GPRS ciphers GEA*x* basically mirror the weaknesses and development of their A5/*x* counterparts. In, the cipher-capabilities of the network and the user device are not integrity-protected and are therefore vulnerable to manipulation. An attacker can change the encryption handshake to block A5/3 and force a downgrade to A5/1.\nIn contrast to GSM, all UMTS and LTE ciphers underwent public development and thus followed Kerckhoffs’ principle. As a consequence, only one attack against the KASUMI based UEA1 algorithm was revealed, but still requires an unpractically large amount of captured data .\nAdditionally, each generation has a null cipher that offers no protection. Since the networks select the encryption algorithm, the user is unaware of sending data in clear text. A ciphering indicator should warn the user on the. However, just few vendors implement such a ciphering indicator .\n#### Interoperability of Access Technologies\nInteresting problems arise due the usage of same key material within the same generation or due to the interconnection of network generations. Barkan et al. describe that it is possible to downgrade to a less secure cipher for a short period of time or to reconstruct the key passively from over-the-air communication and later use it for all the other (secure) ciphers. In order to allow a GSM SIM to connect to UMTS, the key is extended to meet the UMTS key length. Also, a USIM operating on GSM will simply use a shortened key. Thus, an attack on the much weaker A5/*x* series reveals parts of the key information for other access technologies during handover. Additionally, the handover is vulnerable to the so-called “desynchronization” attack . As shown in simulations, an attacker can desynchronize the used key with the core network and, thus, an old session key is reused.\n#### Key Derivation\nWeak cryptography is also used for the initial key derivation algorithms. By reverse-engineering and breaking the COMP128v1 key derivation algorithm of GSM an attacker can reconstruct the shared secret key. In combination with side-channels of some poorly protected implementations, COMP128v1 attacks can be brought down to nearly instant key recovery (see Section [sec:leakyImplementation]). Another attack by Nohl et al. on SIM cards can remotely reconstruct the SIM’s software update key based on weak DES encryption or poorly implemented 3DES (proper implementations are safe). They leveraged the fact that the error messages concerning ciphering are sent encrypted with a known plaintext. As this attack is delivered via SMS, there are no proximity limitations to the attacker. A reconstructed OTA key enables the attacker to install new applications on the SIM card, subsequently accessing secrets stored in other applications (see Section [sec:insecureImplementation]).\n### Assessment\nAll the attacks based on weak cryptography primarily undermine the data confidentially aim of mobile networks. The attacker might also pursue a secondary aim. For example, the shared key obtained through SIM attacks can later be used to decode encrypted transmissions or write the keys on a freely programmable SIM card. Such cards can be used to impersonate a subscriber, redirect calls, change settings, or commit fee fraud. While attacks on the air interface can be executed by an attacker with passive radio capabilities, the attacks on the SIM card require physical access— thereby are thereby either limited to the radio transmission range or to the physical range. The attacks on the session keys are possible using affordable methods such as rainbow tables on an ordinary PC. is especially prone to cryptography attacks. In contrast, newer generations rely on secure algorithms following the Kerckhoffs’ principle such that these attacks are not known for now.\n### Defenses\nBecause the specification follows the best practices in newer generations, most of the defenses concentrate on the weaknesses of GSM. After weaknesses of A5/{1,2} became apparent and attacks were feasible, two new ciphers were added which are backports of the UMTS KASUMI cipher, whereas A5/3 simply pads the GSM 64-bit key to 128 bits and A5/4 uses the full 128 bits. The A5/3 usage is increasing , but as of 2017, there is no known network supporting A5/4. GSMA finally mandated the removal of A5/2 support from phones. Besides A5/2, the A5/0 was a useful downgrade target, so some networks disabled both of them . Disabling A5/1 is still not a viable option for operators.\nThe key derivations and authentication algorithms are exchangeable and also follow the best practice. MILENAGE is based on AES and replaces COMP128. TUAK, based on SHA-3 (Keccak), is another option for the authentication. It is important the algorithms is provably secure and hold strong security assumptions .\nThe introduction of new encryption algorithms for old access technologies decreases the effectiveness of attacks. However, this introduction takes a long time as software/hardware needs to be updated and new algorithms must be specified.\n### Research Questions and Challenges\nStandardization and implementation of cryptographic protocols for handover, initial key derivation, or encryption did not follow best practices at all times for various reasons. Advances in cryptanalysis have revealed various vulnerabilities in these algorithms. In the future, new algorithms need to be carefully analyzed following Kerckhoffs’ principle. Furthermore, as advances in cryptanalysis and computational power need to be factored in, provisions for possible updates of security-relevant algorithms must be build into standards while simultaneously employing anti-downgrade methods in case old and new methods or key lengths need to coexist.\nThis is especially the case for the newly introduced device-to-device (D2D) communication. In the case of direct device communication, two devices agree on a common key by using two protocols that provide no forward secrecy and rely on one master key . Such a scheme has different security implications . If asymmetric cryptography is added to 5G or future generation, this might introduce new attack surfaces if not designed and implemented carefully. Additionally, the to-be-introduced embedded SIM card comes with a complex security infrastructure and with protocols that have not yet been analyzed with respect to security. A vulnerability in the draft of the 5G allowing an attacker to impersonate a victim to the network has been found with a formal symbolic analysis . The latter example shows that new security schemes must be carefully analyzed. Future research should prove the security of all the used cryptographic techniques in the mobile context to ensure overall security.\n[backgroundcolor=blue!05,nobreak] **Summary:** Weak cryptography has led to many attacks against the data confidentiality aim. In future generations, structural changes such as device-to-device communication challenge the use of secure algorithms. The system’s security should continue to rely on well-known and proven secure cryptographic algorithms. Future research must match specified cryptographic algorithms with a realistic scope of attacker capabilities. Additionally, measures to protect against downgrade attacks to older and less secure ciphers and protocols must be developed.\nCause: Insecure Inter-Network Protocols\n---------------------------------------\nNowadays the telecommunication industry is deregulated and has been ported to an IP-based network. Both developments make easily accessible. Thus, an attacker with capabilities becomes more likely. However, for interconnectivity with the networks, messages are translated to Diameter. This makes Diameter also vulnerable to attacks, as this inter-working function does not provide authenticity. Even though Diameter was designed with security features based on protocols like TLS and IPsec, researchers found vulnerabilities in the dedicated Diameter protocol that do not rest upon the inter-working function of.\n### Attacks\nThe general idea of attacks is to request services on different layers of the home network or the serving network. As offers no authentication mechanisms, the network entities cannot decide if the request is legitimate. Thus, the entity replies properly, even though the request might not be legitimate.\nAn attacker can determine the user location on different levels of granularity—in the range of cells up to exact GPS coordinates . Additionally, an attacker can map the temporary identity () to the permanent identity () of a victim by using the system. The permanent identity can then be mapped to the public telephone number. Both attacks are not compliant with the identity confidentiality aim. The misuse of can also lead to attacks that undermine the confidentiality of calls or of text messages . This can be done by rerouting calls or by requesting the over-the-air encryption key. Besides this, the insecurity of can also be exploited for fraud attacks by unblocking a stolen device . Additionally, an attacker can run a precise attack against a distinct user by deleting subscriber data in the VLR . Attacks that are possible due to the inter-working function between Diameter and are discussed by Holtmanns et al. and Rao et al.. Even Diameter has been found vulnerable and allows to intercept text messages .\n#### Assessment\nThe insecurity of leads to a wide range of attacks. Most of them aim to undermine the (location) privacy of the user. Even commercial services were built upon the insecurity of allowing to pinpoint and track a victim. This shows that the SS7 vulnerabilities are actively used and are thereby a serious threat to users. Most of the attacks require capabilities of the attacker. However, some attacks can be accomplished by using passive radio capabilities, e.g., an attacker can decrypt the traffic as soon as the over-the-air encryption key is revealed.\n### Defenses\nThe most sustainable long-term solution is the complete elimination of SS7. With the specification of Diameter in, a more secure protocol is used for inter-networking functions. However, even Diameter is not free of flaws . Additionally, the inter-working function between SS7 and Diameter still allows attacks via Diameter based on vulnerabilities, as long as not all the network providers migrate to Diameter.\nTherefore, short-term solutions to mitigate the threats of and Diameter insecurity have been proposed. Most of them are based on validating the legitimacy of the request and then blocking the request itself or blacklisting certain classes of message types. For example, a request for the over-the-air encryption key is only allowed by a network that proves the user’s registration within its range. Furthermore, certain requests are merely of network-internal interest and are discarded at the network border, e.g., the charging of the prepaid credit. The industry provides solutions for the mobile network operators ranging from scans to stateful firewalls . Peeters et al.suggest a detection mechanism of intercepted phone calls by an redirection attack using distance bounding and timing information .\n### Research Questions and Challenges\nBy now, it is known that is an insecure protocol and the backward-compatibility of Diameter rendering also newer systems vulnerable to attacks. The exclusive use of Diameter in the (inter)-core network communication would be a step forward in terms of security, but it will not entirely solve the security problems.\nThus, the open research question is to design a protocol that is proven secure and that holds the security requirements, especially the privacy requirements in the (inter)-core network while maintaining the functionality of the mobility management. Such a protocol must withstand an exhaustive security analysis. For example, such a protocol should enforce a proof from the remote network that the subscriber is actually present and only authorize such transactions. A solution explored for 5G is to bind keys to a public key identity of the serving network. Both would prevent attacks in which an attacker sends unauthorized requests to the home network, e.g., for the session key. The means of a privacy-preserving protocol are open topics for research.\n[backgroundcolor=blue!05,nobreak] **Summary:** Insecure inter-network protocols (e.g., ) allow privacy and fraud attacks, and will not be entirely switched off in the near future. Firewalls constitute only temporary solutions to the problem. Future research is challenged to design privacy-preserving inter-network protocols that keep the maintenance overhead low.\nCause: Resource Usage Asymmetry\n-------------------------------\n*Resource usage asymmetry* occurs when an simple action on one side triggers a computationally or resource-wise expensive reaction on the other side. This—for example—leads to signaling attacks, during which an attacker misuses signaling/control messages to trigger an expensive action. Thus, the network allocates the resources within different components and may eventually run out of them after repeated or coordinated requests.\n### Attacks\nUnauthenticated messages like those used in the *attach procedure* can be utilized to overload the core network components . Additionally, they can impersonate legitimate subscribers. Similarly, Lee et al. have presented signaling attacks for 3G networks and argue that low-volume but well-timed signaling attacks can have a major impact on the network components. By misusing multiple messages for establishment and release of radio connections, the authors caused a significant increase of message load in the network. Traynor et al. evaluated network attacks targeting the 2G,3G. They found an effective method to tear down an by frequently switching the call forwarding service on and off. They suggest that a mobile phone botnet can disable the service of an.\nSimilarly, a mobile phone botnet could attack a 911 response center, which would result in an outage of emergency services . While this is not exclusively related to mobile-phone networks, the elevated priority of emergency calls makes it a unique mobile network problem: The network will drop other connections in favor of emergency calls if necessary. Enck et al. evaluated attacks considering the to open SMS functionality on the Internet. They analyze an attacker model that uses open SMS centers on the Internet to saturate the wireless link downstream from the base stations, obstructing the service in the whole cell.\n#### Assessment\nAll the attacks based on resource usage asymmetry focus on an exhaustive denial-of-service of the network. However, the impact of these attacks vary. While some attacks require active radio attacker capabilities, others already work with Internet capabilities. Besides intentional disturbance of the service, similar problems can occur due to misconfigured mobile apps or unexpected user behavior.\n### Defenses\nSo far, most suggested detection and protection methods are statistical approaches . Random connection drops might protect the network functionality as a whole, but inevitably they also deny legitimate requests. Even good statistical methods come with a non-negligible false-positive rate. The suggested protocol changes are unrealistic for currently rolled out networks.\n### Research Questions and Challenges\nAll the defenses suggest reactive schemes that come with a certain false-positive rate and do not prevent attacks. Future research should explore how to prevent resource exhaustion in the first place. This could require protocol changes and is, thus, only viable for new network generations.\nPossible approaches can be borrowed from similar problems in the context of other computer networks. RFC5013 proposes a TCP cookie against connection flood attacks. In contrast, Dwork and Back suggested a proof-of-work-based method against flooding and email spam. Before the server or network processes a request, the client has to solve a (moderately hard) computational puzzle, proving its commitment. These puzzles have to be easy to generate, easy to check, but parametrizable hard to solve (e.g., finding bits of a hash-collision). Thus, equalizing the computational load on both sides and making flood-based DoS attacks much more resource-intense for the attacker. However, such schemes have to be adopted to and evaluated in the context of mobile networks. Challenges include the limited resources on mobile devices and low-latency requirements on some operations.\n[backgroundcolor=blue!05,nobreak] **Summary:** Resource usage asymmetry allows to flood networks with signaling messages and eventually a denial of service. Future research must aim for complete attack prevention, as current state of the art research can only provide probabilistic detection. This is possible through protocol designs with balanced resource usage.\nRoot Cause: Implementation Issue\n================================\nDeviations of the implementation from the original specification can open attack vectors and, thus, can have a security impact on otherwise securely defined systems. Such deviations can be introduced on purpose, e.g., for compatibility trade-offs, or result from faulty implementations. In the following, we discuss the implications of insecure and leaky implementations.\nCause: Insecure Implementation\n------------------------------\nWhile insecure implementations can open attack vectors in deployed systems, current research mainly focuses on attacks on the baseband and SIM cards. By sending malicious data to vulnerable devices, an adversary can exploit implementation issues. In the following, we discuss how attacks undermine the system integrity, availability, secrecy, and privacy including potential countermeasures.\n### Attacks\nThe lower layers of the protocol stack run on distinct baseband processors in the. Parser errors within the baseband processor can occur due to faulty implementations of parsing modules or libraries threatening the device’s integrity. In 2016, a heap overflow in a widely used ASN.1 compiler was discovered affecting baseband implementations of multiple manufacturers. Weinmann and Golde demonstrated how to use baseband exploits to further target the application processor and its operating system.\nCrashing-only flaws in the parsing and decoding stage of text messages make the phone inoperable until the next reboot. Similar flaws on SMS parsing have been found on other processing levels .\nApart from attacks on the baseband, Nohl et al. showed that the application isolation on the SIM card is so weak that processes can access foreign data including authentication credentials. Such applications can be remotely installed after reconstructing the over-the-air (OTA) update key (see Section [sec:weakSIMcrypto]).\nImplementation flaws in the protocol state machines of the baseband result in the acceptance of a fake base station as a genuine network endangering data secrecy and privacy .\n### Assessment\nOn the one hand, we see that attacks can be launched globally and in a targeted manner that makes the impact of these flaws very high. On the other hand, the most dangerous ASN.1 heap overflow and the staged baseband-to-application-processor attacks required a fake base station with active radio capabilities and is thus locally bounded. The danger lies in the potential to take over the device at the lowest level.\n### Defenses\nIntermediate workarounds for multiple of the aforementioned attacks are based on operator-side filtering. For example, operators filter out messages that might be used to infer the OTA-key of SIM cards. Such filtering can be easily and quickly deployed by the operator. However, intermediate workarounds are typically only effective against known attacks and, thus, are not very sustainable. Furthermore, network filtering only prevents attacks coming through the network. An attacker with active radio capabilities operating a fake base station can still deliver these exploits directly to the phone, albeit with reduced range.\nMore generic defenses in the field of insecure implementation focus on the detection and prevention of insecure implementations. For SMS parsing errors as well as for state machine errors, various security testing frameworks have been proposed . These frameworks automatically test for known vulnerability patterns based on predefined test cases. While automated approaches were used to find SMS parsing errors and state machine errors, many memory corruption vulnerabilities were manually found through reverse engineering .\n### Research Questions and Challenges\nAll the attacks and mitigations stemming from *insecure implementation* have similarities to classic system security. We distinguish research questions between detection and prevention of vulnerabilities. Additionally, we discuss the shortcomings of the existing work which is the current scope.\n#### Detection of Vulnerabilities\nAlthough testing frameworks have been proposed, they usually focus on one particular type of flaw, such as SMS parsing errors, state machine failures, or particular memory vulnerabilities. Basebands have complex state machines and exhibit a fragile behavior , thus, automated testing tools based on fuzz testing have problems achieving higher levels of code and state coverage. However, alternatives such as manual reverse engineering of the baseband scale poorly and are expensive.\nTherefore, reliable detection methods for vulnerabilities in the decoding functions and state machines are needed. The decoding functions are important to protect against integrity and availability attacks. This can be supported by data for security testing that would allow better corner case testing, e.g., error states and illegal state machine transitions.\n#### Prevention of Vulnerabilities\nControl-flow hijacking, memory corruption, and state machine failures are well-known problems in the context of classic system security . However, in mobile security, classic system security defenses face certain challenges. Most notably, the real-time capability is a hard requirement for the baseband, as it needs to stay synchronized with the radio transmissions. In addition to the run-time overhead, many modern countermeasures come with a certain overhead, unreasonable for the baseband. Adapting classic system security countermeasures like memory-safe languages, memory address randomization, or control-flow integrity solutions in this constrained environment remain an open challenge .\nAnother way to reduce implementation bugs is to carefully choose the development framework based on their intrinsic security properties. Additionally, machine-readable protocol specifications and state machines would allow to generate parsers and state machines directly from the specification, cutting out the error-prone human interpretation of the specification. For parsing, part of the 3GPP specification already employs ASN.1. However, the parser libraries and compilers must be thoroughly tested and audited to avoid the fallout an ASN.1 compiler bug caused in 2016 .\n#### Current Scope\nWithin implementation security, the research community focuses mainly on the user equipment. However, it is very likely that other network components, e.g., the core network or base stations, suffer from similar vulnerabilities. For example, ASN.1 parsing is also implemented on the network side. Thus, it is not unlikely that the known ASN.1 vulnerabilities may also be present in network components. We therefore suggest the examination of network components as well.\n[backgroundcolor=blue!5,nobreak] **Summary:** Insecure implementations open attack vectors for adversaries with active radio capabilities or direct network access. Future research must provide more sustainable defenses of the classical system security context, e.g., control-flow integrity protection for basebands.\nCause: Leaky Implementation\n---------------------------\nImplementations in software and hardware can leak information about internal states in surprising or non-obvious ways. Besides using a provable secure, an implementation might leak enough information to circumvent the strong security measures due to the implementation insufficiencies.\n### Attacks\nThe SIM card stores the secret key for authentication and key derivation. Gaining access to this information breaks the security concept at its very core enabling decryption and impersonation.\nRao et al. and Zhou et al. have built a key reconstruction attack upon the cryptanalysis of Comp128v1 on SIM cards with chosen plaintexts and by using electromagnetic field probes. In 2015, Liu et al. found that the AES-based *MILENAGE* algorithm on USIM implementations is susceptible to power-based side-channel analysis and were thus able to extract the secret key.\n### Assessment\nThe primary aim of such attacks is gaining access to the secret key and, thereby, undermining the confidentially requirement. However, once the key is known to the attacker, he/she might fulfill secondary attack aims. It may enable him/her to decrypt the radio communication with passive radio capabilities or to impersonate a subscriber by cloning the SIM card.\nEven though the aforementioned attacks reveal one of the most valuable secrets in mobile networks, the attacks require temporary physical access to the SIM card. Thus, forging SIM card clones is more likely to happen through an internal attacker or through the device owners themselves than through external attackers.\n### Defenses\nThe proposed defenses for side-channel attacks are implementation-specific . The common ground for all known defenses is to have constant time and power properties, thus not leaking information about the internal state and making it unfeasible to derive the secret key by non-invasive methods.\n### Research Questions and Challenges\nThe proposed countermeasures need to be adopted to SIM cards by the industry. Clear requirements for constant time and constant power properties in the specification would help to accelerate the process of adoption. Additionally, it could be helpful to require a third party certification regarding attack resistance. If asymmetric cryptography should make it into 5G USIMs, than this will pose additional challenges to side-channel prevention.\nAn upcoming technology in the field of SIM cards is the *embedded SIM card* . Embedded SIM cards enable the configuration of the users’ credentials via the Internet and are permanently soldered into the user device. From a research perspective it is interesting to examine how embedded SIM cards are secured against side-channel attacks.\n[backgroundcolor=blue!5,nobreak] **Summary:** Leaky implementations reveal the secret key of the SIM card via unintended side-channel attacks. With the leaked key, an attacker can passively decrypt the communication or impersonate a victim. Future research needs to investigate new technologies with respect to their side-channel resistance, e.g., embedded SIM cards or asymmetric cryptography implemented on SIM cards.\nRoot Cause: Protocol Context Discrepancy\n========================================\nThis root cause is based on protocol context issues that are due to deploying a protocol that is not originally intended for the mobile network environment. Protocol properties are not harmful in a non-mobile network environment, but may be exploitable in a mobile environment if not adjusted properly.\nCause: Cross-Layer Information Loss\n-----------------------------------\nThe layering of network stacks serves multiple important purposes such as implementation transparency (e.g., upper layers do not have to care about details of lower layers) and interoperability (e.g., upper layer applications can span or exchange data over multiple networks). However, such layering also means loss of information that might be needed at higher levels, e.g., at some point, IP addresses or connections need to be mapped to the subscriber identity.\n### Attacks\nThe lack of a strong binding between radio-level authentication and IP-service authentication is the source for multiple vulnerabilities. The literature show that the implementation of such mapping is vulnerable and can be tricked with simple IP-based attacks, such as spoofing of IP addresses. IP address spoofing can be exploited for over- and under-billing attacks and to reverse the isolation of the internals to the Internet network. IPv4 and IPv6 middleboxes pose a threat to the users as well as for the mobile network operator . Similarly to the middleboxes, the rooting configuration seems to be a problem in cases that allow direct communication between two phones . Another related problem is the lack of security checks within the SIP-protocol. Manipulated SIP headers can be used to fake the caller ID with UE-originated SMS messages.\n### Assessment\nAll these attacks consider an attacker able to initiate user traffic and optional Internet traffic capabilities. Hence, all the attacks can be easily realized. The range of those attacks is network-wide, thus an attacker can be anywhere in the network and exploit the flaw. We see the trend that newer generations—especially LTE—are more prone to attacks that are based on cross-layer information loss. This happens because LTE aims to be a general-purpose network providing normal Internet connectivity, and the layering of stacks is more prominent in those networks.\n### Defenses\nHigher-level services cannot solely rely on the transport layer security measures of the lower layers and their authentication. Since an attacker can access any network communication on the device, no data from the device should be trusted. A *secure binding* between the user’s charging ID and the established connection suppresses any possible misuse . Such a secure binding operates across the separated layers. Additionally, Peng et al.suggest active *de-authorization* of a connection and a *feedback-based mischarge correction* scheme for misbilling attacks . Other mitigations built upon well-configured and maintained stateful firewalls to encounter threats due to misconfigured routers and NAT middleboxes. All defenses must be implemented at the core network by operator. While firewalls and the secure binding can by simply implemented, more advanced misbilling countermeasures, e.g., deauthorization or feedback-based mischarge correction need to be specified.\n### Research Questions and Challenges\nThe research question is, how protocols that were not originally designed for the use in mobile networks can be adapted in such a way that they prevent possible information loss across layers. For instance, instead of making it a duty of the higher service to connect the IP identity with the radio identity, some part of the core network could inject the radio identity into the IP stream. For all the countermeasures, it is important that no data from the user should be trusted, as it could be forged. However, such extensions must be carefully evaluated with respect to sustainability and performance. Additionally, currently discussed 5G additions such as software-defined networking and network virtualization, can introduce new ways for cross-layer information loss. Future research should evaluate whether new protocols introduce information losses.\n[backgroundcolor=blue!5,nobreak] **Summary:** Cross-layer information loss causes fraud or attacks and is especially exploitable within newer generations. Countermeasures propose a secure binding between the separated network layers and firewalls. Future research must carefully observe new technological proposals, e.g., 5G network virtualization to avoid cross-layer information loss in the future generations to come.\nCause: Accounting Policy Inconsistency\n--------------------------------------\nMobile networks come with a variety of billing methods. Some services are charged by time and geographical distance, others by data volume. In earlier networks, the different billing methods were straightforward to distinguish as they were based on different network services. However, data networks —such as the Internet—were originally not in mind when earlier networks were built. Another problem are transmission artifacts that occur on lower layers without the knowledge or control of higher layers, such as data retransmissions because of bad connectivity or packet loss. For example, some providers charge for TCP retransmissions while others do not. In addition, some providers have special charging policies for extra services such as music streaming. These policy inconsistencies lead to hidden channels that can be exploited for billing attacks (fraud attacks).\n### Attacks\nHidden channels for different protocols have been found, e.g., in the protocol or in TCP retransmissions , both leading to under billing attacks. Additionally, TCP retransmission can also be exploited for over-billing attacks . In this case, the attacker uses an existing connection to send unwanted TCP retransmissions to increase the victim’s data usage. With the shift from the circuit voice to a packet-based voice switching, introduced a new attack surface for under-billing attacks using the and the protocol. As voice is traditionally charged according to call duration, the voice-related channels can be misused as a hidden channel to transport data and thereby circumvent the accounting mechanism.\n### Assessment\nMost hidden channels are still exploited by an attacker with user traffic capabilities and optional Internet capabilities. Similar to the cross-layer information loss, these attacks are exploitable in the latest network generations and can be exploited everywhere in the network.\n### Defenses\nTo encounter the threat of accounting policy inconsistency, most countermeasures suggest the use of improved filtering at the gateway to detect possible misuses based on technologies like deep packet inspection, stateful protocol monitoring or ratio detection (of DNS packets or TCP retransmission) . These countermeasures need to be installed by the operators in the core network at the packet gateways to protect against revenue losses.\n### Research Questions and Challenges\nIn the future, more applications will utilize the IP connectivity for their service instead of using the special purpose services such as text messages and voice. These special purpose services originally generated a large proportion of the operators revenue. To encounter revenue losses, operators have established new accounting policies, e.g., fixed rates for music or video streaming. Future research should evaluate how such new accounting policies lead to inconsistency and thus open hidden channels for billing attacks. Effective countermeasures against these hidden channels and thus a prevention of billing attacks are remaining challenges.\n[backgroundcolor=blue!05,nobreak] **Summary:** Inconsistent accounting policies open up the possibility for hidden channels, which allow to consume resources without being charged for the service. Future research must exclude the possibility for hidden channels by an early detection of conspicuous behavior, e.g., through anomaly detection.\nRoot Cause: Wireless Channel\n============================\nThe wireless channel is essential for realizing mobility in mobile networks. However, this versatility makes the channel also easily accessible by unauthorized persons within the range of the radio transmission. Additionally, the wireless channel has limited resources. Over time more effective modulations and transmission schemes have been developed to improve the wireless transmission performance by reducing transmission redundancies. The easy access to the wireless channel makes mobile networks prone to jamming attacks for which an attacker disturbs the communication between two parties in a targeted manner. Jamming attacks are attacks and require an active radio attacker. As a result, the wireless channel is prone to several attacks and exhibits fundamental limitations such that we define it as a root cause.\nAttacks\n-------\nJamming attacks disturb the communication by increasing the noise on the wireless channel. Most prior research has concentrated on the evaluation of different constant jamming strategies and their effectiveness . While constant jamming attacks jam the entire communication bandwidth over time, smart jamming attacks are protocol-aware and intentionally jam certain control information that affect the rest of the communication. In general, smart jamming attacks are more cost-efficient. Lichtman et al. demonstrated that is particularly vulnerable to smart jamming.\n### Assessment\nAll jamming attacks require an active radio attacker who needs to be aware of the used frequencies and the bandwidth. For smart jamming attacks, the attacker requires knowledge of the protocol and needs to be synchronized with the cell to obtain the position of control information. Nevertheless, the hardware for such attacks is easily available , in particular in the form of software defined radios such as USRPs . While jamming attacks disturb the communication of all the victims, smart jamming attacks are more targeted. In all cases, the effective range of the attack is limited by the transmission power and location of the jammer. The motivation for jamming attacks is versatile. Besides simply obstructing the mobile service , jamming attacks can also serve as *downgrade attacks*.\nDefenses\n--------\nDifferent countermeasures against jamming have been proposed by the research community ranging from specification changes to smart implementations using different technologies, e.g., beamforming or spread-spectrum techniques. So far, little effort has been devoted to implement or to evaluate jamming countermeasures in mobile networks. Furthermore, it is little known about jamming countermeasure implementations within commercial products and their deployment.\nResearch Questions and Challenges\n---------------------------------\nEven though different defenses are proposed, none of them have been evaluated in detail for mobile networks. Such measures could negatively impact the transmission speed which is an important selling point for future network generations. Future research should explore the methods that were proposed or adopted by related fields and evaluate their fit and benefits to mobile network setups. The challenge is to integrate efficiently jamming countermeasures which typically linked to performance impairments, into the radio layer, still fulfilling quality of service requirements. This can be achieved by a specification that is dedicated for the use in critical networks with efficiency loss. Additional research should also consider new radio technologies like the narrowband . Hardening new generations against jamming attacks is especially important for the availability of safety-critical applications.\n[backgroundcolor=blue!5,nobreak] **Summary:** The wireless channel is open and easily accessible and, thus, can be exploited by jamming attacks. As a consequence, the adversary can impact the availability of services. So far, no strong defenses exist. Future research is challenged by the trade-off between efficient jamming countermeasures and high data rates to ensure the availability of safety-critical applications.\nRelated Surveys\n===============\nWe finally compare our work to related surveys from a methodological perspective highlighting parallels and differences.\nDifferent survey papers study a wide range of aspects of next-generation mobile networks (5G). For example, an overall survey of the performance requirements and solutions for 5G networks is given by Agiwal et al. . Whereas Taleb et al.focus on the particular use case of mobile edge computing in 5G networks . These surveys lack the focus on security in the field of next generations mobile networks.\nSecurity surveys in (mobile) phone networks focus either on one particular aspect of the system or consider just one type of attack. For example, Unger et al. focus on messaging systems and compare them based on desired security features and usability aspects. In contrast to our work, their methodology does not include attacks. Tu et al. directly map telephone spam attacks and their countermeasures, without an abstraction into causes and root causes. Acer et al. identify research issues in the area of Android security and use a methodology that directly addresses the stakeholders who might fix the issues. Our approach has the most similarities with the recently published work by Sahin et al. since they also categorize attacks and defenses into causes and root causes. However, they limit their considerations to telephony fraud. Unique to our approach is that we abstract attacks and defenses into causes and root causes for all the three mobile network generations and use this approach to derive research questions for future generations of mobile networks.\nConclusion\n==========\nIn this work, we introduced a systematization methodology for attacks and defenses in mobile networks. We derived technical causes and abstract root causes for existing vulnerabilities and discussed the impacts of attacks and defenses. We used this to derive challenges and research questions with respect to shortcomings of existing work and security implications for new 5G technologies. The results of our systematization have implications on future security research in mobile networks. We finally point to the major areas and challenges for future research on this topic.\nVulnerabilities in earlier generations of mobile networks were addressed through improvements in the following generations. However, the backward compatibility of systems and attack vectors for downgrade attacks render such vulnerabilities a continuing problem. Two factors are responsible for downgrade attacks: *unsecured pre-authentication traffic* and *openness of the wireless channel*. While protocol changes and new cryptographic methods (e.g., asymmetric cryptography) can address unsecured pre-authentication, the wireless channel requires more fundamental changes to provide security against jamming attacks. Future research must address the class of downgrade attacks to overcome these issues.\nA related problem are *insecure inter-network protocols* (e.g., or Diameter) in such a way that these legacy systems represent a threat to users as well as network providers. While firewalls constitute a temporary solution, research should develop inter-network protocols that keep the misuse potential as low as possible by minimizing the number of trusted entities.\n*Insecure implementations* of network components (e.g., smartphones or core network) are an attack vector that undermines the system’s integrity and immediately affects many users. Research should focus on securing those implementations by adopting means of classical system security while considering the requirements of the mobile network.\n*Resource usage asymmetry* led to the so-called signaling denial-of-service attacks. In future, the number of subscribers and thus the threat of such an attack increases (e.g., by a mobile phone botnet). Therefore, research should investigate protocol designs in which the resource usage is more balanced to mitigate the threat of signaling denial-of-service attacks.\nAcknowledgment\n==============\nThis work has been supported by the Franco-German BERCOM Project (FKZ: 13N13741) co-funded by the German Federal Ministry of Education and Research (BMBF) and by the DFG Research Training Group GRK 1817/1.\nThis work was also sponsored by the *COMET K1* program by the Austrian Research Promoting Agency (FFG) and the Public Employment Service Austria (AMS).\n[!p] [fig:graphvizdiagramall]\n---\n1. \\* These authors contributed equally to this work.[↩](#fnref1)\n2. David Rupprecht and Thorsten Holz are with Horst Görtz Institute for IT-Security (HGI) at Ruhr-Universität Bochum, Bochum, Germany (email: david.rupprecht@rub.de; thorsten.holz@rub.de).[↩](#fnref2)\n3. Adrian Dabrowski and Edgar Weippl are with SBA Research, Vienna, Austria (email: adabrowski@sba-research.org; eweippl@sba-research.org).[↩](#fnref3)\n4. Christina Pöpper is with New York University Abu Dhabi, Abu Dhabi, United Arab Emirates (email: christina.poepper@nyu.edu).[↩](#fnref4)\n5. Revised Version 2018-03-01[↩](#fnref5)\n6. In LTE some uplink data is integrity protected but not encrypted.[↩](#fnref6)\n\nOn Security Research TowardsFuture Mobile Network Generations\n=============================================================\nOver the last decades, numerous security and privacy issues in all three active mobile network generations have been revealed that threaten users as well as network providers. In view of the newest generation (5G) currently under development, we now have the unique opportunity to identify research directions for the next generation based on existing security and privacy issues as well as already proposed defenses. This paper aims to unify security knowledge on mobile phone networks into a comprehensive overview and to derive pressing open research questions. To achieve this systematically, we develop a methodology that categorizes known attacks by their aim, proposed defenses, underlying causes, and root causes. Further, we assess the impact and the efficacy of each attack and defense. We then apply this methodology to existing literature on attacks and defenses in all three network generations. By doing so, we identify ten causes and four root causes for attacks.\nMapping the attacks to proposed defenses and suggestions for the 5G specification enables us to uncover open research questions and challenges for the development of next-generation mobile networks. The problems of unsecured pre-authentication traffic and jamming attacks exist across all three mobile generations. They should be addressed in the future, in particular to wipe out the class of downgrade attacks and, thereby, strengthen the users’ privacy. Further advances are needed in the areas of inter-operator protocols as well as secure baseband implementations. Additionally, mitigations against denial-of-service attacks by smart protocol design represent an open research question.\nSecurity Research, Mobile Networks, GSM, UMTS, LTE, 5G, Systematization of Knowledge\nIntroduction\n============\nthe past decades, mobile communication has become an integral part of our daily life. For instance, in 2016 the mobile network comprised 4.61 billion users and the revenue of all mobile network operators totaled 1,331 billion USD . In many markets, the number of mobile Internet subscribers has outnumbered the stationary ones. A vast and diverse mobile communication and application ecosystem has emerged. These applications include private as well as business communication, and even critical infrastructures. For example, payment services, energy infrastructure, and emergency services (e.g., FirstNet ) highly depend on mobile networks. As a consequence, the reliability and security of mobile networks have become a substantial aspect of our daily lives.\nHowever, over the last years, a large body of literature has revealed numerous security and privacy issues in mobile networks. There is a broad set of attacks that affect the users’ privacy and data secrecy, the mobile network operators’ revenue, and the availability of the infrastructure. Various countermeasures against these attacks have been proposed, some of which have become security features of new mobile generations. Besides the academic community, the non-academic community also substantially contributed to the comprehension of mobile network security. Unfortunately, attacks and countermeasures were mostly considered in an isolated manner and the research efforts have not been systematized or categorized into a big picture. However, these insights are necessary to develop generic countermeasures instead of separate fixes or mitigations. For example, messages being exchanged before the authentication and key agreement is the cause of multiple attacks. Considering the attacks separately, one might not assume that this is a broader problem present in all three mobile generations.\nAs network standards tend to stay in use for decades, structural or backwards-incompatible changes are only possible for new network generations. We would like to use the window of opportunity with regards to 5G for the development of future mobile security specifications in order to eliminate insecure legacies. While considering the next mobile network generation, we systematized the research efforts of the last decades to improve and provide a basis for future security research and specifications. Since the contributions in mobile network security research are fragmented, we develop a methodology to categorize attacks and their countermeasures and thus provide an abstract overview on the topic. We project the design errors and attacks across the network generations to illustrate the specifications’ development. Furthermore, we give an outlook on future developments in mobile communication and map the extracted issues to them. Finally, we identify open research questions regarding mobile network security and point out challenges for future specifications.\nThe scope of our survey is on the technical side of mobile networks (Figure [fig:genericarchitecture]) to fill a blank space between highly researched topics. There are surveys on mobile applications such as secure messaging systems and mobile operating systems, e.g., Android. On the other hand, there are generic Internet security surveys and telephony security contributions focusing on fraud attacks . Recently, Jover pointed out 5G security challenges, but without systematizing prior work.\nIn summary, the main contributions of this article are as follows:\nWe develop a **systematization methodology** for attacks and defenses in mobile networks. Starting from security requirements, we classify the attacks by their aims. We use attack characteristics for estimating the attack impact, e.g., different attacker capabilities. Defenses characteristics help us to describe the advancement for defenses. To gain an abstract overview on the topic, we logically group technical attacks and defenses into *causes* and high-level *root causes*.\nWe **categorize and systematize** attacks and defenses on mobile networks using our systematization methodology to obtain a comprehensive picture of research in this field. To this end, we incorporate publications from the academic as well as non-academic communities to represent the big picture.\nWe derive **open research questions and challenges** building upon our systematization for further studies in both offensive and defensive work. We do this to shape the future research in the field of mobile network security. In order to achieve this, we investigate the shortcomings of existing work, the implications of future technologies, and the concrete challenges of defenses. We underline the challenges of future technologies by mapping implications of 5G technologies to open research questions.\nMobile Network Background\n=========================\nIn the following, we briefly describe the technical background of mobile networks, including an overview of the currently active generations and a generic overview of the network architecture.\nGenerations\n-----------\nOver the years, the requirements for mobile networks have shifted from rather single-purpose networks (voice service) to multi-purpose networks (data). In the following, we introduce the currently active network generations.\n **(2G, )** is the first digital mobile communication system and was designed for voice transmissions. It uses circuit-switched scheduling in which fixed slots are allocated for transmissions over the air and on network components along the transmission path. The is a packet-switched extension on top of the circuit-switched architecture.\n **(3G, Universal Mobile Telecommunications System)**: In order to meet the increasing demand for data transmissions, the next generation was optimized for data transmission on the radio layer. Additionally, UMTS added new security features such as mutual authentication and new encryption algorithms. Although the network is packet-switched in its core, voice and SMS transmissions are still offered as distinct network services.\n **(4G, Long Term Evolution)** uses a completely redesigned radio layer and a strict IP-based packet-switched architecture with guaranteed classes. In contrast to its predecessors, voice and SMS transmissions are no longer network services, but offered as IP-based services (SIP, VoIP) on top of a general-purpose IP data network. However, fallback options exists for phones or operators which do not support.\nNames and abbreviations for equivalent network components and concepts vary between the different network generations. Overall, we try to stay agnostic to the generations and access technologies. If we need to specially differentiate between different terms, we denote them with a 2G for GSM, 3G for UMTS, and 4G for LTE.\nNetwork Architecture\n--------------------\n [fig:genericarchitecture]\nFigure [fig:genericarchitecture] shows a generic network architecture including the scope of this paper. In general, the architecture consists of the following components:\n### User Equipment\n[sec:backgroundTMSI] [sec:architectureTMSI] The 4G (e.g., smartphone) is the device utilized to communicate with the network and consume its services. It comprises different components, such as the application processor that runs the mobile operating system, the graphical user interface, and all its locally installed applications. The baseband processor implements the mobile protocol stacks for multiple network generations and thereby establishes the communication with the network. The SIM2G/USIM3G,4G () directly identifies a customer and stores the authentication information as a pair of the permanent identity (, ) and the secret long-term symmetric key used for encryption and authentication. From outside, a user is identifiable and thus callable via a public phone number (called ). Besides permanent identities, for privacy reasons temporary identities are dynamically allocated to the, such as a that is used for paging and core network communication.\n### Radio Access Network\nThe purpose of the is to transmit data between the and the core network that provides service to the user. Therefore, the mobile phone establishes a radio connection to the base station (BTS2G, nodeB3G, eNodeB4G) that acts as a network access point. For mobility management, base stations are organized into cells which are in turn grouped for circuit-switched services into 2G,3G, and for packet-switched services into 2G,3G and 4G.\n### Core Network\nThe core network’s task is to manage the connection mobility and to deliver the services, e.g., phone calls and Internet connection. For this mobility management, several core network elements are utilized. A central database, the 2G or 3G,4G, stores the authentication, mapping, and other information about the users. Its security functionality is often referred as. Core network elements manage the mobility, connection, and security establishment. is used within and networks for signaling purposes such as mobility management and call setup as well as externally for roaming. was developed in the mid-1970s for landline networks and was later extended for mobile telephony networks. Unfortunately, the protocol only provides limited security mechanisms. Today, SS7 is mostly used as an SS7-over-IP adaptation. introduced new IP-based protocols for the core-network infrastructure, e.g., the SIP-based handles voice, video, and text messages.\n### Inter Network\n[sec:architecturess7] Many services require a connection to other communication networks such as the or the Internet leading to the introduction of subsystems and gateways.\nMobile networks are connected to each other via or its successor, the Diameter protocol for global inter-network operator roaming, text messages, and call forwarding. Diameter inherited most of the semantics, but offers improved authentication and confidentiality through the use of IPsec and.\n### Radio Channel\nThe radio layer shares some common design choices between GSM, UMTS, and LTE, whereas other characteristics like frequencies, modulation, or access technologies are highly individual. All generations incorporate three main types of logical channels into the physical radio channel: (i) Broadcast control channels carry information about the base station, its neighbors, and the network configuration. (ii) Paging channels are used to call out for specific UEs when the network wants to transmit data to them. (iii) Dedicated channels are used for traffic to and from each single device. These are the only channels that can be encrypted and integrity protected, if initiated by the network.\n### Pre-Authentication Traffic and Security Establishment\nUnless initiated by the network, the traffic is unencrypted, not integrity protected and, thus, not authenticated. This means that only dedicated traffic to and from a specific device is secured. Thus paging, other broadcasts, most of the radio resource allocations, and low-level signaling traffic are always unprotected. All traffic that happens before the setup of an authenticated session is defined as *pre-authentication traffic*.\n[sec:architectureenc] The authenticated session is established via an protocol which is a challenge-response protocol, that authenticates the partner(s) and derives a session key for the communication. While only establishes user authentication, newer generations (, ) establish mutual authentication. The session keys are derived from a common long-term shared secret stored in the (U)SIM. The particular used depend on the deployed SIM, the operators, and the access technology. On GSM, the example algorithm COMP128 became the de-facto standard, albeit operators could issue SIMs with a custom algorithm. In later generation key derivations are split between the and the where the publicly reviewed Milenage (and TUAK) algorithms are used. Operators are still able to customize AuC’s and USIM’s algorithms.\n### Mobility Management and Paging\nWhen no active data transmission or phone call is ongoing, the phone goes into the idle state. In this state, the network only knows the coarse Location Area where the subscriber is located. The phone listens to the paging channel as an incoming phone call, message, or data triggers a paging message of the subscriber in the Location Area. Upon receiving a paging message, the phone contacts the network and requests a dedicated (logical) channel for further communication. Thus, only if the phone switches to another Location Area (circuit switched), it has to inform the network using a *Location Update Request*. Additionally, the phone sends *periodic* location updates at a low interval (typically every 24h) to reassure the network of its continued presence. Analogue semantics exist for Routing Areas and Tracking Areas in packet-switched context. Additionally, each cell broadcasts a list of neighbor cells (e.g., their frequencies) to help the phone find these cells faster.\n### RAN Sessions and Data Tunnels\n[sec:architecturebearers] As most data services need stable addresses, tunnels are used between the UE and an IP endpoint. These tunnels hide the mobile network’s mobility management and also allow to offer multiple connections to different IP networks, such as Internet access or private networks. Tunnels terminate at the packet gateways (, 4G). If necessary, middleboxes separate the mobile network from the Internet by translating the private IP address and port to a public IP address and port. Tunnels aim at guaranteeing certain parameters, such as latency or bandwidth.\n[t!] [fig:methodology]\nMethodology of Systematization\n==============================\nIn this section, we introduce our systematization methodology that we apply to categorize attacks on mobile networks and their defenses. In the course of this article, we recognize and formulate research questions and challenges for future research based on this systematization.\nMethodology\n-----------\nWe structure the systematization process according to the flow depicted in Figure [fig:methodology], beginning with the selection of particular security requirements, continuing with the assessment of recent attacks and defenses, and resulting in a set of research questions and challenges to shape future research in mobile network security. This process allows us to incorporate multiple aspects of offensive and defensive previous work resulting in a high-level perspective on essential root causes of security issues.\n**Security requirements** define concrete features to protect mobile networks and their users. Such requirements are challenged by **attack aims**, i.e., the major interests of an attacker (Table [tab:results]). Accordingly, **attacks** are instantiations of these attack aims and exploit existing vulnerabilities in the definitions and implementations of systems or system components. In order to assess the impact of attacks on the mobile network, we use a set of **attack characteristics** that give a precise definition of an adversary’s technical and organizational capabilities, e.g., the preliminaries for an attack. We assess the scope of existing **defenses** in relation to known attack vectors (Table [tab:countermeasures]) and aggregate **defense characteristics** to assess their research and deployment status. However, multiple flaws are often manifestations of a broader problem that we define as a **cause**. Such causes facilitate the differentiation of attacks into distinct classes, thus allowing to derive open research questions and challenges in relation to the current state of the art. All causes are grouped into four fundamental **root causes**, which form the logical structure of our systematization.\nBelow, we give a short example to illustrate the application of this method.\n#### Example\n=2em *Radio Measurement Reports* are used for the maintenance of radio access networks and can be requested by a base station without authentication. This flaw can be exploited via the *Radio Measurement Report Request Attack* that allows an attacker with *active radio capabilities* to pinpoint a victim. Hence, this attacks the user’s privacy (*attack aim*) as it breaks the location confidentiality *requirement*. The attack can be executed in the non-authenticated state of UEs, allowing an active attacker to fake measurement report requests. Therefore, the *flaw* is that requests for Radio Measurement Reports are part of the unsecured pre-authentication traffic (*cause*). As this is a legitimate request according to the specification, the *root cause* lies in the specification. A proposed *mitigation* is to require this specific request to be authenticated. However, this does not fix the underlying general problem of insecure pre-authentication traffic. A *generic solution* would be to eliminate pre-authenticated traffic completely, as it is the source of many other vulnerabilities as well. An *open research question* is how to develop a privacy-preserving specification while keeping the maintainability of mobile networks.\n### Coverage of Literature\nWe focus on academic research, i.e., scientifically peer-reviewed publications, as a foundation for the assessment. In addition, we use non-academic research (including publications, presentations, and demonstrations at hacker venues) and white papers from industry. Even though these publications are not peer-reviewed, they complement the academic body of the systematization with a comprehensive picture of mobile network security. In particular, the hacker community has contributed a lot to the understanding of mobile protocols and has provided tools that academic researchers have built upon.\nWe require that the literature must present security- or privacy-related attacks and defenses and need to be unique to mobile networks and not focus solely on applications. We specifically exclude mobile operating systems security and common challenges of the public phone network. Preferentially, the literature should have a high impact, e.g., it affects many users, is operable from a large distance, or produces considerable damage.\n### Structure\nWe base the structure of our systematization (Sections [sec:rcsi]–[sec:syswc]) on the root causes, and each section is further grouped into causes. Thus, attacks that are evoked by the same cause and root cause are logically grouped together. Mitigations and solutions are discussed directly within the respective subsections. Regarding the structure of this document, we traverse the systematization process (Figure [fig:methodology]) backwards from root causes, to causes, to attacks. Research questions originate from causes, root causes, and the implications of 5G technologies.\nSecurity Requirements\n---------------------\nSecurity requirements describe the demands that need to be met by the system in order to protect the interests of its stakeholders. For our systematization, we aim to establish generic and long-lasting security requirements spanning all three mobile network generations. However, the standardization bodies, e.g.,, have issued diverging requirements over time, which is why, they do not allow us to provide a holistic view and might not fit modern security concepts. In order to define generic and long-lasting security requirements, we therefore base our work on the publication of Avizienis et al. who define a taxonomy of dependable and secure computing and general security requirements that we underpin with some more concrete requirements published by the 3GPP .\n### Confidentiality\nAvizienis et al. define confidentiality as the “*absence of unauthorized disclosure of information*”. This statement is substantiated by the 3GPP with the following requirement: “*the network shall provide several appropriate levels of user privacy including communication confidentiality, location privacy, and identity protection*”. In detail, this means :\nCommunication confidentiality: “[…] *contents, origin, and destination of a particular communication shall be protected from disclosure to unauthorized parties*”.\nIdentity protection: The network shall “*hide the identities of users from unauthorized third parties*”.\nLocation privacy: The network shall “*hide the user location from unauthorized parties*”.\n### Availability\nAvailability denotes the readiness and the continuity of correct services . With the pervasiveness of mobile communications in our everyday lives, availability becomes a crucial factor for customers as well as part of critical infrastructures .\n### System Integrity\nIn contrast to data or transmission integrity, system integrity focuses on the hard- and software of the network components. Integrity is defined as the absence of unauthorized system alterations . System integrity is an essential security requirement as it is crucial for the proper operation and trustworthiness of the system.\n### Unauthorized Service Access and Correct Charging\nThe service should only be accessible to authorized parties . This requirement includes correct recording and offsetting call data records and other chargeable items . In other words, a proper authorization and charging system should only allow subscribed services to be consumed and it should charge the *right* user for the *correct* volume .\nSubsequently, we will use these high-level security requirements to assign an attack aim to each identified attack that challenges one or more of those requirements.\nAttack Aims\n-----------\nEach attack has a clear primary aim that challenges one of the identified security requirements. An attacker might also pursue a secondary attack aim. For example, using side-channels, an attacker can obtain the shared key on the SIM card that undermines primarily the secrecy aim. However, the attacker might also clone the SIM card for free calls and thereby commit fraud attacks (secondary). We define five distinct attack aims:\n**Attacks on Privacy:** This aim covers all attacks that undermine the privacy of the user, including the *identity protection* and the *location privacy*.\n**Attacks on Secrecy:** This category includes attacks on *communication confidentiality*, e.g., the content of the transmission.\n**Denial of Service:** This attack aim contains all the objectives that impact the *availability* of services, or parts of them. Thus, *downgrade attacks*, such as disabling encryption or stepping back to less secure protocols belong here.\n**Attacks on Integrity:** This category comprises all the attempts which undermine the requirements for system integrity.\n**Fraud Attacks:** This aim covers attacks that aim towards directly or indirectly targeting financial benefits for the attacker or financial losses for others. *Direct under-billing* attacks dodge service charges at the expense of the operator, whereas *direct over-billing* produces financial loss to customers. *Indirect fraud* includes scams or spam via telephone.\nAttacks\n-------\nAttacks exploit system flaws under the defined attack aims. We use the following characteristics for an assessment of the attack impact (see Table [tab:results]). In general, as for Table [tab:results], denotes a fully applicable attack for the characteristic, refers to limitations, and characterizes attacks that are not applicable.\n### Attacker Capabilities\n[sec:attackerCapabilities] For mobile radio attacks, an attacker often combines several capabilities to perform an attack (e.g., retrieving session keys over SS7 and passively monitoring and decoding traffic); thus, we describe the attacker model as a set of distinct capabilities (i.e., building blocks). We assume that the attacker is a-priori not in possession of any *private* information (secret keys) of the victim, but might be in possession of *public* identifiers such as the phone number ().\n**Passive Radio**: An attacker with passive radio capabilities is able to capture radio transmissions, decode signals, and read raw messages. The recent developments of and re-purposed hardware render this type of attack quite affordable .\n[sec:attackerCapabilityRadioPassive]\n**Active Radio**: An attacker with this capability has full control over radio transmissions and is therefore able to put arbitrary messages on the radio channel. This enables an attacker to setup a own base station or a fully controllable phone stack using an. [sec:attackerCapabilityRadioTrasnsmit]\n**User Traffic**: The attacker is able to control or initiate traffic on a commodity mobile phone. The phone performs normal radio emissions, but the attacker accesses the higher (user-land controlled) network layers (e.g., IP) or dedicated network services (e.g., SMS). In most cases, this ability does not require a rooted or jail-broken phone.\n[sec:attackerCapabilityUserTraffic]\n**SS7/Diameter Interface**: An attacker with access to SS7/Diameter is able to send and receive or Diameter messages to and from other networks. Some network providers even sell these services. [sec:attackerCapabilitySS7]\n**Nondestructive Physical**: A nondestructive-physical attacker temporarily has physical access to the victim’s device, but neither destroys nor modifies hardware or software. Thus, the attack leaves no visible or measurable trace. We exclude the destructive attacker, because these visible traces would raise doubts by the users. [sec:attackerCapabilityPhysical]\n**PSTN Interface**: An attacker has access to voice or text services of the. [sec:attackerCapabilityPSTN]\n**Internet Traffic**: An attacker with the ability to access the Internet in a way that can specifically contact the victim’s phone. That can be achieved by knowing the phone’s public IP address and the TCP/UDP port mapping on the operator’s packet gateway. Other possibilities include identifiers of chat services, instant messaging apps, or cloud messaging services (such as or Apple’s Push Notifications ), and the ability to transfer such messages. [sec:attackerCapabilityInet]\n### Limitations of Attacker Capabilities\nFor our systematization, we assume that the operator’s authorized personnel is trusted and thus exclude such attacker capabilities from systematization. However, such attacks have occurred in the past and are a threat to the mobile user’s data secrecy and privacy. For instance, in the 2005 Vodafone Greece incident, a staff technician was suspected to have planted a backdoor in mobile switches that allowed copying traffic on government phones. In the Gemalto SIM key material theft , secret key material for the SIM cards was transferred by the use of unprotected means. However, such attacker capabilities are beyond the scope of this paper, as the attackers had the permissions in the first place and deliberately misused them.\n### Target\nThe target category depicts who is harmed by the attack, and if there is a relation to other categories, e.g., privacy attacks predominantly target the user. We focus on the primary goal and disregard secondary effects such as bad publicity due to data breaches.\n### Technology\nThis category maps the applicability of an attack to the three major access technology generations and assesses if there has been a security development, e.g., if defenses have been introduced in later access technologies, or if new technologies open new attack vectors. The former does not necessarily prevent attacks, as multiple downgrade attacks are known.\nFor example, only lacks mutual authentication (*cause*), hence it is prone to attacks. However, UMTS and LTE are open to various downgrade attacks; therefore the problem will not be resolved until phones stop to (unconditionally) support GSM. Downgrade attacks that trick or force a specific party to fall back on older and less secure access technology must be kept in mind when discussing fixes or mitigations for new access technology generations. We filed downgrade attacks as part of attacks, as they deserve a separate review.\n### Range\nThe range of an attack is an indirect indicator of impact and cost. A higher range (e.g., a globally performable attack) increases its impact and versatility and might justify higher costs for an attacker. In contrast, an attack that requires more physical vicinity increases involvement of the attacker and reduces the set of victims. In Table [tab:results], we classify the range by technical boundaries: Physical access (**Phy**), same radio cell including simulated ones (**Cell**), same location area (**LA**), same network (**Net**), and globally executable attacks (**Glo**).\nFlaw\n----\nFor our systematization, we define a flaw as a specific and distinct vulnerability that is exploited by a particular attack. We coalesced attacks that exploit the same technical flaw or are otherwise very similar in their technical or operational principle. This leads to a 1:1 relationship between flaws and attacks.\n[!pht] [tab:results]\nyes, applicable, needed for attack\npartially/supportive/optional\nno, not applicable, or does not apply\n**?** property unknown\nCause\n-----\nWe group flaws that have similar technical or organizational reasons via a common cause. A cause is a broader technical reason summarizing multiple individual flaws and, if dealt with appropriately, would foil an entire class of attacks.\nRoot cause\n----------\nRoot causes are the underlying reason for certain classes of attacks; they are defined on an abstract level and independent from technology. Each root cause summarizes particular flaws and vulnerabilities according to their structural or causal dependency (Table [tab:rootcauseoverview]) and is completely disjoint from other root causes. We use this abstract structure as the foundation for our systematization.\nDefenses\n--------\nIn order to systematize and assess the defenses presented in the literature, we relate them to attacks, causes, and root causes. We show the coverage of the suggested defenses in Table [tab:countermeasures]. If a defense encounters all the attacks of a cause, it can be considered a *generic* defense, otherwise it is a *specialized* defense. We differentiate between two kinds of defenses, namely, **detections** () and **mitigations** (). While the detection of an attack is an important step to impede losses or disadvantages, it does not ultimately prevent the attack. Mitigations foil attacks by fixing the underlying flaw or by making attacks very unlikely to succeed.\nAdditionally, we evaluated the defenses according to a set of quality characteristics that help to assess the realization effort, the sustainability, and the current research status of a defense.\n### Realization Method\nThe realization method (Table [tab:countermeasures]) specifies *how* a defense is achieved. A **specification defense** (S) needs to pass the change-request process, such as protocol changes, to guarantee the interoperability. It has a higher realization effort time-wise, as it needs to pass the specification process. However, it will potentially reach more people in the long run, as new products are likely to implement such measures. In contrast, **implementation defenses** (I) do not depend on a specification and can be implemented directly into network components. While this task can be accomplished by a vendor, only users of an updated product benefit from it.\nThe realization method is an indicator of the *effort*, the *swiftness* with which it can be rolled out, and the *reachability*.\n### Affected Components\nOn the right hand side of Table [tab:countermeasures], we denote which network components need modifications to implement a particular countermeasure. On the one hand, it indicates who needs to take action to roll-out a defense. On the other hand, as a rough estimate, the less components are affected by a defense measure, the easier it can be implemented in practice, and vice-versa. However, particular updates on the UE are hard to roll out due to the high number of units and manufacturers involved.\n### Deployment Status\nThe deployment status has more a practical than a scientific value: It helps to evaluate the feasibility of a defense, if information about its deployment is available. A defense which has been deployed by at least one operator can naturally be considered feasible. The following notation is used:\n**No information** (?). No information about deployment.\n**Not deployed** (). Not deployed by any operator.\n**Partially deployed** (). At least one operator or phone that partially implements the defense.\n**Fully deployed** (). At least one operator or phone that fully implements the countermeasure.\n### Research Status\nIn the literature, defenses are discussed at vastly different levels of detail. Some papers about attacks conclude with rather vague defense proposals, while other works focus on the concrete realization and evaluation of a defense. The research status of a defense helps us to disclose possible shortcomings of the existing work. For this, we estimated the research status by examining the detail level and evaluation degree of a defense proposal and denote three status levels in Table [tab:countermeasures]:\n**Vague Proposal** (). A vague proposal mentions a defense without any precise scheme or architecture.\n**Concrete Proposal** (). A concrete proposal is a detailed scheme for a defense. However, the scheme is not evaluated.\n**Evaluated Proposal** (). The security of the proposal was comprehensively evaluated—preferably by more then one literature source. The evaluation can be done either theoretically and/or practically.\nResearch Questions and Challenges\n---------------------------------\nFrom the assessment of attacks and countermeasures, we derived open questions and challenges that should shape future security research in context of 5G mobile security. Following the systematization structure, we defined three leading questions or challenges for each cause:\n**Shortcomings of Existing Work**. The limitations of prior research lead to individual sets of shortcomings for the different causes that we identified through our systematization. We define concrete starting points to address these shortcomings in future offensive and defensive work.\n**Concrete Challenge of Defenses**. As we systematize the defenses in the context of causes, we emphasize the problems and challenges that all proposed defenses are trying to solve. We evaluate if the defenses are successful and might encounter multiple attacks in the cause. If they are not already solved successfully or just covering a small amount of attacks in a cause, we give starting points using literature of other related disciplines in which similar problems were solved. **Security Implications of 5G**. When possible, we match these research questions onto the current 5G specification development. As the 5G is currently under development, we briefly refer to discussions and suggestions made within technical reports. This matching assists us in finding the difference between current research and the specification process. From today’s perspective, we point out security challenges of 5G technologies that should be addressed by future security research. To do this, we introduce the new technologies as well as the associated 5G challenges.\n[!t] [tab:rootcauseoverview]\n[!pht] [tab:countermeasures]\nSystematization Overview\n========================\nIn the following, we introduce the abstract *root causes* and their individual *causes* as used in this systematization (see Table [tab:rootcauseoverview]). Later, we present each cause in detail in relation to attacks, defenses, and open research questions, following the reverse process of Figure [fig:methodology]. The results regarding attacks are condensed in Table [tab:results] and as graph in Figure [fig:graphvizdiagramall] (see Appendix). Likewise, defenses are aggregated in Table [tab:countermeasures].\n*Specification Issues* originate from incomplete, inaccurate, or faulty definitions of system behavior and comprise five individual causes: *Unsecured Pre-Authentication Traffic* allows to send messages to the phone or network prior to the key agreement and ciphering setup has taken place. *Non-Existing Mutual Authentication* relates to an attack vector exclusive to networks, but is still an issue in recent technologies due to downgrade attacks. The use of *Weak Cryptography* significantly reduces the required effort for attacks on encrypted data, while *Insecure Inter-Network Protocols* undermine the users’ privacy and confidentiality by poor protocol design choices. *Resource Usage Asymmetry* enables an attacker to perform cheap requests that result in intensive operations on the network side and hence can lead to.\n*Implementation Issues* are either caused by *Insecure Implementations* that open attack vectors in components of the mobile network, which can impair the system’s integrity, or by *Leaky Implementations*, which means that sensitive information can be accessed through unintended side channels.\nThe root cause *Protocol Context Discrepancy* summarizes a class of security issues which use a protocol differently or in another environment than it was originally designed for. *Cross-Layer Information Loss* occurs at the interface of different layers in the network stack, e.g., when necessary, trustworthy security information is lost between network layers. *Accounting Policy Inconsistencies* result from different accounting schemes that can be played against each other, e.g., voice data is charged in minutes, whereas other data is charged by volume.\nThe *Wireless Channel* and its characteristics is essential for the transmission of information in mobile communication and comes with several physical limitations that impact the security.\nFollowing the structure of root causes, we discuss offensive and defensive characteristics of specific causes and derive open research questions and challenges for future mobile network technologies.\nRoot Cause: Specification Issue\n===============================\nSpecifications ensure the interoperability between implementations by specifying protocols, state machines, and interfaces. However, there may exists issues in the specification that might lead to flaws that can be exploited by attackers. In the specification-related root cause, we collate all flaws that are based on specification issues. The reasons for these problems range from technical trade-offs to political motivation.\nCause: Unsecured Pre-Authentication Traffic\n-------------------------------------------\nThe signaling traffic prior the security establishment with the protocol is unprotected: it is neither encrypted nor integrity-protected and thus unauthenticated[6](#fn6). This leads to implicit trust between the phone and the network. In this unauthenticated state, the phone fully obeys the network, even if the latter is not genuine. A malicious usage of messages in this unauthenticated state can serve for downgrade, track, or locate a specific user or handset.\n### Attacks\nOne prominent example for attacks based on unsecured pre-authentication is the deployment of *fake base stations*. Fake base stations (also known as *rogue* or *fraudulent base stations*, *IMSI Catchers*, *cell-site simulators*, a *DRT-Box*, or by product names such as *Stingray*) are active devices simulating a genuine base station to the phone by broadcasting genuine network identifiers. These fake base stations exploit the fact that mobile phones cannot verify the authenticity of the network prior to the protocol.\nIn the unauthenticated state, the attacker is allowed to ask for the permanent identity, such as the or, and can thus undermine the user’s identity and location privacy. Besides obvious requests such as the *identity request*, an attacker can also use more subtle ways to determine the vicinity of a victim, e.g., with the linkability attack. Additionally, an attacker can repeatedly page the victim’s IMSI and, thereby, determine if a user is in radio range. Moreover, an attacker can retrieve a more precise location by requesting measurement reports from the victim’s handset enabling an attacker to track a victim or to request the identity of people within radio range.\nFurthermore, unsecured pre-authentication traffic allows *downgrade attacks* to a less secure access technology by denying service using the *tracking area update reject* or a combination of other messages. This serves as a stepping stone for further attacks such as the attack (Section [sec:MITMimsicatcher]). Additionally, a fake base station can disable the location service on some phones from the late 2000s by sending out the country code of Egypt, because GPS receivers used to be forbidden in Egypt at that time ,. Furthermore, Golde et al. showed that unsecured pre-authentication uplink traffic in can be misused for a attack dropping calls in the entire location area by winning the race answering paging requests. This is a problem of the GSM state-machine specification, as it can not recover once it proceeds to the ciphering setup.\nThe missing protection of broadcast and paging messages also enables attacks that retrieve the temporary identity of a victim by triggering the paging process multiple times and statically analyzing the paged s . An attacker can trigger the paging procedure in multiple ways: for example, with targeted Internet traffic, a short phone call and immediate hang-up, e.g., before the ring starts, or with a *Silent SMS* that is a text message which is silently discarded by the phone.\n### Assessment\nAttacks based on unauthenticated uplink traffic or on passively exploitable downlink traffic are vastly outnumbered by active radio attacks based on pre-authentication traffic. While potentially having a very severe impact, an active radio attacker is limited to his/her radio vicinity. Most of these attacks undermine the victim’s data or location privacy. Many commercially available products exploit unsecured pre-authentication traffic, hence making it a high priority to be addressed.\n### Defenses\nThe research community proposed and implemented multiple detection and mitigation techniques against fake base stations. Detection schemes against the client include geographical mapping (e.g., via GPS) of the network structure to detect new—and possible fraudulent—base stations, finding unusual frequency or cell parameter configurations, and behavioral analysis of base stations. Some are implemented as smartphone apps others use dedicated smartphones with baseband firewalls. Furthermore, a sensor network can detect such changes. Recently, operator-based detection schemes were proposed. Some of these approaches also detect large-scale paging race attacks .\nBesides detection schemes, mitigations and fixes based on protocol changes have been proposed. An ephemeral identifier, e.g., dynamic IMSI or pseudonymic IMSI (P-IMSI) instead of the static IMSI has been proposed . All of them protect against an unauthorized IMSI request by providing a new, seemingly unrelated number as the identifier for each request. However, the ephemeral identifier require severe changes to the network structure as the IMSI is used as a primary key to link all the subscriber-related information in a network. Additionally, protocol changes have been proposed against the paging response race attack and the linkability attacks.\nBesides these specialized solutions, securing the paging and other broadcast traffic would eliminate the cause for all these attacks, e.g., using a source origin authentication scheme. Some proposed options are based upon asymmetric cryptography with a public key infrastructure or upon broadcast authentication schemes .\nAttacks that map a public identifier to the temporary identifier (TMSI de-anonymization) are currently countered by frequently changing TMSIs even after unsuccessful calls. Hong et al.pointed out that TMSI reallocation schemes lack randomness in the real world and remain attackable . A proper reallocation scheme must be eimplemented by the operators for encountering the threat of identity mapping attacks. The GSMMap project provides a rough estimation on the deployment of this workaround.\n### Research Questions and Challenges\nThe detection of fake base stations via the handset (e.g., behavioral analysis) or with externally deployed sensors only benefits a small number of people or a certain geographical area.\nCountermeasures based on protocol changes often have a hard time to get through specifications for current network generations, as they introduce non-backward-compatible changes. For example, dynamic identifiers such as PMSI (as a replacement for TMSI and IMSI) require deep changes in many systems at once (VLR, HLR, Packet Gateways). The importance of such measures influenced the 5G design process and some of them might be included in the new specification . However, they introduce the risk of a permanent SIM card lock if the SIM and home network HLR/HSS should get out of sync. New solutions that address this need to be sought. In contrast, moving measurement reports from an unauthenticated to an authenticated protocol state is possible in current network generations, as the functionality of currently deployed handsets is unaffected (but stay vulnerable). However, in recent years, new vulnerabilities based on pre-authentication traffic have been revealed . In conclusion, none of the proposed protocol changes abandons non-protected pre-authenticated traffic in its entirety.\nThus, a more general approach is based on abandoning pre-authentication traffic in particular – or unprotected signaling in general – completely. Two main ideas have been proposed: Schemes based on asymmetric cryptography and broadcast-authentication schemes such as the TESLA protocol. For example, 5G currently explores ways to protect base station identity signatures using asymmetric cryptography. However, both were not extensively researched in the context of mobile networks with its limited bandwidth, power consumption restrictions, and low computational USIM card. Another related question is, if the protection of signaling uplink traffic from the to the network would increase the over-all security of the system. Such a general solutions would be desirable for future network generations as to remove the entire threat class.\nNevertheless, formal analysis of proposed protocols should prove the authenticity, confidentiality, privacy, and availability requirements . This can be either accomplished by manually proving the protocols or the use of automated tools.\n[backgroundcolor=blue!5,nobreak] **Summary:** Attacks based on pre-authentication traffic affect the privacy and aim to downgrade the service. Such attacks are possible across all three mobile generations. Defenses include either attack detection or protocol changes that aim to mitigate specific attacks or abandon the entire pre-authentication traffic. Future research must focus on completely abandoning pre-authentication traffic, e.g., with asymmetric cryptography. Automated and manual inspection aid the goal of provable security by identifying shortcomings.\nCause: Non-Existing Mutual Authentication\n-----------------------------------------\nThe original specification of does not include network authentication and, thus, allows a attack. While 3G can be used in GSM if supported by all parties, no downgrade prevention exists.\nAlthough the fact of non-existing mutual authentication originally exploits a specific vulnerability of networks, they are still a relevant threat in today’s networks as the weakest-link-principle applies. Downgrade attacks via unsecured pre-authentication traffic on UMTS or LTE (Section [sec:UnsecuredPreAuthTraffic]) still allow to exploit this GSM vulnerability on modern phones. The difference to pre-authentication traffic (Section [sec:UnsecuredPreAuthTraffic]) is the lack of mutual authentication. In that sense, the non-existing mutual authentication is an extension of the unsecured pre-authentication traffic issue and has similarities in attacks and defenses with the former cause.\n### Attacks\nIf the phone cannot verify the authenticity of the network, an unconditional trust of the phone to the network and, thus, to a potential attacker is established. In a network-centric architecture, where most decisions are made by the network, an attacker faking a base station gains excessive power over the handset.\nFake base stations often employ additional techniques to keep a victim in the fake cell, such as not supplying information on neighboring cells or manipulating cell reselection thresholds . The phone behaves inconspicuously and is able to make phone calls as well as send text messages and data to the fake network. However, without any further exploit, the attacker can not gain the possession of the cryptographic keys. Still, the attacker can downgrade the communication to the null-cipher or an easily attackable cipher (see Section [sec:GSMciphers]) for passing it to the real network. In this case, the phone remains reachable for the genuine network. Alternatively, calls, SMS, and data could be forwarded with additional modems or SIP, in which case the original caller-ID is lost, and the phone is not reachable from the outside. The impact of the attacks can be increased by an attacker with SS7 capabilities, e.g., she/he can directly inject the traffic into the phone network.\nSimilar to measurement reports on LTE, the GSM radio resource location service protocol enables the network to request GPS coordinates from the phone . Developed for emergency services, most phones will answer the request even though it is not an emergency call . Additionally, the non-existing mutual authentication has been a rich source for location-based SMS spam—mainly in China . Commercially available fake base stations with ready-to-use exploits are a reminder of the urgency with which this threat should be addressed.\n### Defenses\nSimilar to unsecured pre-authentication traffic, fake base stations that exploit the non-existing mutual authentication can be detected with mobile apps, baseband firewalls, sensors, or network-based sensors . Nevertheless, mobile apps do not sufficiently protect against the threat of fake base stations . Besides that, fake base stations that send out SMS spam can be detected based on the content of the SMS and meta-information (e.g., signal strength, duration of cell contact) . Many defenses (and caveats) from unsecured pre-authentication (Section [sec:PATdefenses]) traffic also apply here.\nWhile and later network generations introduced mutual authentication, the literature urges retrofitting mutual authentication to GSM. Some of the proposed schemes need major changes in the specification , while others focus on minimizing changes to ensure a fast roll-out . Different proposals for UMTS authentication schemes over GSM exist, but the current used one does neither protect against downgrades nor integrity-protect the CipherModeCommand. Thus, giving an MitM-attacker attacker the ability to choose a weak or no cipher.\n### Research Questions and Challenges\nEven the phase-out of by some network providers does not protect users against attacks, as the phone continues to “speak”. Research can thus proceed into several directions. Modern authentication protocols could be retrofitted into GSM with a downgrade protection that prohibits legacy GSM operations if both the phone and the network can support newer methods. A downgrade protection scheme will also benefit future network generations with their security problems. For this, we refer to the causes *unsecured pre-authentication traffic* and *wireless channel*, as they are responsible for downgrade attacks.\n[backgroundcolor=blue!5,nobreak] **Summary:** GSM has no network authentication, which leads to privacy and confidentiality issues. Nevertheless, the GSM specification will not be addressed by any improvements. For future generations downgrade attacks to the insecure GSM standard open old attack vectors. Future research must suggest technologies for prevention downgrade attacks, e.g., by securing the pre-authentication traffic. Furthermore, the retrofitting network authentication to GSM provides protection in case of a downgrade attack.\nCause: Weak Cryptography\n------------------------\nCryptography provides the means to achieve data confidentiality. However, weak cryptography can lead to attacks revealing protected information. This can emerge from intentionally weakened algorithms or by evolving attack methods . Cryptographic systems should be designed following Kerckhoffs’ principle , which states that a system should only rely on the secrecy of the key, while everything else might be known to the attacker (or the public). In the following, we describe attacks that aim specifically at weak cryptography.\n[b] [tab:ciphering]\nnot attackable\nattacks with commodity hardware known\nattacks known, but not practicable or not demonstrated\n### Attacks\nCryptography is used for the encryption algorithms on the air interface, for the handover, and initial key derivation. In all these parts, we identify attacks due to the use of weak cryptography. The found attacks undermine the data confidentially requirement, either by breaking the used session key or the shared key on the SIM card.\n#### Encryption Algorithms\nTable [tab:ciphering] depicts the air interface encryption algorithms for all three generations, in particular the type of cipher, the effective key length, and if the cipher is attackable. All cipher suites in GSM except for A5/{3,4} are breakable within minutes on commodity hardware for different reasons. A5/1 is a 64-bit cipher based on three connected Linear Feedback Shift Registers (LFSR) with major cryptographic vulnerabilities that have led to passive decryption attacks. A5/2 was designed as a stripped-down export version of A5/1 with a shorter effective key length, and Goldberg et al. showed how to break this cipher in near real-time. Rainbow table approaches are further eased by the predictable padding of messages. Theoretical attacks exist against KASUMI but they are impractical in terms of space requirements, as they require 226 captured data frames and do not directly translate into A5/3. A5/3 rainbow tables similar to A5/1 were proposed but never published . GPRS ciphers GEA*x* basically mirror the weaknesses and development of their A5/*x* counterparts. In, the cipher-capabilities of the network and the user device are not integrity-protected and are therefore vulnerable to manipulation. An attacker can change the encryption handshake to block A5/3 and force a downgrade to A5/1.\nIn contrast to GSM, all UMTS and LTE ciphers underwent public development and thus followed Kerckhoffs’ principle. As a consequence, only one attack against the KASUMI based UEA1 algorithm was revealed, but still requires an unpractically large amount of captured data .\nAdditionally, each generation has a null cipher that offers no protection. Since the networks select the encryption algorithm, the user is unaware of sending data in clear text. A ciphering indicator should warn the user on the. However, just few vendors implement such a ciphering indicator .\n#### Interoperability of Access Technologies\nInteresting problems arise due the usage of same key material within the same generation or due to the interconnection of network generations. Barkan et al. describe that it is possible to downgrade to a less secure cipher for a short period of time or to reconstruct the key passively from over-the-air communication and later use it for all the other (secure) ciphers. In order to allow a GSM SIM to connect to UMTS, the key is extended to meet the UMTS key length. Also, a USIM operating on GSM will simply use a shortened key. Thus, an attack on the much weaker A5/*x* series reveals parts of the key information for other access technologies during handover. Additionally, the handover is vulnerable to the so-called “desynchronization” attack . As shown in simulations, an attacker can desynchronize the used key with the core network and, thus, an old session key is reused.\n#### Key Derivation\nWeak cryptography is also used for the initial key derivation algorithms. By reverse-engineering and breaking the COMP128v1 key derivation algorithm of GSM an attacker can reconstruct the shared secret key. In combination with side-channels of some poorly protected implementations, COMP128v1 attacks can be brought down to nearly instant key recovery (see Section [sec:leakyImplementation]). Another attack by Nohl et al. on SIM cards can remotely reconstruct the SIM’s software update key based on weak DES encryption or poorly implemented 3DES (proper implementations are safe). They leveraged the fact that the error messages concerning ciphering are sent encrypted with a known plaintext. As this attack is delivered via SMS, there are no proximity limitations to the attacker. A reconstructed OTA key enables the attacker to install new applications on the SIM card, subsequently accessing secrets stored in other applications (see Section [sec:insecureImplementation]).\n### Assessment\nAll the attacks based on weak cryptography primarily undermine the data confidentially aim of mobile networks. The attacker might also pursue a secondary aim. For example, the shared key obtained through SIM attacks can later be used to decode encrypted transmissions or write the keys on a freely programmable SIM card. Such cards can be used to impersonate a subscriber, redirect calls, change settings, or commit fee fraud. While attacks on the air interface can be executed by an attacker with passive radio capabilities, the attacks on the SIM card require physical access— thereby are thereby either limited to the radio transmission range or to the physical range. The attacks on the session keys are possible using affordable methods such as rainbow tables on an ordinary PC. is especially prone to cryptography attacks. In contrast, newer generations rely on secure algorithms following the Kerckhoffs’ principle such that these attacks are not known for now.\n### Defenses\nBecause the specification follows the best practices in newer generations, most of the defenses concentrate on the weaknesses of GSM. After weaknesses of A5/{1,2} became apparent and attacks were feasible, two new ciphers were added which are backports of the UMTS KASUMI cipher, whereas A5/3 simply pads the GSM 64-bit key to 128 bits and A5/4 uses the full 128 bits. The A5/3 usage is increasing , but as of 2017, there is no known network supporting A5/4. GSMA finally mandated the removal of A5/2 support from phones. Besides A5/2, the A5/0 was a useful downgrade target, so some networks disabled both of them . Disabling A5/1 is still not a viable option for operators.\nThe key derivations and authentication algorithms are exchangeable and also follow the best practice. MILENAGE is based on AES and replaces COMP128. TUAK, based on SHA-3 (Keccak), is another option for the authentication. It is important the algorithms is provably secure and hold strong security assumptions .\nThe introduction of new encryption algorithms for old access technologies decreases the effectiveness of attacks. However, this introduction takes a long time as software/hardware needs to be updated and new algorithms must be specified.\n### Research Questions and Challenges\nStandardization and implementation of cryptographic protocols for handover, initial key derivation, or encryption did not follow best practices at all times for various reasons. Advances in cryptanalysis have revealed various vulnerabilities in these algorithms. In the future, new algorithms need to be carefully analyzed following Kerckhoffs’ principle. Furthermore, as advances in cryptanalysis and computational power need to be factored in, provisions for possible updates of security-relevant algorithms must be build into standards while simultaneously employing anti-downgrade methods in case old and new methods or key lengths need to coexist.\nThis is especially the case for the newly introduced device-to-device (D2D) communication. In the case of direct device communication, two devices agree on a common key by using two protocols that provide no forward secrecy and rely on one master key . Such a scheme has different security implications . If asymmetric cryptography is added to 5G or future generation, this might introduce new attack surfaces if not designed and implemented carefully. Additionally, the to-be-introduced embedded SIM card comes with a complex security infrastructure and with protocols that have not yet been analyzed with respect to security. A vulnerability in the draft of the 5G allowing an attacker to impersonate a victim to the network has been found with a formal symbolic analysis . The latter example shows that new security schemes must be carefully analyzed. Future research should prove the security of all the used cryptographic techniques in the mobile context to ensure overall security.\n[backgroundcolor=blue!05,nobreak] **Summary:** Weak cryptography has led to many attacks against the data confidentiality aim. In future generations, structural changes such as device-to-device communication challenge the use of secure algorithms. The system’s security should continue to rely on well-known and proven secure cryptographic algorithms. Future research must match specified cryptographic algorithms with a realistic scope of attacker capabilities. Additionally, measures to protect against downgrade attacks to older and less secure ciphers and protocols must be developed.\nCause: Insecure Inter-Network Protocols\n---------------------------------------\nNowadays the telecommunication industry is deregulated and has been ported to an IP-based network. Both developments make easily accessible. Thus, an attacker with capabilities becomes more likely. However, for interconnectivity with the networks, messages are translated to Diameter. This makes Diameter also vulnerable to attacks, as this inter-working function does not provide authenticity. Even though Diameter was designed with security features based on protocols like TLS and IPsec, researchers found vulnerabilities in the dedicated Diameter protocol that do not rest upon the inter-working function of.\n### Attacks\nThe general idea of attacks is to request services on different layers of the home network or the serving network. As offers no authentication mechanisms, the network entities cannot decide if the request is legitimate. Thus, the entity replies properly, even though the request might not be legitimate.\nAn attacker can determine the user location on different levels of granularity—in the range of cells up to exact GPS coordinates . Additionally, an attacker can map the temporary identity () to the permanent identity () of a victim by using the system. The permanent identity can then be mapped to the public telephone number. Both attacks are not compliant with the identity confidentiality aim. The misuse of can also lead to attacks that undermine the confidentiality of calls or of text messages . This can be done by rerouting calls or by requesting the over-the-air encryption key. Besides this, the insecurity of can also be exploited for fraud attacks by unblocking a stolen device . Additionally, an attacker can run a precise attack against a distinct user by deleting subscriber data in the VLR . Attacks that are possible due to the inter-working function between Diameter and are discussed by Holtmanns et al. and Rao et al.. Even Diameter has been found vulnerable and allows to intercept text messages .\n#### Assessment\nThe insecurity of leads to a wide range of attacks. Most of them aim to undermine the (location) privacy of the user. Even commercial services were built upon the insecurity of allowing to pinpoint and track a victim. This shows that the SS7 vulnerabilities are actively used and are thereby a serious threat to users. Most of the attacks require capabilities of the attacker. However, some attacks can be accomplished by using passive radio capabilities, e.g., an attacker can decrypt the traffic as soon as the over-the-air encryption key is revealed.\n### Defenses\nThe most sustainable long-term solution is the complete elimination of SS7. With the specification of Diameter in, a more secure protocol is used for inter-networking functions. However, even Diameter is not free of flaws . Additionally, the inter-working function between SS7 and Diameter still allows attacks via Diameter based on vulnerabilities, as long as not all the network providers migrate to Diameter.\nTherefore, short-term solutions to mitigate the threats of and Diameter insecurity have been proposed. Most of them are based on validating the legitimacy of the request and then blocking the request itself or blacklisting certain classes of message types. For example, a request for the over-the-air encryption key is only allowed by a network that proves the user’s registration within its range. Furthermore, certain requests are merely of network-internal interest and are discarded at the network border, e.g., the charging of the prepaid credit. The industry provides solutions for the mobile network operators ranging from scans to stateful firewalls . Peeters et al.suggest a detection mechanism of intercepted phone calls by an redirection attack using distance bounding and timing information .\n### Research Questions and Challenges\nBy now, it is known that is an insecure protocol and the backward-compatibility of Diameter rendering also newer systems vulnerable to attacks. The exclusive use of Diameter in the (inter)-core network communication would be a step forward in terms of security, but it will not entirely solve the security problems.\nThus, the open research question is to design a protocol that is proven secure and that holds the security requirements, especially the privacy requirements in the (inter)-core network while maintaining the functionality of the mobility management. Such a protocol must withstand an exhaustive security analysis. For example, such a protocol should enforce a proof from the remote network that the subscriber is actually present and only authorize such transactions. A solution explored for 5G is to bind keys to a public key identity of the serving network. Both would prevent attacks in which an attacker sends unauthorized requests to the home network, e.g., for the session key. The means of a privacy-preserving protocol are open topics for research.\n[backgroundcolor=blue!05,nobreak] **Summary:** Insecure inter-network protocols (e.g., ) allow privacy and fraud attacks, and will not be entirely switched off in the near future. Firewalls constitute only temporary solutions to the problem. Future research is challenged to design privacy-preserving inter-network protocols that keep the maintenance overhead low.\nCause: Resource Usage Asymmetry\n-------------------------------\n*Resource usage asymmetry* occurs when an simple action on one side triggers a computationally or resource-wise expensive reaction on the other side. This—for example—leads to signaling attacks, during which an attacker misuses signaling/control messages to trigger an expensive action. Thus, the network allocates the resources within different components and may eventually run out of them after repeated or coordinated requests.\n### Attacks\nUnauthenticated messages like those used in the *attach procedure* can be utilized to overload the core network components . Additionally, they can impersonate legitimate subscribers. Similarly, Lee et al. have presented signaling attacks for 3G networks and argue that low-volume but well-timed signaling attacks can have a major impact on the network components. By misusing multiple messages for establishment and release of radio connections, the authors caused a significant increase of message load in the network. Traynor et al. evaluated network attacks targeting the 2G,3G. They found an effective method to tear down an by frequently switching the call forwarding service on and off. They suggest that a mobile phone botnet can disable the service of an.\nSimilarly, a mobile phone botnet could attack a 911 response center, which would result in an outage of emergency services . While this is not exclusively related to mobile-phone networks, the elevated priority of emergency calls makes it a unique mobile network problem: The network will drop other connections in favor of emergency calls if necessary. Enck et al. evaluated attacks considering the to open SMS functionality on the Internet. They analyze an attacker model that uses open SMS centers on the Internet to saturate the wireless link downstream from the base stations, obstructing the service in the whole cell.\n#### Assessment\nAll the attacks based on resource usage asymmetry focus on an exhaustive denial-of-service of the network. However, the impact of these attacks vary. While some attacks require active radio attacker capabilities, others already work with Internet capabilities. Besides intentional disturbance of the service, similar problems can occur due to misconfigured mobile apps or unexpected user behavior.\n### Defenses\nSo far, most suggested detection and protection methods are statistical approaches . Random connection drops might protect the network functionality as a whole, but inevitably they also deny legitimate requests. Even good statistical methods come with a non-negligible false-positive rate. The suggested protocol changes are unrealistic for currently rolled out networks.\n### Research Questions and Challenges\nAll the defenses suggest reactive schemes that come with a certain false-positive rate and do not prevent attacks. Future research should explore how to prevent resource exhaustion in the first place. This could require protocol changes and is, thus, only viable for new network generations.\nPossible approaches can be borrowed from similar problems in the context of other computer networks. RFC5013 proposes a TCP cookie against connection flood attacks. In contrast, Dwork and Back suggested a proof-of-work-based method against flooding and email spam. Before the server or network processes a request, the client has to solve a (moderately hard) computational puzzle, proving its commitment. These puzzles have to be easy to generate, easy to check, but parametrizable hard to solve (e.g., finding bits of a hash-collision). Thus, equalizing the computational load on both sides and making flood-based DoS attacks much more resource-intense for the attacker. However, such schemes have to be adopted to and evaluated in the context of mobile networks. Challenges include the limited resources on mobile devices and low-latency requirements on some operations.\n[backgroundcolor=blue!05,nobreak] **Summary:** Resource usage asymmetry allows to flood networks with signaling messages and eventually a denial of service. Future research must aim for complete attack prevention, as current state of the art research can only provide probabilistic detection. This is possible through protocol designs with balanced resource usage.\nRoot Cause: Implementation Issue\n================================\nDeviations of the implementation from the original specification can open attack vectors and, thus, can have a security impact on otherwise securely defined systems. Such deviations can be introduced on purpose, e.g., for compatibility trade-offs, or result"},"data_id":{"kind":"string","value":"arxiv_0182403"}}},{"rowIdx":45598,"cells":{"content":{"kind":"string","value":"Thermodynamics of light\n=======================\nIntroduction\n============\nThese notes where written as a running documentation of a project whose goal was to explore the thermodynamical properties of light in a cavity. The notes does not make claims of originality with respect to any the topics that is discussed, perhaps with the exception og 2.1.4, 2.1.5 and 5.4.2, but we believe that it is useful to collect together the basic descriptive tools needed to discuss thermal fields in cavities together with a nontrivial example in one place with a consistent notation and point of view. We surely would have been happy to discover these notes while we where starting up our project. The inspiration of our project and hence these notes was papers published on the subject by Masud Masuripur and Pin Han,. In addition to this introduction, the notes consists of four separate sections.\nIn the second section we introduce the thermodynamical formalism that we will be using. The formalism is of course well known and has been around for at least 150 years, but we include a derivation here because we will use the opportunity to remind the readers that the foundation of the subject, which has been argued over for at least 100 years has, starting in the 1950s, largely been resolved.\nIt has of course been known since the seminal work on kinetic theory starting in 1843 with Waterston and continuing by Helmholtz, Clausius and Maxwell, Boltzman and last but not least Gibbs, that the foundation of thermodynamics is statistical mechanics. In fact, it was Gibbs who lay much of the foundation for how we think about the subject today, in particular he was the first to observe that a certain variational principle is hiding at the heart of statistical mechanics.\nThere has however been a long running controversy over exactly how statistical mechanics provide a justification for the thermodynamical formalism. The key problems that have been argued over is how statistical mechanics can be used to justify the assumption of thermodynamic equilibrium and the second law. The problems has been vigorously pursued by scientists of the highest caliber and the controversy has over the years spawned entire new fields of mathematics, like ergodic theory and dynamical system theory. One of the important foundational problem for thermodynamics, the equilibrium assumption, was however not fully resolved until the full ramifications of Shannons seminal paper on a mathematical theory of communication from 1948, was realized. Many scientists took part in the effort of founding thermodynamics on the mathematical theory of information that grew out of Shannon’s 1948 paper. However, in this effort E. T. Jaynes stands out. He both initiated the effort in 1957 and he was also integral to the effort right up until he died in 1998, leaving his monumental book on the subject unfinished. There however exists a well written two-volume textbook on statistical mechanics by W. T. Grandy which is using the information theoretical foundation throughout the text.\nA careful reader might register that I am a little guarded when I speak about the resolution of the foundational problems of thermodynamics through information theory; I say that the nature of the equilibrium assumption has been fully understood through information theory, I don’t say that the whole foundational problem of thermodynamics has been resolved through information theory. This is because in my opinion the foundation of the second law of thermodynamics has not been resolved in this way. Papers by both Jaynes, Grandy and also others introduce arguments that purport to show that the second law can indeed be derived from information theory, but I find these arguments unconvincing.\nIn the third section we describe the quantization of free electromagnetic fields in a class of cavities that will be the focus of these notes. Field quantization is in general an immensely technically demanding and subtle part of physics. However, most of these technicalities and subtleties only come into play for interacting gauge fields, like the fields defining the Standard Model of elementary particle physics. The quantization of the electromagnetic field in a cavity is by comparison technically and conceptually simple. In fact this was the first kind of field quantization that was achieved and it’s basic idea of identifying the shape of photons, the quanta of the electromagnetic field, with classical electromagnetic cavity modes is an idea that permeate the whole area of quantum field theory. We include a detailed, but simple, account of the quantization of the electromagnetic field in cavities because it is beautiful and fundamental, but also in order to fix our notation.\nIn the fourth section our focus is on the thermodynamics of the free quantum electromagnetic field in the class of cavities described in section three. The state of the field will thus always be given by the density operator for a noninteracting bosonic many particle system. The mathematical description of this density operator is well known from the literature, but we nevertheless include a detailed description of the density operator. The reason we do this is both to fix the notation we will be using when we study specific cavities in section five, but also because, in our humble opinion, the notation used to describe the bosonic many particle density operator is often excessive and confusing. We introduce a somewhat unusual notation for the number state basis for the bosonic Fock space which we believe is more compact and less confusing than the standard one.\nIn the fifth and last section we apply the formalism developed in the four first section to investigate thermodynamical properties of the free electromagnetic field in certain specific cavities.\nThermodynamics\n==============\nThe maximum entropy principle for classical systems\n---------------------------------------------------\nLet *x*1, ..., *x**n* be random variables with an associated probability distribution *ρ*(*x*1, ..., *x**n*). Let *f*1(*x*1, ..., *x**n*), ..., *f**p*(*x*1, ..., *x**n*) be functions defined on the space of random variables Ω = {(*x*1, *x*2, ..., *x**n*)} where the variables *x**n* can run over a finite set, an infinite discrete set, for example a set indexed by a finite set of integers, or the variables can run over the real numbers. We will usually think about the real number case and will therefore write integrals instead of sums. The functions *f**j* are our *observables*. Their $\\it{expectation}$ values are as usual defined by\n \n$$\\begin{aligned}\n\\left<f\\_j\\right> \\; = \\int\\_{\\mathbf{R}^n} dV \\; f\\_j(x\\_1,...,x\\_n) \\; \\rho(x\\_1,...,x\\_n). {\\label{eq: 75}} \\end{aligned}$$ \nThe expectation value of a given observable of course depends on which probability distribution, *ρ*, we use. The challenge in statistics is to figure out which probability distribution one should use in any given situation. Let us say that we for some reason, (expert knowledge,guesswork, hearsay,...) believe that a probability distribution *ρ*0 accurately represents what we currently know about a given system. The probability distribution *ρ*0 is called the *prior distribution*, or just the *prior*.\nLet us next assume that we measure the mean values of the observables *f*1, ..., *f**p* and find the values *c*1, ..., *c**p*. If\n \n$$\\begin{aligned}\n\\left<f\\_j\\right>\\_0 \\; = \\int\\_{\\mathbf{R}^n} dV \\; f\\_j \\; (x\\_1,...,x\\_n) \\; \\rho\\_0 (x\\_1,...,x\\_n) = c\\_j, {\\label{eq: 76}} \\end{aligned}$$ \nwe are satisfied with our choice of prior. It predicts exactly the mean values that are observed. \nBut we might not be so lucky. Perhaps\n \n$$\\begin{aligned}\n\\left<f\\_j\\right>\\_0 \\; \\ne c\\_j, {\\label{eq: 77}} \\end{aligned}$$ \nfor at least one *j*. Our selected *ρ*0 is then not the correct one, it predicts expectation values that are not observed. The challenge is to modify *ρ*0 into a new distribution *ρ* that is consistent with *all* the observed mean values.\nFor this purpose we define a functional *S*(*ρ*) by\n \n$$\\begin{aligned}\nS(\\rho) = - \\int\\_{\\mathbf{R}^n} dV \\; \\rho \\; \\ln (\\frac{\\rho}{\\rho\\_0}). {\\label{eq: 78}} \\end{aligned}$$ \n*S* is by definition the *relative entropy* of the probability distribution *ρ* with respect to *ρ*0. We will see later that our use of the word entropy here is consistent with its usage in thermodynamics.\nThe *maximum entropy* principle states that one should choose the probability distribution that maximizes the functional\n \n$$\\begin{aligned}\nS(\\rho) = - \\int\\_{\\mathbf{R}^n} dV \\; \\rho \\; \\ln (\\frac{\\rho}{\\rho\\_0}), {\\label{eq: 79}} \\end{aligned}$$ \nsubject to the constraints\n \n$$\\begin{aligned}\n\\left<f\\_j\\right> \\; = \\int\\_{\\mathbf{R}^n} dV \\; f\\_j \\; \\rho = c\\_j,\\;\\; j=1,2,...,p. {\\label{eq: 80}}\\end{aligned}$$ \n \n### The general thermodynamical formalism\nIn this section we will solve the maximum principle stated in the previous section using the calculus of variations. The problem will initially be solved in the general setting described in the previous section but we will eventually specialize to the case of statistical mechanics.\nIn order to proceed we must first recognize that in additional to the *p* constraints ([eq: 80]), we have one more constraint that simply expresses the fact that *ρ* is a probability distribution.\n \n$$\\begin{aligned}\n\\left<1\\right> \\; = \\int\\_{\\mathbf{R}^n} dV \\; \\rho (x\\_1,...,x\\_n) = 1, {\\label{eq: 206}} \\end{aligned}$$ \nand we thus have *p* + 1 constraints and therefore introduce an extended functional\n \n$$\\begin{aligned}\nT(\\rho) = S(\\rho) - \\lambda\\_0 \\left<1\\right> - \\mathlarger{\\sum}^p\\_{j=1} \\lambda\\_j \\; \\left<f\\_j\\right>, {\\label{eq: 207}} \\end{aligned}$$ \nNote that we could have written\n \n$$\\begin{aligned}\nT (\\rho) = S(\\rho) - \\lambda\\_0 \\; ( \\left<1\\right> - 1) - \\mathlarger{\\sum}^p\\_{j=1} \\lambda\\_j \\; ( \\left<f\\_j\\right> - c\\_j), {\\label{eq: 208}} \\end{aligned}$$ \nin order to make the values of the constraints explicit. However, all constant terms vanish when we take variational derivative, so we might as well drop the constant terms. Also note that our choice of minus sign in front of the Lagrange multiplier terms in ([eq: 207]) and ([eq: 208]) is a convention inspired by the application of this formalism to the case of statistical mechanics.\nThe integral density corresponding to the extended functional *T*(*ρ*) is\n \n$$\\begin{aligned}\n\\mathcal{L} = - \\rho \\; \\ln (\\frac{\\rho}{\\rho\\_0}) - \\lambda\\_0 \\; \\rho - \\mathlarger{\\sum}^p\\_{j=1} \\lambda\\_j \\; f\\_j \\; \\rho {\\label{eq: 209}} \\end{aligned}$$ \nObserve that L does not depend on any derivatives of *ρ*. The Euler-Lagrange equation for *T* is therefore simply\n \n$$\\begin{aligned}\n\\frac{\\partial \\mathcal{L}}{\\partial \\rho} &= 0, {\\label{eq: 210}} \\\\ \n&\\Updownarrow \\nonumber \\\\\n& - \\ln (\\frac{\\rho}{\\rho\\_0}) - 1 - \\lambda\\_0 - \\mathlarger{\\sum}^p\\_{j=1} \\lambda\\_j f\\_j=0, \\nonumber\\end{aligned}$$ \nwhose solution is\n \n$$\\begin{aligned}\n\\rho &= \\frac{\\rho\\_0}{Z} \\; \\exp{-\\mathlarger{\\sum}\\_j \\lambda\\_j \\; f\\_j}, \\nonumber \\end{aligned}$$ \nwhere we have defined *Z* = exp(1 + *λ*0). In order for the constraint $\\left<1\\right> = 1$ to be satisfied, we must have\n \n$$\\begin{aligned}\n\\left<1\\right> &= 1, \\nonumber \\\\\n&\\Updownarrow \\nonumber \\\\ {\\int\\_{\\mathbf{R}^n}}dV \\; \\frac{\\rho\\_0}{Z} \\; \\exp{-\\mathlarger{\\sum}\\_j \\lambda\\_j \\; f\\_j} &= 1, \\nonumber \\\\ \n&\\Updownarrow \\nonumber \\\\ Z = Z(\\lambda\\_1,...,\\lambda\\_p) &= {\\int\\_{\\mathbf{R}^n}}dV \\; \\rho\\_0 \\; \\exp{-\\mathlarger{\\sum}\\_j \\; \\lambda\\_j \\; f\\_j}, {\\label{eq: 211}} \\end{aligned}$$ \nand the stationary distribution is\n \n$$\\begin{aligned}\n\\rho(x\\_1,...,x\\_n) = \\frac{\\rho\\_0(x\\_1,...,x\\_n)}{Z(\\lambda\\_1,...,\\lambda\\_n)} \\; \\exp{-\\mathlarger{\\sum}^p\\_{j=1} \\lambda\\_j \\; f\\_j (x\\_1,...,x\\_n)}. {\\label{eq: 212}} \\end{aligned}$$ \n*ρ* is called the *maximum entropy distribution* and *Z* is the *p**a**r**t**i**t**i**o**n**f**u**n**c**t**i**o**n*. Note that we have not proved that the distribution ([eq: 212]) in fact gives a maximum value for *S*, but this can be done. \nThe Lagrange multipliers *λ*1, ..., *λ**p* are chosen so that all the constraints are satisfied\n \n$$\\begin{aligned}\n\\left<f\\_j\\right> \\; = {\\int\\_{\\mathbf{R}^n}}dV \\; f\\_j (x\\_1,...,x\\_n) \\; \\rho(x\\_1,...,x\\_n) = c\\_j && j=1,...,p\\;. {\\label{eq: 213}} \\end{aligned}$$ \nThe system of equations ([eq: 213]) consists of *p* equations for the *p* unknown quantities *λ**j*.\nAs it turns out, we almost never need to know the distribution *ρ* from ([eq: 212]), it is enough to know the partition function. Observe that\n \n$$\\begin{aligned}\n\\left<f\\_j\\right> \\; &= {\\int\\_{\\mathbf{R}^n}}dV \\; f\\_j \\; \\rho\\nonumber\\\\ \n&= {\\frac{1}{Z}} \\; {\\int\\_{\\mathbf{R}^n}}dV \\; f\\_j \\; \\rho\\_0 \\; \\exp(-\\mathlarger{\\sum}^p\\_{i=1} \\lambda\\_{i} f\\_{i}) \\nonumber \\\\ \n&= -{\\frac{1}{Z}} \\; {\\int\\_{\\mathbf{R}^n}}dV \\; \\partial\\_{\\lambda\\_j}\\{ \\rho\\_0 \\; \\exp(-\\mathlarger{\\sum}^p\\_{i=1} \\lambda\\_{i} f\\_{i})\\} \\nonumber \\\\ \n&= -{\\frac{1}{Z}} \\; \\partial\\_{\\lambda\\_j} {\\int\\_{\\mathbf{R}^n}}dV \\; \\rho\\_0 \\; \\exp(-\\mathlarger{\\sum}^p\\_{i=1} \\lambda\\_{i} f\\_{i}) \\nonumber \\\\\n&= -{\\frac{1}{Z}} \\; {\\partial\\_{\\lambda\\_j}} Z =- {\\partial\\_{\\lambda\\_j}} \\ln Z \\nonumber \\\\\n&\\Downarrow\\nonumber\\\\\n\\left<f\\_j\\right> \\; &=- {\\partial\\_{\\lambda\\_j}} \\ln Z,{\\label{eq: 214}} \\end{aligned}$$ \nand thus we can find the mean of all the quantities *f**j* by taking partial derivatives of the natural logarithm of the partition function with respect to the Lagrangian multipliers. Moreover, we also have\n \n$$\\begin{aligned}\n{\\partial\\_{\\lambda\\_j \\lambda\\_k}}\\ln Z &= {\\partial\\_{\\lambda\\_j}} ({\\frac{1}{Z}} \\; {\\partial\\_{\\lambda\\_k}} Z) {\\label{eq: 215}} \\\\\n&= - {\\frac{1}{Z^2}} \\; {\\partial\\_{\\lambda\\_j}} Z \\; {\\partial\\_{\\lambda\\_k}} Z + {\\frac{1}{Z}} \\; {\\partial\\_{\\lambda\\_j \\lambda\\_k}} Z \\nonumber \\\\\n&= - {\\partial\\_{\\lambda\\_j}} \\ln Z \\; {\\partial\\_{\\lambda\\_k}} \\ln Z + {\\frac{1}{Z}} \\; {\\int\\_{\\mathbf{R}^n}}dV \\; f\\_j \\; f\\_k \\; \\rho\\_0 \\; \\exp(-\\mathlarger{\\sum}\\_i \\lambda\\_i \\; f\\_i) \\nonumber \\\\\n&= - {\\partial\\_{\\lambda\\_j}}\\ln Z \\; {\\partial\\_{\\lambda\\_k}} \\ln Z + \\left<f\\_j \\; f\\_k\\right>. \\nonumber \\end{aligned}$$ \nThus\n \n$$\\begin{aligned}\n\\left<f\\_j \\; f\\_k\\right> \\; = {\\partial\\_{\\lambda\\_j}} \\ln Z \\; {\\partial\\_{\\lambda\\_k}} \\; \\ln Z + {\\partial\\_{\\lambda\\_j \\; \\lambda\\_k}} \\ln Z {\\label{eq: 216}} \\end{aligned}$$ \nIn a similar way *all* correlation coefficients $\\left<f\\_1^{n\\_1}... f\\_p^{n\\_p}\\right>$ can be expressed through partial derivatives of the partition function.\nInserting the maximum entropy distribution ([eq: 212]) into the entropy functional ([eq: 79]) gives us the following expression for the maximal value of the entropy\n \n$$S=\\ln Z+\\mathlarger{\\sum}\\_j \\; \\lambda\\_j \\;\\left< f\\_j\\right>.{\\label{eq: MaxEntropy}}$$ \nFrom a mathematical point of view we now have two sets of variables $\\{\\left< f\\_1\\right>,...,\\left< f\\_p\\right>\\}$ and {*λ*1, ..., *λ**p*}. Geometrically we imagine that these two pairs of variables, together with *S*, defines a space Ω of odd dimension 2*p* + 1 with coordinates $\\{S,\\left< f\\_1\\right>,...,\\left< f\\_p\\right>,\\lambda\\_1,...,\\lambda\\_p\\}$. The *p* identities ([eq: 213]) defines a *p* + 1 dimensional surface Λ in Ω.\nTaking the differential of the identity ([eq: MaxEntropy]) we get\n \n$$dS=\\mathlarger{\\sum}\\_j\\; \\frac{\\partial\\ln Z}{\\partial\\lambda\\_j}d\\lambda\\_j+\\mathlarger{\\sum}\\_j \\;\\{\\left< f\\_j\\right>\\;d\\lambda\\_j+\\lambda\\_jd\\left< f\\_j\\right>\\}.$$ \nRestricting this differential to the surface Λ, and thus using the identities ([eq: 213]), gives us the following expression for the differential *d**S* restricted to the surface Λ\n \n$$dS=\\mathlarger{\\sum}\\_j \\;\\lambda\\_jd\\left< f\\_j\\right>{\\label{eq: dS1}}.$$ \nThe identity ([eq: MaxEntropy]) defines the entropy as a function depending on all 2*p* variables in Ω. We therefore have\n \n$$dS=\\mathlarger{\\sum}\\_j\\;\\frac{\\partial S}{\\partial\\lambda\\_j}d\\lambda\\_j+\\mathlarger{\\sum}\\_j\\;\\frac{\\partial S}{\\partial\\left< f\\_j\\right>}d\\left< f\\_j\\right>{\\label{eq: dS2}}.$$ \nComparing ([eq: dS1]) and ([eq: dS2]) we conclude that on the surface Λ we must have the identities\n \n$$\\begin{aligned}\n\\frac{\\partial S}{\\partial\\lambda\\_j}&=0,\\nonumber\\\\\n\\frac{\\partial S}{\\partial{\\left< f\\_j\\right>}}&=\\lambda\\_j.{\\label{eq: SRelation1}}\\end{aligned}$$ \nThus, on the surface Λ, the entropy depends only on the variables $\\{\\left< f\\_1\\right>,...,\\left< f\\_p\\right>\\}$ and the derivative with respect to these variables determines the values of the Lagrange multipliers in terms of the data {*c*1, ..., *c**p*} of the problem.\nIt is frequently the case that in addition to the variables {*x*1, ..., *x**n*}, the observables depends on parameters. For notational simplicity, let us assume that there is only one parameter denoted by the symbol *α*. Thus we have observables {*f*1(*x*1, ...*x**n*; *α*), ..., *f**p*(*x*1, ...*x**n*; *α*)}. The presence of the parameter does not change the argument leading up to the maximum entropy distribution ([eq: 212]) and thus we have the formulas\n \n$$\\begin{aligned}\n\\rho(x\\_1,...,x\\_n;\\alpha)& = \\frac{\\rho\\_0(x\\_1,...,x\\_n)}{Z(\\lambda\\_1,...,\\lambda\\_p;\\alpha)} \\; \\exp{-\\mathlarger{\\sum}^p\\_{j=1} \\lambda\\_j \\; f\\_j (x\\_1,...,x\\_n;\\alpha)},\\nonumber\\\\\n Z(\\lambda\\_1,...,\\lambda\\_p;\\alpha) &= {\\int\\_{\\mathbf{R}^n}}dV \\; \\rho\\_0 \\; \\exp{-\\mathlarger{\\sum}\\_j \\; \\lambda\\_j \\; f\\_j(x\\_1,...,x\\_n;\\alpha)}.{\\label{eq: parameterZ}}\\end{aligned}$$ \nDifferentiation of the partition function ([eq: parameterZ]) with respect to the parameter *α* gives us the expression\n \n$$\\begin{aligned}\n \\frac{\\partial Z}{\\partial \\alpha}&=-{\\int\\_{\\mathbf{R}^n}}dV \\; \\rho\\_0 \\;{\\mathlarger{\\sum}\\_j \\; \\lambda\\_j \\;\\frac{\\partial f\\_j}{\\partial \\alpha}} \\exp{-\\mathlarger{\\sum}\\_j \\; \\lambda\\_j \\; f\\_j},\\nonumber\\\\\n &\\Downarrow\\nonumber\\\\\n \\frac{\\partial \\ln Z}{\\partial \\alpha}&=-\\mathlarger{\\sum}\\_j \\; \\lambda\\_j \\;\\left<\\frac{\\partial f\\_j}{\\partial \\alpha}\\right>.{\\label{eq: DParameterZ}}\n \\end{aligned}$$ \nIf we repeat the calculation leading from ([eq: MaxEntropy]) to ([eq: dS1]) for the case when the observables depends on a parameter *α*, we now get instead of ([eq: dS1]) the following more general expression for the differential of the entropy\n \n$$dS=\\mathlarger{\\sum}\\_j \\;\\lambda\\_jd\\left< f\\_j\\right>-\\mathlarger{\\sum}\\_j \\;\\lambda\\_j \\;\\left<\\frac{\\partial f\\_j}{\\partial \\alpha}\\right>d\\alpha{\\label{eq: dS3}},$$ \nwhere we have used the identity ([eq: DParameterZ]).\nNote that this differential identity can be written in the form\n \n$$\\begin{aligned}\ndS=\\mathlarger{\\sum}\\_j \\;\\lambda\\_jdQ\\_j{\\label{eq: dS3}},\\end{aligned}$$ \nwhere we have introduced the quantities *d**Q**j* representing *generalized heat* associated with the observables\n \n$$\\begin{aligned}\ndQ\\_j&=d\\left< f\\_j\\right>-\\left<\\frac{\\partial f\\_j}{\\partial \\alpha}\\right>d\\alpha\\nonumber\\\\\n&=d\\left< f\\_j\\right>-\\left<\\frac{\\partial f\\_j}{\\partial \\alpha}d\\alpha\\right>\\nonumber\\\\\n&=d\\left< f\\_j\\right>-\\left<df\\_j\\right>.{\\label{eq: GeneralizedHeat}}\\end{aligned}$$ \nFormula ([eq: GeneralizedHeat]) tells us what heat actually represents. Physical systems on the human scale, these are evidently the ones of most immediate interest to us, consists of an immense number of elementary subsystems. The detailed configurational variables for all these elementary systems defines the *microscopic* degrees of freedom of the human scale system. The state of these microscopic degrees of freedom are unknown to us and our ability manipulate then directly is entirely lacking. The few degrees of freedom of the system whose state we *can* know and which we have the means to manipulate defines the *macroscopic* degrees of freedom for the system. In our description of thermodynamics these are the observables *f**j*. A change in the mean value of a macroscopic degrees of freedom,$d\\left<f\\_j\\right>$, comes from two sources. The first source is a change in the observable representing the said macroscopic degree of freedom, this is the kind of change that we have the ability to induce by direct manipulation. This quantity is represented by $\\left<df\\_j\\right>$ in formula ([eq: GeneralizedHeat]). When this quantity is subtracted from $d\\left<f\\_j\\right>$, what remains is the second source of change of the mean. This second source is a change in the underlying probability distribution which represents a change in our information about the microscopic state of the system. When our ignorance about the microscopic state of a system increase the system grows “hotter”, corresponding to an increase in *d**Q**j*.\nAs is usual in thermodynamics, the formalism is misleading in the sense that *d**Q**j* merely denote an infinitesimal amount of generalized heat and is *not* the differential of some function *Q**j*. No such function exists. The proper mathematical way to think about the identity ([eq: dS3]) is that *d**S* and *d**Q**j* are differential forms where *d**S* is an exact differential forms, meaning it is the differential of a function, and *d**Q**j* are inexact differential forms and thus not the differential of a function. However, the mathematical formalism of differential forms must be introduced in the very large context of differential geometry and we will not digress into this area of mathematics.\nThe above explanation of the nature of heat referred to the original application of the thermodynamical formalism where the systems has an immense number of microscopic degrees of freedom which are in principle knowable and controllable but in praxis not. We however know that the thermodynamical formalism can be applied to any situation where systems has more degrees of freedom than the ones we chose to observe. This might be because the underlying degrees of freedom are unknown but it could also be the case that they are known but that we for various reasons choose to ignore them. In both cases the argument above stands and the existence of the unknown or ignored degrees of freedom manifest as heat in the theory.\nWe will now derive a generalized version of identity ([eq: DParameterZ]) that plays a crucial role when the thermodynamical formalism is applied to the special case for which the underlying space is the state space for a physical system. The system could be a classical mechanical system consisting of a finite number of particles, a system of classical fields or, which will be our main application, the Fock state space for a quantum mechanical many particle system.\nIn all these cases one consider systems that are confined to a bounded spatial domain *D* which is defined by its bounding surface Γ. Thus all observables for the system will typically depend on the bounding surface Γ, *f**j* = *f**j*(*x*1, ...*x**n*; Γ). We will now consider a small deformation, *δ*Γ, of the bounding surface Γ. Thus Γ → Γ + *δ*Γ. This deformation leads to variations\n \n$$\\begin{aligned}\n \\delta\\_\\Gamma f\\_j(x\\_1,...x\\_n;\\Gamma)=f\\_j(x\\_1,...x\\_n;\\Gamma+\\delta\\Gamma)-f\\_j(x\\_1,...x\\_n;\\Gamma),\\nonumber\\\\\n \\delta\\_\\Gamma Z(\\lambda\\_1,...,\\lambda\\_p; \\Gamma)=Z(\\lambda\\_1,...,\\lambda\\_p; \\Gamma+\\delta\\Gamma)-Z(\\lambda\\_1,...,\\lambda\\_p; \\Gamma).{\\label{eq: variations}}\n \\end{aligned}$$ \nArguing exactly like we did for the simple case of a single parameter we now find the important identity\n \n$$\\begin{aligned}\n \\delta\\_\\Gamma \\ln Z=-\\mathlarger{\\sum}\\_j \\; \\lambda\\_j \\;\\left< \\delta\\_\\Gamma f\\_j\\right>.{\\label{eq: FundamentalVariationalIdentity}}\n \\end{aligned}$$ \nThis identity will, for the special cases mentioned above, lead to the definition of the thermodynamic pressure and related quantities. Corresponding to the *differential* identity for the entropy ([eq: dS3]) we now get the following more general *variational* identity\n \n$$\\begin{aligned}\n\\delta S=\\mathlarger{\\sum}\\_j \\;\\lambda\\_jd\\left< f\\_j\\right>-\\mathlarger{\\sum}\\_j \\;\\lambda\\_j \\;\\left< \\delta\\_\\Gamma f\\_j\\right>{\\label{eq: dS5}}.\\end{aligned}$$ \n### The thermodynamic formalism in statistical physics\nLet us now consider the special case when our underlying space is the classical state space for a mechanical system with *n* degrees of freedom. This could for example consist of *n* mass points. We will assume that the system is confined to a bounded domain *D* in **R**3 defined by a bounding surface Γ. The state space is thus a subset of the euclidean space **R**6*n* with coordinates $(\\vb{q},\\vb{p})=(\\vb{q}\\_1,...,\\vb{q}\\_n,\\vb{p}\\_1,...,\\vb{p}\\_n)$, since we need 3 position coordinates and 3 momentum coordinates for each particle in order to uniquely specify the state of the system. \nLet $\\mathcal{H} = \\mathcal{H}(\\vb{q},\\vb{p})$ be the Hamiltonian for the system of mass points. Recall that the value of the Hamiltonian on any given state, $(\\vb{q},\\vb{p})$, is the energy, *E*, of that state. \nWhen *n* is large it is very hard, and also mostly useless, to try to track the exact state $(\\vb{q}(t),\\vb{p}(t))$ of a system of mass points. \nFor such a large system it is more useful to consider a probability distribution $\\rho(\\vb{q},\\vb{p})$ on the state-space. This is the point of view introduced by Gibbs. We will first consider the simplest, and by far the most common situation, where the Hamiltonian, $\\mathcal {H}=\\mathcal {H}(\\vb{q},\\vb{p})$ is the only observable. The maximum entropy distribution for this case is\n \n$$\\begin{aligned}\n\\rho (\\vb{q},\\vb{p}) = \\frac{\\rho\\_0(\\vb{q},\\vb{p})}{Z} \\; \\exp(-\\frac{\\mathcal {H}(\\vb{q},\\vb{p})}{kT}), {\\label{eq: 217.1}} \\end{aligned}$$ \nwhere the partition function is given by\n \n$$\\begin{aligned}\nZ=Z(T)=\\int\\_{\\mathcal{R}^{6n}}d\\vb{q}d\\vb{p}\\;\\rho\\_0(\\vb{q},\\vb{p})\\exp(-\\frac{\\mathcal {H}(\\vb{q},\\vb{p})}{kT}),{\\label{eq: statmechZ1}}\\end{aligned}$$ \nand where we have redefined the single Lagrange multiplier using\n \n$$\\begin{aligned}\n\\lambda = {\\frac{1}{k \\; T}}. {\\label{eq: 218}} \\end{aligned}$$ \nIn this formula, *k* is the Boltzmann constant and *T* is a new parameter which by definition is the thermodynamic temperature. The parameter *T* is determined by\n \n$$\\begin{aligned}\n E&= \\left<\\mathcal{H}\\right>, \\nonumber \\\\\n&\\Updownarrow \\nonumber \\\\ \nE&=k \\; T^2 \\; {\\partial\\_{T}} \\ln Z,{\\label{eq: 220}}\\end{aligned}$$ \nwhere we have used the chain rule\n \n$$\\begin{aligned}\n{\\partial\\_{\\lambda}} =- k \\; T^2 \\; {\\partial\\_{T}}, {\\label{eq: 221}} \\end{aligned}$$ \nin the general formula ([eq: 214]).\nFormula ([eq: 220]) is in statistical mechanics and thermodynamics called the *equation of state*, and all thermodynamic statements that can be made about the system flows from this formula. The formula for equation of state may look innocent, in order to find it you merely need to take the derivative of the partition function, and partition function also looks innocent, after all it is just a function of one variable, the kind of function we study in first year calculus. However, in order to actually find an expression for this single variable function one needs to do the integral in formula ([eq: statmechZ1]), and this is a multiple integral involving something like 1027 integration variables in typical situations! Clearly, an exact formula for the partition function can rarely be found. Approximate expressions where the large number of particles are used to ones advantage can more frequently be found, but pushing through calculations like these are as a rule extremely technical. More than one Nobel price has been handed out for developing feasible schemes for calculating the partition function. Given the level of complexity involved in calculating the partition function from the defining formula ([eq: statmechZ1]), and the fact that the partition function simply is a function of one or a few variables, it should come as no surprise that the most common approach to finding the equation of state is to fit parametrized functions to experimental data.\nThe maximum entropy distribution ([eq: 217.1]) is recognized to essentially be the *Gibb’s Canonical ensemble* from statistical physics.\nThe Gibb’s ensemble is the foundation of statistical physics. All results in statistical physics flows from formula ([eq: 217.1]). Statistical physics is also the foundation of thermodynamics so all conclusions from that subject also flow from the Gibb’s ensemble ([eq: 217.1]). In the thermodynamics context, ([eq: 220]) is, as we have already remarked, nothing but the *equation of state*.\nAn interesting insight here is that the temperature of a thermodynamic system is in fact a Lagrange multiplier!! This is a profound insight that to this day has not been fully understood or explored.\nFrom this example, it appears useful to think of any application of the maximal entropy principle as an extension of the methods of statistical mechanics to systems that has absolutely nothing to do with the motion of mass points.\nThis wide general applicability of the methods of statistical physics has lead to deep questions and insights into the nature and significance of the assumption of equilibrium that appears to underline the application of the Gibb’s ensemble in statistical physics.\nThere is also the intriguing fact that the very same functional ([eq: 79]) used in the maximum entropy principle, is also the foundation of information theory which was discovered by Shannon in 1948. This connection between information theory and statistical mechanics (and thermodynamics) has lead to deep insights into the role of information in our fundamental physical theories. \nAs already discussed in the introduction, the general nature and wide applicability of the maximum entropy principle has been described well by E.T. Jaynes in many papers and the (unfinished) monumental book “Probability theory: The Logic of Science”.\nAs if all this is not impressive enough for one single principle, it is also a very intriguing fact that when one looks deep into the heart of fundamental physics, in the form of quantum field theory, one again finds an appropriately generalized form of the partition function ([eq: 211]). The whole computational engine in the theory of quantum fields revolve around a generalized Gibb’s ensemble!\nWhat on earth is going on...\n### The problem of prior\nNote that formula ([eq: 217.1]) does not uniquely define Gibbs ensemble because of the presence of the prior distribution *ρ*0. The actual Gibbs ensemble corresponds to the choice *ρ*0 = 1. When using the information theoretical approach to statistical mechanics and thermodynamics, like we do here, one should be very wary when it comes to the choice of the prior distribution. It is simply the most contentious issue in the whole theory. One should ask pointed questions of justification for any proposed choice. What kind of information about the system is it based on, and is it the correct embodiment of said information?\nIn fact, if one study expositions of statistical mechanics and thermodynamics which is based on the traditional objective dynamical approach to the subject, one finds that the choice of what from the information theoretical point of view is the prior distribution, is much discussed. The reasons for choosing *ρ*0 = 1 that have appeared through these discussions are, in our humble opinion, not very convincing.\nThe problem of determining the prior distribution has been at the center of probability theory and statistics from the very start. The general rules of probability theory tells us how to compute probabilities for derived events from probabilities of primary events. The problem of prior is concerned with the problem of assigning probabilities to primary events. The assignment is supposed to reflect an observers state of knowledge about the primary events. The assignment should be the same for different observers with the same state of knowledge but can be different for observers with different states of knowledge. In this sense probability assignments are subjective,,. The problem of the prior is how to turn states of knowledge into probability assignments. The first solution to this problem was used by the very founders of probability theory (Bernoulli and Laplace). If the observers only knowledge of the primary events are their number, then a uniform probability assignment should be used. This idea was later named the principle of indifference by J. M. Keynes. Generalizing this idea to countably infinite or even continuous spaces of primary events has turned out to be very problematic. Laplace himself used such a generalization is his work on probability theory. His probability distribution was uniform and not normalizable since it was defined on the whole real line. Using a uniform distribution for representing indifference about a random variable on a finite interval on the real line would seem to be more reasonable, at least it is normalizable. However even in this case serious problems arise as the well known Bertrand’s paradox shows. Problems and paradoxes arising from the various generalizations of the principle of indifference to continuous random variables played no small part in the creation and for a long time complete dominance of the frequency interpretation of probability theory.\nThe principle of maximum entropy appears first in the writings of W. Gibbs on thermodynamics and statistical physics and later in the fundamental work on information theory by Shannon. However it was E. T. Jaynes who realized the real importance and general nature of the principle of maximum entropy. In his hands it turned into a general method for turning prior knowledge in the form of mean values for observables defined on finite state spaces, into prior probability assignments.\nLet us consider this simplest case in more detail. Let Ω = {*x*1, *x*2, ...., *x**n*} be a finite space of primary events. The algebra of possible events is the set of all subsets of Ω. A probability assignment on the set of primary events is a finite set of numbers *p* = {*p**i*} such that 0 ≤ *p**i* ≤ 1 and ∑*i* = 1*n**p**i* = 1. Let *f*1, ..., *f**k* be real valued functions on Ω. The principle of maximum entropy states that if the means of the functions *f*1, .., *f**k* are known, $\\left<f\\_{i}\\right>=c\\_{i}$, one should, among all probability assignments that satisfy the constraints, pick the one that maximizes the entropy *S* = − ∑*i* = 1*n**p**i*ln*p**i*. The solution to this constrained maximization problem is, as we have seen, the maximum entropy distribution\n \n$$\\begin{aligned}\np=\\frac{1}{Z(\\lambda \\_{1},...,\\lambda\n\\_{k})}\\exp\\left(-\\sum\\_{j=1}^{k}\\lambda \\_{j}f\\_{j}\\right),\\end{aligned}$$ \nwhere *Z* is the partition function and is given by\n \n$$\\begin{aligned}\nZ(\\lambda \\_{1},...,\\lambda \\_{k})=\\sum\\_{i=1}^{n}\\exp\\left(-\\sum\\_{j=1}^{k}\\lambda \\_{j}f\\_{j}(i)\\right).\\end{aligned}$$ \nObserve that for the particular situation where there are no constraints, the principle gives *Z* = *n* and the maximum entropy distribution is uniform\n \n$$\\begin{aligned}\np\\_{i}=\\frac{1}{n}.\\end{aligned}$$ \nThus, for an observer that only know that there are *n* possible primary events, the maximum entropy distribution is exactly the one suggested by the principle of indifferent! The conclusion appeared to be that not only could the maximum entropy distribution tell us how to choose the best distribution in the presence of observed means of a finite number of observables, it could also tell us which distribution to choose when our ignorance is so profound that the only thing we know about a situation is the number of possible primary events. This distribution is of course exactly what we have called the prior distribution. For a time it looked as if the problem of prior was essentially solved. However continuous valued random variables again turned out to be the Achilles heel. For finite spaces of events the principle will give a unique probability assignment, but when generalizing it to continuous random variables by taking a continuum limit of the finite discrete expression for the entropy, an unknown probability density appears. The density appears because the continuum limit is not unique. Different limiting expressions are found depending on how one approach the continuum through a countable set of discrete spaces. The unknown probability distributions that appears essentially depends on how the discrete points bunch up in the limit. The meaning of this probability density became clear when it was realized that it is the maximum entropy distribution corresponding to no constraints. Thus it was understood that in order to apply the principle of maximum entropy one must start with a prior distribution. The principle of maximum entropy could not determine the prior, it could only tell us how to modify an already existing prior in order to satisfy constraints in the form of mean values. It seemed as if one were back to square one.\nThere *does* however exist a systematic way to turn prior information on means of observables into prior distributions, and it *does* involve the maximum entropy principle, but not in the direct way just described. In fact, after a certain reformulation it will become evident that the problem of selecting a prior is not merely a side issue that has to be resolved in order to proceed with the real work of applying the maximum entropy principle, the problem of prior is the *only* issue as far as the maximum entropy principle is concerned.\nIn order to describe this reformulation of the principle of maximum entropy, we will return to the special case of statistical mechanics. In the previous section we discussed the problem of specifying the prior in the context of statistical mechanics and expressed our doubt as to the justifications for making the standard choice $\\rho\\_0(\\vb{q},\\vb{p})=1$. Even if we are doubtful about the justification for this particular choice, it is clear that when we apply the maximum entropy principle in statistical mechanics there is a physical context that certainly makes some choices of the prior more reasonable than othersr. By picking the Hamiltonian function as our observable we must also acknowledge that the system evolve according to the corresponding Hamiltonian equations. It is always the case that the the Hamiltonian function, $\\mathcal{H}(\\vb{q},\\vb{p})$, which represents the energy, is a constant of the motion. Depending on the symmetries of the interaction, Hamiltonian systems of equations may also have other conserved quantities. The generic situation is however that the energy is the only conserved quantity. We will assume that this is the case and let the corresponding Hamiltonian function H be our only observable. The maximum entropy distribution is now given by expression ([eq: 217]) where *ρ*0 is the prior distribution. It is in the current context reasonable to impose the condition that the prior is a stationary solution to the corresponding Liouvillian equation. But this means that the prior distribution, $\\rho\\_0=\\rho\\_0(\\vb{q},\\vb{p})$, is a conserved quantity for the Hamiltonian system and since the Hamiltonian is the only independent conserved quantity for our generic Hamiltonian system we must have\n \n$$\\begin{aligned}\n\\rho\\_0(\\vb{q},\\vb{p})=f\\_0(\\mathcal{H}(\\vb{q},\\vb{p})),\\nonumber\\end{aligned}$$ \nwhere *f*0 is an arbitrary function defined on the positive real line. Using this fact we have from ([eq: statmechZ1])\n \n$$\\begin{aligned}\nZ(T)&=\\int\\_{\\mathcal{R}^{6n}}d\\vb{q}\\;d\\vb{p}\\rho\\_0(\\vb{q},\\vb{p})\\exp(-\\frac{\\mathcal {H}(\\vb{q},\\vb{p}}{kT})\\nonumber\\\\\n&=\\int\\_{\\mathcal{R}^{6n}}d\\vb{q}\\;d\\vb{p}f\\_0(\\mathcal{H}(\\vb{q},\\vb{p}))\\exp(-\\frac{\\mathcal {H}(\\vb{q},\\vb{p}}{kT})\\nonumber\\\\\n&=\\int\\_0^{\\infty} dE\\;\\exp(-\\frac{E}{kT})\\;f\\_0(E)\\;\\int\\_{\\mathcal{H}=E}d\\vb{q}\\;d\\vb{p}\\nonumber\\\\\n&=\\int\\_0^{\\infty} dE\\;\\exp(-\\frac{E}{kT})\\;\\rho\\_0(E),{\\label{eq: statmechZ2}}\\end{aligned}$$ \nwhere we have defined\n \n$$\\begin{aligned}\n\\rho\\_0(E)=f\\_0(E)\\;\\int\\_{\\mathcal{H}=E}d\\vb{q}\\;d\\vb{p}.{\\label{eq: macroPrior}}\\end{aligned}$$ \nThe constraints on the *microscopic* prior distribution $\\rho\\_0(\\vb{q},\\vb{p})$ has reduced our original maximum entropy principle on the extremely high dimensional space R6*n* with the Hamiltonian as our observable, to a maximum entropy problem on the real line where the coordinate on the line, *E*, is the observable and the *macroscopic* prior is given by ([eq: macroPrior]). The maximum entropy distribution for this case is\n \n$$\\begin{aligned}\n\\rho(E)=\\frac{\\rho\\_0(E)}{Z(T)}\\exp(-\\frac{E}{kT}),{\\label{eq: MaxEntOnLine}}\\end{aligned}$$ \nwhere the partition function is given by\n \n$$\\begin{aligned}\nZ(T)=\\int\\_0^{\\infty}dE\\;\\rho\\_0(E)\\exp(-\\frac{E}{kT}).{\\label{eq: StatMechReducedPartitionFunction}}\\end{aligned}$$ \nThis simple situation where we apply the entropy principle to a low dimensional state space R*p* and where the observables are the coordinate functions, *x*1, ..., *x**p* on the space is not special at all, in fact this is the most common situation when we apply the maximum entropy principle and other applications can almost always be reduced to this situation using an approach similar to the reduction from R6*n* to R described for the case of statistical mechanics.\nIn most applications of probability theory in statistics there is no underlying high dimensional space of primary events, Ω, like in statistical mechanics and other areas of physics, and the random variables are not some functions, like the Hamiltonian, defined on this space.\nThus in the typical case one can assume that Ω = R*p*, where *p* is a fairly small number, and that the random variables are just the coordinate function on R*p*. The prior probability distribution is then a function, *ρ*0 = *ρ*0(*x*1, ..., *x**p*), on R*p*, and the partition function is given by the formula\n \n$$\\begin{aligned}\nZ(\\lambda \\_{1},..,\\lambda \\_{p})=\\int\\_{\\mathbb{R}^p}dx\\_1dx\\_2...dx\\_p\\;\\rho\\_0(x\\_1,...,x\\_p)\\exp(-\\sum\\_{j=1}^{p}\\lambda \\_{j}x\\_{j}).{\\label{eq: FromRho0ToZ}}\\end{aligned}$$ \nThe partition function is thus nothing but the multi dimensional Laplace transform of the prior distribution. This relation can be inverted, using analytical continuation and the multidimensional Fourier transform on the imaginary *λ**j* axes, and thereby expressing the prior in terms of the partition function\n \n$$\\begin{aligned}\n\\rho \\_{0}(x\\_{1},..x\\_{p})=\\frac{1}{(2\\pi )^{p}}\\int\\_{\\mathbb{R}^{p}}d\\lambda \\_{1}..d\\lambda \\_{p}Z(i\\lambda \\_{1},..,i\\lambda \\_{p})\\exp(i\\sum\\_{j=1}^{p}\\lambda\n\\_{j}x\\_{j}).{\\label{eq: FromZToRho0}}\\end{aligned}$$ \nThe whole content of the maximum entropy principle is contained in the integral transforms ([eq: FromRho0ToZ]) and ([eq: FromZToRho0]) connecting the partition function and the prior distribution. This is the promised reformulation of the maximum entropy principle, and we understand now that the prior distribution is not merely a bit player in this drama, it is the *only* player. In the next section we will show how this reformulation of the maximum entropy principle gives us a method for solving the problem of prior, which we now see is the only remaining fundamental problem in the statistical modeling of natural or artificial systems. The material on the problem of prior in the next section has been previously published in a slightly different form.\n### Solving the problem of prior using stochastic relations\nIn probability theory and statistics, random variables are often grouped into statistical quantities. These are certain algebraic combinations of means of functions of the random variables. A large set of such statistical quantities are in use, some simple examples are\n \n$$\\begin{aligned}\n&\\left<x\\right>\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{ The mean of x.}\\\\\n&\\left<x^{2}\\right>-\\left<x\\right>^{2}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\; \\text{ The variance of x.}\\\\\n&\\left<x^{3}\\right>-3\\left<x\\right>\\left<x^{2}\\right>+2\\left<x\\right>^{3}\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{The third cumulant.} \\\\\n&\\left<xy\\right>-\\left<x\\right>\\left<y\\right>\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\text{The cross variance of x and y.}\\end{aligned}$$ \nAll such quantities can systematically be expressed as functions of the form *F*(*q*1, .., *q**k*) where the variables *q**j* are means of monomials in the random variables. We will define *stochastic relations* to be systems of equations for the quantities *q**j*.\n \n$$\\begin{aligned}\nF\\_{i}(q\\_{1},..,q\\_{k}) =0\\text{ \\ \\ \\ \\ }i=1,...,s.{\\label{eq: StochasticRelation}}\\end{aligned}$$ \nSuch relations are common in probability and statistics. Examples are zero mean, fixed variance, uncorrelated variables and identities expressing higher order cumulants in terms of lower ones. Identities such as the last ones in the previous list are the fundamental tools used to construct theories of turbulence in fluid, gases and elsewhere. They are also, in their quantum incarnations,the key tools used to find viable simplified models in solid state physics and material science.\nIn the previous section we have seen that the maximum entropy principle defines a Laplace transform that map the prior distribution to a partition function. As a direct consequence of this transformation we can express means of monomials in the random variables in terms of partial derivatives of the partition function. For example we have\n \n$$\\begin{aligned}\n\\left<f\\_i\\right>&=-\\frac{1}{Z}\\partial\\_{\\lambda\\_i}Z,\\nonumber\\\\\n\\mathop{var}(f\\_i)&=\\left<f\\_i^2\\right>-\\left<f\\_i\\right>^2=-\\frac{1}{Z^2}\\partial\\_{\\lambda}Z^2+\\frac{1}{Z}\\partial\\_{\\lambda\\lambda}Z.\\nonumber\\end{aligned}$$ \nThis means that the maximum entropy principle turns stochastic relations into systems of partial differential equations for the partition function and therefore imposes constraints on the prior distribution.\nThe problem is now how to describe the space of solutions of these systems of partial differential equations. In general, not all solutions to the equations can correspond to prior probability distributions. From the definition of the partition function it is for example clear that *Z*(0) = 1 must hold for any acceptable solution. Finding necessary and sufficient conditions for functions to be the Laplace transform of a probability distribution, and thus be acceptable solutions of the systems of differential equations corresponding to stochastic relations, is not a simple matter, but some results are known. We will not discuss this problem but rather try to explicitly construct the solution space or to say something useful about the structure of the solution space using methods from the formal theory of differential equations. Typically, the solution space is not a linear space and even when it is, the dimension could easily be infinite. However, depending on the number and types of stochastic relations the solution space can end up being parametrized by a finite set of parameters or even be a single point. In this last situation the stochastic relations determine the prior uniquely. Note that in ordinary (parametric) statistics finite parameter families of probability distributions (Gaussian, Poissont, Bernoulli, t-distribution, etc) are assumed to apply in given situations. From the point of view discussed in these notes, this means that in ordinary statistics, stochastic relations constrain the solution space enough for it to be parameterized in terms of a finite number of parameters. Nonparametric statistics correspond to the situation when the solution space is so weakly constrained that it can not be parameterized in terms of a finite number of parameters. Methods from the theory of partial differential equations can in some cases parameterize such weakly constrained solution spaces, not in terms of real numbers, but in terms of arbitrary functions. However for such weakly constrained solution spaces there is another powerful tool available. This is the formal theory of partial differential equations. The main object of study in this theory is the infinitely prolonged hierarchy of a given systems of differential equations. Thus one studies the infinite set of all differential consequences of a given system of equations. Each such differential consequence can be converted back into a stochastic relation by using the relation between mean of monomials and partial derivatives in reverse. One therefore gets the corresponding hierarchy of stochastic relations that are consequences of the original relations induced by the maximum entropy principle and implemented through the Laplace transform.\nIn the remaining part of this subsection we will discuss several examples that illustrate the method that has been outlined.\n#### Stochastic relations for one random variable\nEssentially all families of distribution in use in parametric statistics can be derived from simple stochastic relations involving the mean, variance and skewness. In this section we show some examples that support this statement.\n##### Delta distribution\nLet us consider the stochastic relation corresponding to a fixed mean. It is\n \n$$\\begin{aligned}\n\\left<x\\right>-q=0.\\end{aligned}$$ \nThe Laplace transform convert this into the ordinary differential equation\n \n$$\\begin{aligned}\nZ\\_{\\lambda }=-qZ.\\end{aligned}$$ \nFor this simple stochastic relation our system of partial differential equations is a single linear ordinary differential equation. The solution space is linear and parameterized by a single parameter\n \n$$\\begin{aligned}\nZ(\\lambda )=ae^{-q\\lambda }.\\end{aligned}$$ \nThe condition *Z*(0) = 1 fixes the parameter *a* to be one and we have a unique solution. It is a simple matter to apply the inverse transform ([eq: FromZToRho0]) to find the corresponding prior distribution\n \n$$\\begin{aligned}\n\\rho\\_0(x)& =\\frac{1}{2\\pi}\\int\\_{-\\infty}^{\\infty}d\\lambda Z(i\\lambda)\\exp(i\\lambda x)\\nonumber\\\\\n&=\\frac{1}{2\\pi}\\int\\_{-\\infty}^{\\infty}d\\lambda\\;\\exp(-iq\\lambda)\\exp(i\\lambda x)\\nonumber\\\\\n&=\\frac{1}{2\\pi}\\int\\_{-\\infty}^{\\infty}d\\lambda\\;\\exp(i(x-q)\\lambda)\\nonumber\\\\\n&=\\delta (x-q).\\end{aligned}$$ \n##### Normal distribution\nThe stochastic relation corresponding to constant variance is\n \n$$\\begin{aligned}\n\\mathop{var}(x)=q,\\end{aligned}$$ \nand the corresponding differential equation is\n \n$$\\begin{aligned}\nZZ\\_{\\lambda \\lambda }-Z\\_{\\lambda }^{2}-qZ^{2}=0.\\end{aligned}$$ \nThis is a second order nonlinear ordinary differential equation. The general solution of the nonlinear equation that satisfies the requirement *Z*(0) = 1 is\n \n$$\\begin{aligned}\nZ(\\lambda )=e^{-a\\lambda +\\frac{1}{2}q\\lambda ^{2}}\\text{, \\ \\ \\ \\\n\\ \\ \\ }a\\in\\mathbb{R}.\\end{aligned}$$ \nUsing this partition function we can predict the mean of the random variable *x* to be\n \n$$\\begin{aligned}\n\\left<x\\right)&=-\\frac{1}{Z(\\lambda)}\\frac{\\partial Z(\\lambda)}{\\partial \\lambda}\\nonumber\\\\\n&=a\\end{aligned}$$ \nand the corresponding prior distribution is found, using ([eq: FromZToRho0]), to be\n \n$$\\begin{aligned}\n\\rho\\_0 (x)&=\\frac{1}{2\\pi }\\int\\_{\\mathbb{R}}d\\lambda\\;Z(i\\lambda )\\exp(i\\lambda\\;x)\\nonumber\\\\\n&=\\frac{1}{2\\pi}\\int\\_{-\\infty}^{\\infty}d\\lambda\\;\\exp(-ia\\lambda -\\frac{1}{2}q\\lambda ^{2})\\exp(i\\lambda\\;x)\\nonumber\\\\\n&=\\frac{1}{\\sqrt{2\\pi q}}e^{-\\frac{(x-a)^{2}}{2q}}.\\end{aligned}$$ \nwhich is the normal distribution.\n##### Poisson distribution\nLet us consider the stochastic relation\n \n$$\\begin{aligned}\n\\mathop{var}(x)=<x>.\\end{aligned}$$ \nThe corresponding differential equation is\n \n$$\\begin{aligned}\nZZ\\_{\\lambda \\lambda }-Z\\_{\\lambda }^{2}+ZZ\\_{\\lambda }=0.\\end{aligned}$$ \nThis equation and most equations derived from stochastic relations simplify considerably if we introduce a new function *φ* through *Z* = *e**φ*. The equation for *φ* is\n \n$$\\begin{aligned}\n\\varphi \\_{\\lambda \\lambda }=-\\varphi \\_{\\lambda }.\\end{aligned}$$ \nThis equation is easy to solve and the corresponding family of partition functions satisfying, as always, the constraint *Z*(0) = 1 is\n \n$$\\begin{aligned}\nZ(\\lambda )=e^{a(e^{-\\lambda }-1)}.\\end{aligned}$$ \nThe corresponding prior distribution is found using ([eq: FromZToRho0]) to be supported on Ω = {0, 1, 2, ....} and is of the form\n \n$$\\begin{aligned}\n\\rho\\_0 (k)=\\frac{e^{-a}a^{k}}{k!}.\\end{aligned}$$ \nThis is the Poisson distribution.\n##### Gamma distribution\nLet us consider a stochastic relation\n \n$$\\begin{aligned}\n\\mathop{var}(x)=\\frac{1}{k}{\\left<x\\right>}^{2}\\text{ \\ \\ }k>0.\\end{aligned}$$ \nExpressed in terms of *φ* the corresponding differential equation is\n \n$$\\begin{aligned}\n\\varphi \\_{\\lambda \\lambda }=\\frac{1}{k}\\varphi \\_{\\lambda }^{2}.\\end{aligned}$$ \nThe general solution of this equation gives the following family of partition functions\n \n$$\\begin{aligned}\nZ(\\lambda )=(1-a\\lambda )^{-k}\\text{ \\ \\ }a>0.\\end{aligned}$$ \nThe correspondingprior distribution is supported on Ω = (0, ∞) and is given by\n \n$$\\begin{aligned}\n\\rho\\_0 (x)=x^{k-1}\\frac{e^{-\\frac{x}{a}}}{a^{k}\\Gamma (k)}.\\end{aligned}$$ \nThis is the Gamma distribution\n##### Bernoulli and Binomial distribution\nLet the variance be the following quadratic function of the mean\n \n$$\\begin{aligned}\n\\mathop{var}(x)=\\left<x\\right>(1-\\left<x\\right>).\\end{aligned}$$ \nThe corresponding differential equation for *φ* is\n \n$$\\begin{aligned}\n\\varphi \\_{\\lambda \\lambda }=-\\varphi \\_{\\lambda }(1+\\varphi \\_{\\lambda }).\\end{aligned}$$ \nThe solution of the equation gives the following family of partition functions\n \n$$\\begin{aligned}\nZ(\\lambda )=p+qe^{-\\lambda }\\text{ \\ \\ \\ \\ \\ }p+q=1.\\end{aligned}$$ \nThe corresponding distribution is supported on Ω = {0, 1} and is given by *ρ*(0) = *p*, *ρ*(1) = *q*. This is the Bernoulli distribution. If we generalize the stochastic relation to\n \n$$\\begin{aligned}\n\\mathop{var}(x)=\\left<x\\right>(1-\\frac{1}{n}\\left<x\\right>).\\end{aligned}$$ \nwhere *n* is a natural number we get the differential equation\n \n$$\\begin{aligned}\n\\varphi \\_{\\lambda \\lambda }=-\\varphi \\_{\\lambda }(1+\\frac{1}{n}\\varphi\n\\_{\\lambda }).\\end{aligned}$$ \nThe partition function is found to be\n \n$$\\begin{aligned}\nZ(\\lambda )=(p+qe^{-\\lambda }\\text{ )}^{n}\\text{\\ \\ \\ \\ \\ }p+q=1.\\end{aligned}$$ \nThe corresponding prior distribution is now found to be supported on Ω = {0, 1, ...*n*} and is on this domain given by\n \n$$\\begin{aligned}\n\\rho\\_0 (k)=\\binom{n}{k}p^{k}q^{n-k}.\\end{aligned}$$ \nThis is the Binomial distribution.\n#### Stochastic relations for more than one random variable\nWhen the number of random variables become larger than one, stochastic relations in general leads to systems of nonlinear partial differential equations. Unless the number and type of relations is right, it is impossible to describe the solution space in terms of a finite number of parameters. This lead us into the domain of nonparametric statistics. This is the domain where the methods from the formal theory of differential equations comes into play. It is not possible to give nontrivial applications of the theory here and we will therefore limit ourselves to two simple examples.\n##### The Multinomial distribution\nLet *x*1, ...*x**n* be *n* random variables and consider the following system of stochastic relations\n \n$$\\begin{aligned}\n\\mathop{var}(x\\_{i}) &=\\left<x\\_{i}\\right>(1-\\frac{1}{n}\\left<x\\_{i}\\right>)\\text{ \\ }\ni=1,..n\\;, \\nonumber\\\\\n\\mathop{cov}(x\\_{i},x\\_{j}) &=-\\frac{1}{n}\\left<x\\_{i}\\right>\\left<x\\_{j}\\right>\\text{ \\ \\ }\ni,j=1,...n,\\text{ \\ \\thinspace }i\\neq j.\\end{aligned}$$ \nThe corresponding system of partial differential equations is\n \n$$\\begin{aligned}\n\\varphi \\_{\\lambda \\_{i}\\lambda \\_{i}} &=-\\varphi \\_{\\lambda \\_{i}}(1+\\frac{1}{n}\n\\varphi \\_{\\lambda \\_{i}}), \\nonumber\\\\\n\\varphi \\_{\\lambda \\_{i}\\lambda \\_{j}} &=-\\frac{1}{n}\\varphi \\_{\\lambda\n\\_{i}}\\varphi \\_{\\lambda \\_{j}}.\\end{aligned}$$ \nThe second part of the system of equations has general solutions of the form *φ* = *n*ln(*θ*) where *θ*(*λ*1, .., *λ**n*) = ∑*i* = 1*n**θ**i*(*λ**i*). Inserted into the first part of the system this form of *φ* easily gives the partition function corresponding to the multinomial distribution. This system of relations thus constrained the space of solutions so much that it could be describes in terms of a finite number of parameters.\n##### Stochastic relations for the mean\nFor a single random variable, stochastic relations involving only the mean gives distributions located on a finite number of points. For more than one random variable such relations gives rise to nonparametric statistics, or solution spaces parameterized by functions. The theory of partial differential equations can be used to give a full description of these solution spaces. As an example of such a relation consider the case of two random variables whose means are constrained to be on a circle of radius *r*.\n \n$$\\begin{aligned}\n{\\left<x\\right>}^{2}+{\\left<y\\right>}^{2}=r^{2}.\\end{aligned}$$ \nThe corresponding partial differential equation is i terms of *φ*\n \n$$\\begin{aligned}\n\\varphi \\_{\\lambda }^{2}+\\varphi \\_{\\mu }^{2}=r^{2},\\end{aligned}$$ \nand is known from optics as the *Eiconal* equation.The following *Z* is in the solution space\n \n$$\\begin{aligned}\nZ=e^{r\\sqrt{\\lambda ^{2}+\\mu ^{2}}}.\\end{aligned}$$ \nThis partition functions predicts that the following stochastic relation should hold\n \n$$\\begin{aligned}\n\\mathop{var}(x)=\\left( \\frac{\\left<y\\right>}{\\left<x\\right>}\\right) ^{2}\\mathop{var}(y).\\end{aligned}$$ \nThe partial differential equation has, however, infinitely many solutions. The method of characteristics can be used to describe the complete solution space. In order to derive stochastic relations that holds for all *Z* in the solution space, these are the ones that can be said to be consequences of the of the circle constrain, we should consider differential prolongations of the original differential equation. The first prolongation is the system\n \n$$\\begin{aligned}\n\\varphi \\_{\\lambda }^{2}+\\varphi \\_{\\mu }^{2} &=r^{2}, \\\\\n\\varphi \\_{\\lambda }\\varphi \\_{\\lambda \\lambda }+\\varphi \\_{\\mu }\\varphi \\_{\\mu\n\\lambda } &=0, \\\\\n\\varphi \\_{\\lambda }\\varphi \\_{\\lambda \\mu }+\\varphi \\_{\\mu }\\varphi \\_{\\mu \\mu\n} &=0,\\end{aligned}$$ \nand this system implies that\n \n$$\\begin{aligned}\n\\varphi \\_{\\lambda \\lambda }=\\left( \\frac{\\varphi \\_{\\mu }}{\\varphi \\_{\\lambda }}\\right) ^{2}\\varphi \\_{\\mu \\mu }.\\end{aligned}$$ \nTranslated into stochastic relations this is exactly the one we derived for the special solution $\\varphi =r\\sqrt{\\lambda ^{2}+\\mu ^{2}}$ and it thus holds for all solutions. It is of considerable interest to find a finite set of basic stochastic relations that through some construction procedure implies all consequences of some given system of stochastic relations. This is exactly the kind of question addressed in the formal theory of partial differential equations and the tools developed there can now through the maximum entropy principle be brought into the area of nonparametric statistics.\n### Thermodynamic pressure and its cousins\n[eq: pressure] We will now investigate an important consequence of the fundamental variational identity ([eq: FundamentalVariationalIdentity]) for the thermodynamic case when the total energy is the only observable. For this special case the variational identity turns into\n \n$$\\begin{aligned}\n\\left< \\delta\\_\\Gamma\\mathcal{ H} \\right>&= -kT\\;\\delta\\_\\Gamma \\ln Z.{\\label{eq: FundamentalVariationalIdentity1}}\n \\end{aligned}$$ \nThe force acting at a boundary point $\\vb{x}$ can, taking into account the fact that the state of the system is determined by the position and momenta of all the *n* particles comprising the system, be given by a function\n \n$$\\begin{aligned}\n \\vb{F}&=\\vb{F}(\\vb{p},\\vb{q},\\vb{x}),\\;\\;\\;\\;\\;\\vb{x}\\in\\Gamma,\\;(\\vb{p},\\vb{q})\\in\\mathbb{R}^{6n}.{\\label{eq: BoundaryForce}}\n \\end{aligned}$$ \nA small deformation of the boundary is determined by an infinitesimal deformation vector field $d\\vb{r}\\_{\\Gamma}$ defined on the boundary Γ. The change in total energy induced by this deformation is given by\n \n$$\\begin{aligned}\n\\delta\\_{\\Gamma}\\mathcal{H}(\\vb{p},\\vb{q};\\Gamma) &=-\\int\\_{\\Gamma}\\;d\\vb{x}\\;\\vb{F}(\\vb{p},\\vb{q},\\vb{x})\\cdot d\\vb{r}\\_{\\Gamma}.\n \\end{aligned}$$ \nThe fundamental variational identity ([eq: FundamentalVariationalIdentity]) now gives\n \n$$\\begin{aligned}\n\\int\\_{\\Gamma}\\;d\\vb{x}\\;\\left<\\vb{F}(\\vb{p},\\vb{q},\\vb{x}) \\right>\\cdot d\\vb{r}\\_{\\Gamma}&= kT\\;\\delta\\_\\Gamma \\ln Z.{\\label{eq: FundamentalVariationalIdentity2}}\n \\end{aligned}$$ \nWe will in the following only consider the common situation defined by\n \n$$\\begin{aligned}\n \\left<\\vb{F}(\\vb{p},\\vb{q},\\vb{x}) \\right>&=p(\\vb{x})\\;\\vb{n},{\\label{eq: PressureAssumption}}\n \\end{aligned}$$ \nwhere $\\vb{n}$ is the unit normal for the surface Γ. Note that by definition, *p* is now the pressure for the system. For this case ([eq: FundamentalVariationalIdentity2]) turns into the identity\n \n$$\\begin{aligned}\n\\int\\_{\\Gamma}\\;d\\vb{x}\\;p(\\vb{x})\\;\\vb{n}\\cdot d\\vb{r}\\_{\\Gamma}&= kT\\;\\delta\\_\\Gamma\\ln Z.{\\label{eq: FundamentalVariationalIdentity2}}\\end{aligned}$$ \nLet us first consider the case of a smooth surface, and for this kind of surface, let us consider an infinitesimal variation of the surface that is a pure expansion or contraction. This means that $d\\vb{r}\\_{\\Gamma}=\\vb{n}ds$. For this kind of variation the fundamental variational identity ([eq: FundamentalVariationalIdentity2]) takes the form\n \n$$\\begin{aligned}\nds\\;\\int\\_{\\Gamma}\\;d\\vb{x}\\;p(\\vb{x})&= kT\\;\\delta\\_\\Gamma\\ln Z.{\\label{eq: FundamentalVariationalIdentity3}}\\end{aligned}$$ \nUsing the fact that the volume spanned by the deformation is *d*Γ*V* = *A*(Γ)*d**s*, where *A*(Γ) is the area of the surface, we have\n \n$$\\begin{aligned}\n\\left<p\\right>\\_{\\Gamma}= kT\\;\\frac{\\delta\\_\\Gamma\\ln Z}{d\\_{\\Gamma}V},{\\label{eq: FundamentalVariationalIdentity3.5}}\\end{aligned}$$ \nwhere $\\left<p\\right>\\_{\\Gamma}$ is the average of the pressure over the surface of the cavity.\nFor some important cases the partition function depends on the surface only through the volume. For this situation we have\n \n$$\\begin{aligned}\n\\delta\\_\\Gamma\\ln Z=\\frac{\\partial\\ln Z}{\\partial\\;V}d\\_{\\Gamma}V,\\nonumber\\end{aligned}$$ \nso that\n \n$$\\begin{aligned}\n\\left<p\\right>\\_{\\Gamma}= kT\\;\\frac{\\partial\\ln Z}{\\partial\\;V}.{\\label{eq: FundamentalVariationalIdentity4}}\\end{aligned}$$ \nThis is the standard formula for the thermodynamic pressure that one finds in any textbook. It is very frequently true that, independently of the shape, the partition function for large cavities depends only on the volume of the cavity. This may however not be the case for smaller cavities and for such cases we must retreat to the more general identity ([eq: FundamentalVariationalIdentity3.5]). It is easy to verify that the indentity ([eq: FundamentalVariationalIdentity4]) holds for any surface smooth or not. If the deformation is a pure expansion or contraction of a part of the surface defined by Γ0 ⊂ Γ, we also get the identity ([eq: FundamentalVariationalIdentity4]), but now with Γ → Γ0.\nThe general variational identity for the entropy ([eq: dS5]) takes for the particular case discussed in this section the form\n \n$$\\begin{aligned}\n\\delta S&=\\frac{1}{kT}d\\left<\\mathcal{H}\\right>-\\frac{1}{kT}\\left< \\delta\\_\\Gamma\\mathcal{ H} \\right>\\nonumber\\\\\n&=\\frac{1}{kT}d\\left<\\mathcal{H}\\right>+\\delta\\_\\Gamma \\ln Z\\nonumber\\\\\n&=\\frac{1}{kT}d\\left<\\mathcal{H}\\right>+\\frac{1}{kT}\\left<p\\right>\\_{\\Gamma}d\\_{\\Gamma}V,\\nonumber\\end{aligned}$$ \nwhich can be rewritten as\n \n$$\\begin{aligned}\nkTdS=dE+\\left<p\\right>\\_{\\Gamma}d\\_{\\Gamma}V,{\\label{eq: dS6}}\\end{aligned}$$ \nwhere we have used ([eq: FundamentalVariationalIdentity3.5]) and where now $E=\\left<\\mathcal{H}\\right>$ is the energy of the system. We recognize ([eq: dS6]) as one of the fundamental formulas from conventional thermodynamics.\nIn this section we have done the derivation of the formulas for the Thermodynamical pressure for the case of a classical system. However, the derivation of the pressure formula for the case of quantum systems leads the exact same formulas. If there are more observables in addition to the energy, for example total momentum and/or total angular momentum, the pressure formulas must be generalized. The derivation of the generalizations follow the pattern laid down in this section.\nThe maximum entropy principle for quantum systems\n-------------------------------------------------\nWe have in the first section of these notes introduced the thermodynamical formalism in the context of classical physics and classical observables. It involved a state space that was finite or at least finite dimensional, and the challenge was to determine which probability distribution on the state space is the best to use, given the means of a finite number of observables of the system. The solution to this problem was to choose the probability distribution that maximized the entropy functional ([eq: 78]) under the constraints determined by the given means. At the face of it, for a quantum system, the situation appears to be very different. For this case the state space is an infinite dimensional Hilbert space and the full information that an observer has is encoded in the *density operator* for the system. This is a self-adjoint positive operator $\\hat\\rho$, on the Hilbert space with trace equal to one.\n \n$$\\begin{aligned}\n\\mathrm{Tr}(\\hat\\rho)=1.\\nonumber\\end{aligned}$$ \nThe expectation value of any quantum observable, $\\hat A$ is by definition\n \n$$\\begin{aligned}\n\\left<\\hat A\\right>=Tr(\\hat\\rho\\hat A).{\\label{eq: QuantumMean}}\\end{aligned}$$ \nThe question one poses is which density operator should be used if we only know the expectation value of a finite number of quantum observables $\\hat A\\_i$.\n \n$$\\begin{aligned}\n\\left<\\hat A\\_i\\right>&=a\\_i,\\;\\;\\;i=1,2,...p.{\\label{eq: Constraints}}\\end{aligned}$$ \nEven though there are real differences between the classical and the quantum case, much is also the same.\nThe analog of the Gibbs entropy measure ([eq: 79]) is the Von Neuman entropy measure for density operators given by\n \n$$\\begin{aligned}\nS(\\hat\\rho)=-Tr(\\hat\\rho\\ln\\hat\\rho).{\\label{eq: NeumanEntropy}}\\end{aligned}$$ \nThe solution to the question posed on the previous page proposed by the Maximum entropy method is to choose the density operator that maximize the Von Neuman entropy while satisfying the constraints ([eq: Constraints]). It is a simple exercise to show that the unique solution to this maximization problem is the following density operator\n \n$$\\begin{aligned}\n \\hat\\rho=\\frac{1}{Z(\\lambda\\_1,...,\\lambda\\_p)}\\exp(-\\sum\\_i\\;\\lambda\\_i\\hat A\\_i).{\\label{eq: MaxEntDensityOperator}}\n \\end{aligned}$$ \nThis operator is the *maximum entropy density operator*. The function *Z* is the partition function and is given by\n \n$$\\begin{aligned}\n Z(\\lambda\\_1,...,\\lambda\\_p)=Tr\\left\\{\\exp(-\\sum\\_i\\;\\lambda\\_i\\hat A\\_i)\\right\\}.{\\label{eq: PartitionFunction}}\n \\end{aligned}$$ \nArguing like in the classical case we find that much of the thermodynamic formalism is the same as before. Spesifically we have\n \n$$\\begin{aligned}\n S&=\\ln Z+\\sum\\_{i=1}^p\\;\\lambda\\_i\\left<\\hat A\\_i\\right>,\\nonumber\\\\\n \\left<\\hat A\\_i\\right>&=-\\frac{\\partial \\ln Z}{\\partial \\lambda\\_i},\\nonumber\\\\\n \\lambda\\_i&=\\frac{\\partial S}{\\partial \\left<\\hat A\\_i\\right>},\\\\\n\\sum\\_i\\lambda\\_i \\left<\\frac{\\partial \\hat A\\_i}{\\partial \\alpha}\\right>&=-\\frac{\\partial \\ln Z}{\\partial \\alpha},{\\label{eq: QThermodynamicFormalism}}\\end{aligned}$$ \nwhere in the last identity we assume that all the observables depend on some parameter *α*. In our application of this formalism the operators will depend on the surface enclosing a domain *D*, and for this case we get a quantum analog to the classical formula ([eq: FundamentalVariationalIdentity])\n \n$$\\begin{aligned}\n \\delta\\_\\Gamma\\ln Z=-\\mathlarger{\\sum}\\_j \\; \\lambda\\_j \\;\\left< \\delta\\_\\Gamma \\hat A\\_j\\right>.{\\label{eq: QuantumFundamentalVariationalIdentity}}\n \\end{aligned}$$ \nAs we can see, much of the thermodynamic formalism is the same for the classical and the quantum case. However some things are different, or they at least appear to be different. In the classical case we can find correlations between different observables by computing mixed partial derivatives of the partition function as shown in ([eq: 215]) and ([eq: 216]). In the quantum case this is problematic unless the operators commute. For the case of two observables $\\hat A$ and $\\hat B$ we have for example\n \n$$\\begin{aligned}\n \\frac{\\partial^2 Z}{\\partial\\_{\\mu}\\partial\\_{\\lambda}}= \\frac{\\partial^2 Z}{\\partial\\_{\\lambda}\\partial\\_{\\mu}}=2\\left<\\hat A\\hat B+\\hat B\\hat A\\right>-\\left<\\hat A\\right>\\left<\\hat B\\right>.\\nonumber\n \\end{aligned}$$ \nIn a sense this should not come as a surprise. The reason for this is that $\\hat A\\hat B$ is not in general Hermitian even if both $\\hat A$ and $\\hat B$ are. Thus $\\left<\\hat A\\hat B\\right>$ is not something that you could ever measure, so it does not make sense to try to predict it. However, $\\hat A\\hat B+\\hat B\\hat A$ \\it is a Hermitian operator and thus $\\left<\\hat A\\hat B+\\hat B\\hat A\\right>$ is something one would want to predict. And, this is exactly what you would be able to predict using the thermodynamical formalism.\nThere is another important way in which the classical and quantum cases are different; namely the question of how to include prior information about the system into the thermodynamical formalism. In the classical case this was taken care of by using the entropy measure ([eq: 79]) that included the prior distribution *ρ*0. We have seen how maximization the entropy in the context of statistical mechanics leads to the distribution\n \n$$\\begin{aligned}\n\\rho ({\\vb{x}}\\_1,...,{\\vb{x}}\\_n, \\vb{p}\\_1,..., \\vb{p}\\_n) = \\frac{\\rho\\_0}{Z} \\; \\exp(-\\frac{\\mathcal{H}}{kT}), {\\label{eq: 217}} \\end{aligned}$$ \nwhich we recognized to be the Gibbs canonical ensemble. This is however not entirely correct, the canonical ensemble corresponds to the case when we have a uniform prior. We could have gotten this distribution directly from a maximization of the functional\n \n$$\\begin{aligned}\nS(\\rho) = - \\int\\_{\\mathbf{R}^n} dV \\; \\rho \\; \\ln (\\rho). {\\label{eq: GibbsEntropy}} \\end{aligned}$$ \nThis is in fact the functional used by Gibbs in his foundational studies of thermodynamics. It is this functional that is called the Gibbs entropy measure. The functional we introduced in ([eq: 79]) measure by definition the relative entropy of *ρ* with respect to *ρ*0. It is also denoted by other names in the research literature.\nThe Von Neuman entropy introduced in ([eq: NeumanEntropy]) is the direct analog of the Gibbs entropy measure ([eq: GibbsEntropy]). However, in contrast to the classical case, there does not exists a measure of information for quantum systems that is universally recognized to be the best measure to use in all cases where there is prior information to take into account. Many different kind of measures has been studied by physicists and mathematicians over the years. Today these questions are intensely pursued in the topical field of quantum information theory.\nNote that the classical relative entropy measure can be written in the form\n \n$$\\begin{aligned}\nS(\\rho|\\;\\rho\\_0) = - \\int\\_{\\mathbf{R}^n} dV \\; \\rho \\; (\\ln (\\rho-\\ln\\rho\\_0)). {\\label{eq: RelativeEntropy}} \\end{aligned}$$ \nOne approach to a quantum version of relative entropy is to directly generalize ([eq: RelativeEntropy]) to the quantum case. This gives us for any pair of density operators $\\hat\\rho$ and $\\hat\\rho\\_0$ the relative quantum entropy in the form\n \n$$\\begin{aligned}\nS\\_Q(\\hat \\rho|\\;\\hat\\rho\\_0) = - Tr( \\hat \\rho \\; (\\ln \\hat \\rho-\\ln\\hat\\rho\\_0)). {\\label{eq: QRealtiveEntropy}} \\end{aligned}$$ \nOne could now guess that the corresponding maximal entropy distribution, when the only observable is the total energy, will take the form\n \n$$\\begin{aligned}\n\\hat\\rho = \\frac{\\hat\\rho\\_0}{Z} \\; \\exp(-\\frac{\\mathcal{\\hat H}}{kT}), {\\label{eq: QMaxEntDensityOperator}} \\end{aligned}$$ \nHowever, a formula like this can not possibly be correct because the right hand side of ([eq: QMaxEntDensityOperator]) is not a Hermitian operator unless the prior $\\hat\\rho\\_0$ commutes with the total energy operator \\hat H.\nHere we will not pursue these matters, but rather use the Neuman entropy measure as a basis for the theory. But we should keep in mind that there is a real issue here concerning the general validity of the thermodynamical formalism for quantum systems in cases when there is prior information present.\nCanonical quantization of electromagnetic fields in a cavity\n============================================================\nQuantum theory, as we know it today, originated from the investigation of electromagnetic fields confined to a cavity. At first however, the insights gained from the study of this system by Planck, Einstein and others did not lead to a quantum description of electromagnetic fields in cavities, but rather to the quantum description of atoms and molecules by Schrødinger and Heisenberg. This is the theory that today is known as *Quantum Mechanics*. Compared to the quantum description of fields(QFT), the electromagnetic field in particular, the quantum theory of atoms and molecules is conceptually and technically much simpler. Of course, calculating quantum properties of atoms and molecules is not in any way simple, it is just that QFT is so very hard.\nBy the end of the 1920s, quantum mechanics was well established and widely regarded as the correct theory of atoms and molecules. The general principles and abstract mathematical structures underlying quantum mechanics was at this time known, if not entirely understood. Books published by P. A. M. Dirac in 1930 and John von Neuman in 1932 gave a precise description of these underlying structures and principles. What they describe in these two books is what today is known as *Quantum Theory*. It has been realized over time that this theory, Quantum Theory, is not actually a physical theory at all, but rather a general mathematical scheme that *any* fundamental physical theory must conform to.\nW. Heisenberg, Max Born and Pascual Jordan were in 1926 the first physicists to construct a viable quantum theory of the electromagnetic field. The theory they constructed is what we today call a quantum theory of *free fields*.\nTheir approach was to expand the electromagnetic field in modes in such a way that the total energy of the system, which in classical mechanics is known as the *Hamiltonian* for the system, became a sum of an infinite set of independent harmonic oscillator Hamiltonians. Each of these harmonic oscillators could then be quantized using the Heisenberg canonical quantization rules for bosonic particle systems that that were well understood at the time.\nOf course, for a quantum theory of the electromagnetic field to become a useful and flexible tool for the description of nature, the interaction between electrons and the electromagnetic fields had to be taken into account and given a description that conformed to the rules of Quantum Theory. This was achieved by Dirac in 1927.\nIn these notes we will follow the simple approach introduced by Heisenberg, Born and Jordan in 1926, and we will start by considering the very simplest case of an empty cavity with perfectly reflecting walls. Maxwell’s equations for the electromagnetic field in an empty cavity, *D* are\n \n$$\\begin{aligned}\n\\curl \\bf{E}+\\mu\\_0\\partial\\_t\\vb{H}&=0,\\nonumber\\\\\n\\curl\\vb{H}-\\epsilon\\_0\\partial\\_t\\vb{E}&=0,\\nonumber\\\\\n\\div\\vb{E}&=0,\\nonumber\\\\\n\\div\\vb{H}&=0.{\\label{eq: Maxwell1}}\\end{aligned}$$ \nThese equations are supplemented with boundary conditions on the perfectly reflecting bounding surface Γ defining the cavity *D*\n \n$$\\begin{aligned}\n\\vb{n}\\times\\vb{E}(\\vb{x},t)&=0,\\nonumber\\\\\n\\vb{n}\\cdot\\vb{H}&=0,\\;\\;\\;\\;\\forall \\bf{x}\\in\\Gamma.{\\label{eq: BC1}}\\end{aligned}$$ \nFrom the Maxwell equations ([eq: Maxwell1]) it is easy to show that the the electric field solves the equation\n \n$$\\begin{aligned}\n\\curl(\\curl\\vb{E})&=(\\frac{\\omega}{c})^2\\vb{E},\\nonumber\\\\\n\\div\\vb{E}=0.{\\label{eq: AProblem}}\\end{aligned}$$ \nThe magnetic field solves the exact same equation. In order to solve this problem we introduce an eigenvalue problem\n \n$$\\begin{aligned}\n\\curl(\\curl\\vb{u})&=(\\frac{\\omega}{c})^2\\vb{u},\\nonumber\\\\\n\\div\\vb{u}=0,\\nonumber\\\\\n\\vb{n}\\times\\vb{u}=0.{\\label{eq: EigenvalueProblem}}\\end{aligned}$$ \nOne can show that this eigenvalue problem has a complete set of orthogonal eigenfunction $\\vb{u}\\_{\\vb{m}}$ and corresponding eigenvalues $(\\frac{\\omega\\_{\\vb{m}}}{c})^2$. We will assume that the eigenfunctions has been normalized so that\n \n$$\\begin{aligned}\n \\int\\_Dd\\vb{x}\\;\\vb{u}\\_{\\vb{m}}\\cdot\\vb{u}\\_{\\vb{m}'}=\\delta\\_{\\vb{m}\\;\\vb{m}'}.\\nonumber\n \\end{aligned}$$ \nObserve however, that there appears to be some kind of problem here. For a particular eigenfunction we have a corresponding electric field mode\n \n$$\\begin{aligned}\n \\vb{E}\\_{\\vb{m}}(\\vb{x},t)&=\\vb{u}\\_{\\vb{m}}(\\vb{x})e^{-i\\;\\omega\\_{\\vb{m}}\\; t},\\nonumber\\end{aligned}$$ \nand the boundary conditions on the eigenfunctions $u\\_{\\vb{m}}$ ensure that the field mode satisfy the boundary condition $\\vb{n}\\times\\vb{E}\\_{\\vb{m}}=0$. However it is *not* obvious that the corresponding magnetic field satisfy the boundary condition $\\vb{n}\\cdot\\vb{H}=0$. The reason for this is that it is not in general true that $\\vb{n}\\cdot\\curl\\vb{f}=0$ is implied by $\\vb{n}\\times\\vb{f}=0$ for a vector field $\\vb{f}$ defined on a domain *D* bounded by a surface whose unit normal is $\\vb{n}$. The resolution of this apparent problem is somewhat subtle. The fact of the matter is that the result derived is a pure *existence* result, it is not constructive. It only ensures the existence of a set of electric field modes\n \n$$\\begin{aligned}\n \\vb{E}\\_{\\vb{m}}(\\vb{x},t)&=\\vb{u}\\_{\\vb{m}}(\\vb{x})e^{-i\\;\\omega\\_{\\vb{m}}\\; t},\\nonumber\n \\end{aligned}$$ \nsuch that any solution to the problem ([eq: Maxwell1]),([eq: BC1]) can be written as an expansion in $\\vb{E}\\_{\\vb{m}}$ and $\\curl\\vb{E}\\_{\\vb{m}}$. In order to actually *construct* the modes in any particular case we will end up with an situation where one of the two boundary conditions ([eq: BC1]) is automatically satisfied and where imposing the second one will determine the eigenvalues.\nUsing the modes $\\vb{u}\\_{\\vb{m}}$ we can write down following expansions for the electric and magnetic field\n \n$$\\begin{aligned}\n \\vb{E}(\\vb{x},t)&=-\\sum\\_{\\vb{m}}\\frac{\\dot q\\_{\\vb{m}}(t)}{\\sqrt{\\epsilon\\_0}}\\;\\vb{u}\\_{\\vb{m}}(\\vb{x}),{\\label{eq: EExpansion1}}\\\\\n \\vb{H}(\\vb{x},t)&=\\sum\\_{\\vb{m}}\\frac{q\\_{\\vb{m}}(t)}{\\mu\\_0\\sqrt{\\epsilon\\_0}}\\;\\curl \\vb{u}\\_{\\vb{m}}(\\vb{x}),{\\label{eq: HExpansion1}}\n \\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n q\\_{\\vb{m}}''(t)=-\\omega\\_{\\vb{m}}q\\_{\\vb{m}}(t).{\\label{eq: HarmonicOscillator1}}\n \\end{aligned}$$ \nThis is the equation for a harmonic oscillator, and it, together with ([eq: EigenvalueProblem]) ensure that the expansions ([eq: EExpansion1]),([eq: HExpansion1]) for the electric and magnetic fields represents the general solution of the Maxwell equations in a cavity with perfectly conducting walls.\nThe total electromagnetic energy in the cavity is\n \n$$\\begin{aligned}\n\\mathcal{H}= \\frac{1}{2}\\;\\int\\_Dd\\vb{x}\\left(\\epsilon\\_0\\vb{E}\\cdot\\vb{E}+\\mu\\_0\\vb{H}\\cdot\\vb{H}\\right).{\\label{eq: TotalEnergy}}\n \\end{aligned}$$ \nInserting the expansions for the electric and magnetic fields, and using the orthonormality of the modes and their defining equation ([eq: EigenvalueProblem]), we immediately get\n \n$$\\begin{aligned}\n \\mathcal{H}=\\sum\\_{\\vb{m}}\\frac{1}{2}\\left(\\dot q\\_{\\vb{m}}^2+\\omega\\_{\\vb{m}}^2q\\_{\\vb{m}}^2\\right){\\label{eq: OscillatorExpansion1}}\n \\end{aligned}$$ \nWe observe that each term in this Hamiltonian is the Hamiltonian for a harmonic oscillator of unit mass. We have now realized a description of the electromagnetic field as an infinite system of uncoupled harmonic oscillators. This is the starting point for the Heisenberg, Born, Jordan approach to quantizing free fields in a cavity.\nHowever, before we proceed with the quantization of the harmonic oscillators, we will generalize the situation in a way that will cover all cases of interest to us in these notes.\nAs a matter of fact, the Heisenberg, Born, Jordan approach to quantizing the electromagnetic field does not only work for an empty cavity, it also works for a cavity containing electrons and atoms. The restriction is that there are no free electrons; all electrons are bound to atoms. Furthermore, we require that the interaction between the electromagnetic field and and the atoms can be accurately modeled by a macroscopic *real* refractive index, thus no absorption is allowed. A further essential restriction is that the material response is *local in time*; thus no material dispersion is allowed. We will however allow for the possibility that the refractive index vary from point to point in the cavity. It is not an essential requirement, but we will in these notes also assume that the material is nonmagnetic.\nThe assumptions made amounts the following relations connecting the fields $\\vb{B}$ and $\\vb{D}$ to $\\vb{H}$ and $\\vb{E}$\n \n$$\\begin{aligned}\n \\vb{B}&=\\mu\\_0\\vb{H},\\nonumber\\\\\n \\vb{D}&=\\epsilon\\_0n(\\vb{x})^2\\vb{E}.{\\label{eq: DERealtions}}\n \\end{aligned}$$ \nWe will be considering a situation where one or several separate objects with different dielectric constants are contained in a cavity which has some constant background index. Let the different objects occupy domains *D*1, ...*D**p* in the cavity with refractive indices *n*1, ..., *n**p*. The boundary of object *j* is denoted by Γ*j*. Let *D*0 be the domain inside the cavity that remains after the domains *D*1, ...*D**p* has been subtracted from the domain defining the whole cavity, which is denoted by *D*. The background refractive index in *D*0 is denoted by *n*0. The outer boundary of the cavity is denoted by Γ0. By common convention the unit normal for Γ0, Γ1, ...Γ*p* are pointing out of the domains they define. The geometry of the situation is summarized in the following formulas\n \n$$\\begin{aligned}\nD&=\\cup\\_{j=0}^p D\\_j,\\nonumber\\\\\n\\partial D\\_j&=\\Gamma\\_j,\\;\\;\\;j=1,...p,\\nonumber\\\\\n\\partial D\\_0&=\\Gamma\\_0-\\cup\\_{j=1}^p\\Gamma\\_j,{\\label{eq: Geometry}}\\end{aligned}$$ \nwhere the minus sign signify that Γ*j*, as part of the boundary of *D*0, has a unit normal that points into the domain *D**j* since the inside of *D**j* is part of the outside of *D*0.\nIt is easy to show that the electric field satisfy the equation\n \n$$\\begin{aligned}\n\\curl(\\curl\\vb{E})&=(\\frac{\\omega}{c})^2n\\_j^2\\vb{E}\\;\\;\\;\\;\\;\\vb{x}\\in D\\_j,\\nonumber\\\\\n\\div\\vb{E}=0,{\\label{eq: EProblem}}\\end{aligned}$$ \nas does the magnetic field. With this equation in mind we introduce the following eigenvalue problem\n \n$$\\begin{aligned}\n\\left. \\begin{array}{ccc}\n&\\curl(\\curl\\vb{u})=(\\frac{\\omega}{c})^2n\\_j^2\\vb{u}\\\\\n & \\\\\n&\\div\\vb{u}=0\n\\end{array} \\right\\} \\vb{x}\\in D\\_j\\;\\;\\;\\;j=1,...p.{\\label{eq: EigenValueProblem2}}\\end{aligned}$$ \nIn order to complete the description of the eigenvalue problem we must supply boundary conditions on all bounding surfaces Γ0, Γ1, ..., Γ*p*.\nFor the boundaries internal to the cavity Γ1, ..., Γ*p* we will assume the usual electromagnetic interface conditions for dielectric materials. For the external boundary Γ0 we will either assume that the boundary is perfectly reflecting and thus use the same boundary conditions as in ([eq: EigenvalueProblem]), for this case the bounding surface can take on any shape, or for the special case of a rectangular cavity we will use a combination of periodic boundary conditions connecting fields on all or some pairs of opposing surfaces, and use perfectly reflecting surface conditions for the remaining faces. The eigenvalue problem is now fully specified and we will assume that this eigenvalue problem has a complete set of orthonormal eigenfunctions $u\\_{\\vb{m}}$, where the orthonormalization condition is defined by\n \n$$\\begin{aligned}\n&\\int\\_{D}d\\vb{x\\;}n(\\vb{x})^2\\vb{u}\\_{\\vb{m}}\\cdot\\vb{u}\\_{\\vb{m'}}\\nonumber\\\\\n&=\\sum\\_{j=0}^p\\int\\_{D\\_j}d\\vb{x}\\;n\\_j^2\\vb{u}\\_{\\vb{m}}\\cdot\\vb{u}\\_{\\vb{m'}}=\\delta\\_{\\vb{m}\\vb{m'}}.{\\label{eq: Normalization}}\\end{aligned}$$ \nIt is simple to show that the general solution of the Maxwell equations for this more complicated cavity situation still are of the form\n \n$$\\begin{aligned}\n \\vb{E}(\\vb{x},t)&=-\\sum\\_{\\vb{m}}\\frac{\\dot q\\_{\\vb{m}}(t)}{\\sqrt{\\epsilon\\_0}}\\;\\vb{u}\\_{\\vb{m}}(\\vb{x}),\\nonumber\\\\\n \\vb{H}(\\vb{x},t)&=\\sum\\_{\\vb{m}}\\frac{q\\_{\\vb{m}}(t)}{\\mu\\_0\\sqrt{\\epsilon\\_0}}\\;\\curl \\vb{u}\\_{\\vb{m}}(\\vb{x}),{\\label{eq: HEexpansion2}}\n \\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n q\\_{\\vb{m}}''(t)=-\\omega\\_{\\vb{m}}q\\_{\\vb{m}}(t),{\\label{eq: HarmonicOscillator2}}\\end{aligned}$$ \nand where now, of course, the functions $\\{\\vb{u}\\_{\\vb{m}}\\}$ are solutions to the more complicated eigenvalue problem ([eq: EigenValueProblem2]). In Appendix A we show that also for this more complicated situation the total energy of the system can be written in the form\n \n$$\\begin{aligned}\n\\mathcal{H}=\\sum\\_{\\vb{m}}\\frac{1}{2}\\left(\\dot q\\_{\\vb{m}}^2+\\omega\\_{\\vb{m}}^2q\\_{\\vb{m}}^2\\right).{\\label{eq: OscillatorExpansion1}}\\end{aligned}$$ \nIn orther to arrive at this expression for the total energy we must use both the boundary conditions at the outer surface, Γ0, as for the empty cavity, but also use the boundary conditions at the internal boundaries Γ*j*, *j* = 1, ...*p*.\nThe Heisenberg, Born, Jordan quantization now proceeds by first introducing the canonical momentum variables\n \n$$\\begin{aligned}\n p\\_{\\vb{m}}=\\dot q\\_{\\vb{m}},\\nonumber\n \\end{aligned}$$ \nso that the Hamiltonian take the form\n \n$$\\begin{aligned}\n\\mathcal{H}=\\sum\\_{\\vb{m}}\\mathcal{H}\\_{\\vb{m}},{\\label{eq: OscillatorExpansion2}}\n \\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n\\mathcal{H}\\_{\\vb{m}}=\\frac{1}{2}\\left(p\\_{\\vb{m}}^2+\\omega\\_{\\vb{m}}^{2}q\\_{\\vb{m}}^2\\right).{\\label{eq: SingleOscillator}}\n \\end{aligned}$$ \nThe expression ([eq: OscillatorExpansion2]) is now the Hamiltonian for an uncoupled system of independent harmonic oscillators, one oscillator ([eq: SingleOscillator]) for each mode index $\\vb{m}$. We now perform canonical quantization to each of these harmonic oscillators by letting\n \n$$\\begin{aligned}\n q\\_{\\vb{m}}\\longrightarrow\\hat q\\_{\\vb{m}},\\nonumber\\\\\n p\\_{\\vb{m}}\\longrightarrow\\hat p\\_{\\vb{m}},\\nonumber\n \\end{aligned}$$ \nand imposing the bosonic commutation rules on the operators $\\hat q\\_{\\vb{m}}$ and $\\hat p\\_{\\vb{m}}$\n \n$$\\begin{aligned}\n \\left[\\hat q\\_{\\vb{m}},\\hat p\\_{\\vb{m}'}\\right]&=i\\hbar\\delta\\_{\\vb{m},\\vb{m}'},\\nonumber\\\\\n \\left[\\hat q\\_{\\vb{m}},\\hat q\\_{\\vb{m}'}\\right]&=0,\\nonumber\\\\\n \\left[\\hat p\\_{\\vb{m}},\\hat p\\_{\\vb{m}'}\\right]&=0.{\\label{eq: HarmonicOscillatorCommutationRules}}\n \\end{aligned}$$ \nThe Hamiltonian operators corresponding to the classical Hamiltonian ([eq: OscillatorExpansion1]) is now\n \n$$\\begin{aligned}\n\\mathcal{\\hat H}=\\sum\\_{\\vb{m}}\\frac{1}{2}\\left(\\hat p\\_{\\vb{m}}^2+\\omega\\_{\\vb{m}}^{2}\\hat q\\_{\\vb{m}}^2\\right).{\\label{eq: HOperator1}}\\end{aligned}$$ \nWe now introduce annihilation and creation operators for each harmonic oscillator\n \n$$\\begin{aligned}\n\\hat a\\_{\\vb{m}}&=\\frac{1}{\\sqrt{2\\hbar\\omega\\_{\\vb{m}}}}\\left(\\omega\\_{\\vb{m}}\\hat q\\_{\\vb{m}}+i\\hat p\\_{\\vb{m}}\\right),\\nonumber\\\\\n\\hat a\\_{\\vb{m}}^{\\dagger}&=\\frac{1}{\\sqrt{2\\hbar\\omega\\_{\\vb{m}}}}\\left(\\omega\\_{\\vb{m}}\\hat q\\_{\\vb{m}}-i\\hat p\\_{\\vb{m}}\\right).{\\label{eq: CreationAnnihilationOperators}}\\end{aligned}$$ \nFrom the canonical commutation rules ([eq: HarmonicOscillatorCommutationRules]) it is easy to show that\n \n$$\\begin{aligned}\n[\\hat a\\_{\\vb{m}},\\hat a\\_{\\vb{p}}^{\\dagger}]&=\\delta\\_{\\vb{m}\\vb{p}},\\nonumber\\\\\n[\\hat a\\_{\\vb{m}},\\hat a\\_{\\vb{p}}]&=0,\\nonumber\\\\\n[\\hat a\\_{\\vb{m}}^{\\dagger},\\hat a\\_{\\vb{p}}^{\\dagger}]&=0.{\\label{eq: BosonicCommutationRules}}\\end{aligned}$$ \nInverting the identities ([eq: CreationAnnihilationOperators]) we have\n \n$$\\begin{aligned}\n\\hat q\\_{\\vb{m}}&=\\sqrt{\\frac{\\hbar}{2\\omega\\_{\\vb{m}}}}\\left(\\hat a\\_{\\vb{m}}+\\hat a\\_{\\vb{m}}^{\\dagger}\\right),\\nonumber\\\\\n\\hat p\\_{\\vb{m}}&=i\\sqrt{\\frac{\\hbar\\omega\\_{\\vb{m}}}{2}}\\left(\\hat a\\_{\\vb{m}}-\\hat a\\_{\\vb{m}}^{\\dagger}\\right).{\\label{eq: qInTermsOfAC}}\\end{aligned}$$ \nUsing the annihilation and creation operators ([eq: CreationAnnihilationOperators]), the expression for the total energy operator ([eq: HOperator1]) takes the form\n \n$$\\begin{aligned}\n\\mathcal{\\hat H}=\\sum\\_{\\vb{m}}\\;\\hbar\\;\\omega\\_{\\vb{m}}\\;\\mathcal{\\hat N}\\_{\\vb{m}},{\\label{eq: EnergyOperator}}\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\n\\mathcal{\\hat N}\\_{\\vb{m}}=\\hat a\\_{\\vb{m}}^{\\dagger}\\hat a\\_{\\vb{m}},{\\label{eq: NumberOperator}}\\end{aligned}$$ \nis the number operator for the oscillator corresponding to index $\\vb{m}$. In this formula we have dropped the infinite contribution from the vacuum energy.\nIt is easy to verify, using the Heisenberg equations of motion, that the time dependence of the annihilation and creation operators are simply\n \n$$\\begin{aligned}\n\\hat a\\_{\\vb{m}}(t)&=\\hat a\\_{\\vb{m}}\\;e^{-i\\;\\omega\\_{\\vb{m}}\\; t},\\nonumber\\\\\n\\hat a\\_{\\vb{m}}^{\\dagger}&=\\hat a\\_{\\vb{m}}^{\\dagger}\\;e^{i\\;\\omega\\_{\\vb{m}}\\; t}.{\\label{eq: ACTimeDependence}}\\end{aligned}$$ \nUsing the identities ([eq: ACTimeDependence]) and ([eq: qInTermsOfAC]) in ([eq: HEexpansion2]) we find the following expressions for the quantized electric and magnetic field in the cavity\n \n$$\\begin{aligned}\n\\vb{\\hat E}(\\vb{x},t)&=-\\sum\\_{\\vb{m}}\\;i\\sqrt{\\frac{\\hbar\\omega\\_{\\vb{m}}}{2\\epsilon\\_0}}\\left(\\hat a\\_{\\vb{m}}e^{-i\\;\\omega\\_{\\vb{m}}\\; t}-\\hat a\\_{\\vb{m}}^{\\dagger}e^{i\\;\\omega\\_{\\vb{m}}\\; t}\\right)\\;\\vb{u}\\_{\\vb{m}}(\\vb{x}),\\nonumber\\\\\n\\vb{\\hat H}(\\vb{x},t)&=\\sum\\_{\\vb{m}}\\frac{1}{\\mu\\_0}\\sqrt{\\frac{\\hbar}{2\\omega\\_{\\vb{m}}\\epsilon\\_0}}\\left(\\hat a\\_{\\vb{m}}e^{-i\\;\\omega\\_{\\vb{m}}\\; t}+\\hat a\\_{\\vb{m}}^{\\dagger}e^{i\\;\\omega\\_{\\vb{m}}\\; t}\\right)\\;\\curl\\vb{u}\\_{\\vb{m}}(\\vb{x}),{\\label{eq: QuantumCavityFields}}\\end{aligned}$$ \nThe non-interacting many-particle bosonic density operator\n==========================================================\nFor a non-interacting bosonic many-body system in thermal equilibrium at a temperature *T*, the maximum entropy principle tell us that the following density operator\n \n$$\\begin{aligned}\n\\hat\\rho = \\frac{1}{Z} \\; \\exp(-\\frac{\\mathcal{\\hat H}}{kT}), {\\label{eq: QMaxEntDensityOperator}} \\end{aligned}$$ \nis the one that in the best way represent the limited information\n \n$$\\begin{aligned}\n \\left< \\mathcal{\\hat H}\\right>=E,\\nonumber\n \\end{aligned}$$ \nthat we have about the state of the many-body system. Using the expression for \\hat H from the previous section we have\n \n$$\\begin{aligned}\n\\hat\\rho = \\frac{1}{Z} \\; \\exp(-\\frac{1}{kT}\\sum\\_{\\vb{m}}\\;\\hbar\\;\\omega\\_{\\vb{m}}\\;\\mathcal{\\hat N}\\_{\\vb{m}}). {\\label{eq: BosonicDensityOperator}} \\end{aligned}$$ \nFor any observable $\\hat A$ for the bosonic system we compute the expectation value in the usual way\n \n$$\\begin{aligned}\n\\left<\\hat A\\right>&=Tr(\\hat\\rho\\;\\hat A)\\nonumber\\\\\n&=\\sum\\_i\\;<\\psi\\_i\\;| \\hat\\rho\\;\\hat A\\;|\\psi\\_i>,\\nonumber\\end{aligned}$$ \nwhere the vectors *ψ**i* is a basis for the state space. For the current situation the state space is the bosonic Fock-space. The basis for this space that we will use is the occupation number basis. The standard notation for this basis appear to us to be unwieldy and overpowering. We therefore introduce a different notation that is simpler and more compact.\nIn the current case and other cases that we will be interested in, the complete set of electromagnetic modes are indexed by a countable discrete set of indices, *I*, whose elements will be denoted by $\\vb{m},\\vb{n},\\vb{p}$ etc. We use fat type for the indices since most of the time the set of indices forms a countable subset of Z*r* for some natural number *r*. Let us denote the set of finite subsets of the index set, I, by P(*I*)\n \n$$\\begin{aligned}\n\\mathcal{P}(I)=\\{U\\subset I\\;|\\;| U|<\\infty\\}.\\end{aligned}$$ \nFor any set in *S* ∈ P(*I*), define\n \n$$\\begin{aligned}\n\\mathcal{F}(S)=\\{n:I\\rightarrow\\mathbb{N}\\cup\\{0\\}\\;|\\;n(\\vb{p})\\neq\\;0\\;\\Leftrightarrow\\;\\vb{p}\\in\\;S\\}.\\nonumber\\end{aligned}$$ \nThus, F(*S*) is the set of non-negative, integer valued functions with support *S* defined on the index set *I*. We are now ready to describe the occupation number basis for the many-body Bosonic Fock space. Denoting the orthonormal basis by B we have\n \n$$\\begin{aligned}\n\\mathcal{B}=\\{\\left|n\\right>\\;|\\;n\\in\\mathcal{F}(S),\\;S\\in\\mathcal{P}(I)\\;\\}{\\label{eq: FockBasis}}\\end{aligned}$$ \nThus any state in the many-body bosonic Fock space can be written as\n \n$$\\begin{aligned}\n\\left|\\psi\\right>=\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\;a\\_{S,n}\\;\\left|n\\right>.{\\label{eq: FockBasisExpansion}}\\end{aligned}$$ \nThis represents an expansion of the state vector $\\left|\\psi\\right>$ into a countable sum of basis vectors where each basis vector is determined by a finite subset of indices such that the corresponding set of oscillators are above the ground state. Observe that a state vector $\\left|n\\right>$ depends on the \\it set of mode indices *S* that defines the domain of the function *n*. Since a set, by definition, has no order assumed among it’s elements it follows that the state vector ([eq: FockBasisExpansion]) is fully symmetric with respect to permutation of the oscillator indices $\\vb{m}$. In the usual notation for state vectors one first assume an ordering of the indices $\\vb{m}\\_1,\\vb{m}\\_2,...$, write down an expression for the state vectors of the form $\\left|n\\_{\\vb{m}\\_1},n\\_{\\vb{m}\\_2},..\\right>$ and then remove the order by imposing permutation symmetry $\\left|n\\_{\\vb{m}\\_{\\sigma(1)}},n\\_{\\vb{m}\\_{\\sigma(2)}},..\\right>=\\left|n\\_{\\vb{m}\\_1},n\\_{\\vb{m}\\_2},..\\right>$. Here *σ* is an arbitrary permutation of the natural numbers.\nThe occupation number basis is orthonormal\n \n$$\\begin{aligned}\n\\left<n\\;|\\;n'\\right>=\\delta\\_{n,n'}.{\\label{eq: Normalization}}\\end{aligned}$$ \nCompleteness is expressed in terms of the identity\n \n$$\\begin{aligned}\n\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\left|n\\right>\\left<n\\right|=\\vb{1}.\\end{aligned}$$ \nWe will in the following find it useful to introduce a special type of Kronecker delta\n \n$$\\begin{aligned}\n\\delta\\_{\\vb{m},S}=\\left\\{ \n\\begin{tabular}{l}\n1 \\ \\ $\\mathbf{m}\\in S$ \\\\ \n0 \\ \\ $\\mathbf{m}\\notin S$\n\\end{tabular}.\n\\right. \\end{aligned}$$ \nFor each $\\vb{m}\\in I$ it is also useful to define a function $\\delta\\_{\\vb{m}}:I\\rightarrow\\{0,1\\}$ by\n \n$$\\begin{aligned}\n\\delta\\_{\\vb{m}}(\\mathbf{l})=\\left\\{ \n\\begin{tabular}{l}\n1 \\ \\ $\\mathbf{m}=\\mathbf{l}$ \\\\ \n0 \\ \\ $\\mathbf{m}\\neq\\mathbf{l}$\n\\end{tabular}\n\\right. \\;.\\end{aligned}$$ \nThe action of the annihilation and creation operators are then given by\n \n$$\\begin{aligned}\n\\hat a\\_{\\vb{m}}\\;\\left|n\\right>&=\\delta\\_{\\vb{m},S}\\;\\sqrt{ n(\\vb{m})}\\;\\left|n-\\delta\\_{\\vb{m}}\\right>,\\nonumber\\\\\n\\hat a\\_{\\vb{m}}^{\\dagger}\\;\\left|n\\right>&=\\sqrt{ n(\\vb{m})+1}\\;\\left|n+\\delta\\_{\\vb{m}}\\right>.{\\label{eq: CreationAnnihilationAction}}\\end{aligned}$$ \nFor the number operators $\\mathcal{\\hat N}\\_{\\vb{m}}$ we then as expected have\n \n$$\\begin{aligned}\n\\mathcal{\\hat N}\\_{\\vb{m}}\\;\\left|n\\right>&=\\hat a\\_{\\vb{m}}^{\\dagger}\\hat a\\_{\\vb{m}}\\;\\left|n\\right>=\\delta\\_{\\vb{m},S}\\;\\sqrt{ n(\\vb{m})}\\hat a\\_{\\vb{m}}^{\\dagger}\\;\\left|n-\\delta\\_{\\vb{m}}\\right>\\nonumber\\\\\n&=\\delta\\_{\\vb{m},S}\\; n(\\vb{m})\\;\\left|n\\right>.{\\label{eq: NumberOperatorAction}}\\end{aligned}$$ \nThermodynamical formalism for electromagnetic fields in a general cavity.\n=========================================================================\nIn the previous sections of this document we have developed the mathematical tools that we need to investigate the thermodynamics of electromagnetic fields for a large class of cavities.In this section we use these tools to calculate thermodynamical quantities and investigate relations between them.\nDirect verification of the thermodynamical formalism.\n-----------------------------------------------------\nLet us start with finding an expression for the partition function under the assumption that the only observable is the total energy. Using the notation from the previous section and the expression for the density operator for a non-interacting many-body bosonic system ([eq: BosonicDensityOperator]) we have\n \n$$\\begin{aligned}\nZ&=Tr(\\hat\\rho)\\nonumber\\\\\n&=\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\left<n\\right|\\; \\exp{-\\frac{1}{kT}\\sum\\_{\\vb{l}\\in I}\\;\\hbar\\;\\omega\\_{\\vb{l}}\\;\\mathcal{\\hat N}\\_{\\vb{l}}}\\;\\left|n\\right>\\nonumber\\\\\n&=\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\left<n\\right|\\;\\Pi\\_{\\vb{l}\\in I}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}}{kT}\\mathcal{\\hat N}\\_{\\vb{l}}}\\left|n\\right>\\nonumber\\\\\n&=\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\left<n\\right|\\;\\Pi\\_{\\vb{l}\\in I}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}\\delta\\_{\\vb{l},S}n(\\vb{l})}{kT}}\\left|n\\right>\\nonumber\\\\\n&=\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\Pi\\_{\\vb{l}\\in I}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}\\delta\\_{\\vb{l},S}n(\\vb{l})}{kT}}\\nonumber\\\\\n&=\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}.{\\label{eq: PartitionFunctionFormula1}}\\end{aligned}$$ \nRecall that the system we are discussing is contained in a cavity defined by a closed surface Γ in R3. Thus, the operator for the total energy depends on this surface and so does the partition function. From the expression ([eq: PartitionFunctionFormula1]), that we just derived, it is clear that the partition function depend in an implicit way on the surface through the spectrum, $\\omega\\_{\\vb{m}}$, of eigenvalues of the operator $L=\\curl(\\curl\\;)$ with the appropriate boundary conditions. Thus, in terms of direct variable dependence, the partition function is of the form\n \n$$\\begin{aligned}\nZ=Z(T,\\{\\omega\\_{\\vb{m}}\\}\\_{m\\in I})\\end{aligned}$$ \nLet us next calculate an expression for the expected number of photons in the electromagnetic mode indexed by $\\vb{m}\\in I$.\n \n$$\\begin{aligned}\nN\\_{\\vb{m}}&\\equiv\\left<\\mathcal{\\hat N}\\_{\\vb{m}}\\right>=Tr(\\rho\\;\\mathcal{\\hat N}\\_{\\vb{m}})\\nonumber\\\\\n&=\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\left<n\\right|\\;\\frac{1}{Z}\\; \\exp{-\\frac{1}{kT}\\sum\\_{\\vb{l}\\in I}\\;\\hbar\\;\\omega\\_{\\vb{l}}\\;\\mathcal{\\hat N}\\_{\\vb{l}}}\\;\\mathcal{\\hat N}\\_{\\vb{m}}\\;\\left|n\\right>\\nonumber\\\\\n&=\\frac{1}{Z}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\left<n\\right|\\;\\Pi\\_{\\vb{l}\\in I}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}}{kT}\\mathcal{\\hat N}\\_{\\vb{l}}}\\mathcal{\\hat N}\\_{\\vb{m}}\\;\\left|n\\right>\\nonumber\\\\\n&=\\frac{1}{Z}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\delta\\_{\\vb{m},S}\\;n(\\vb{m})\\;\\left<n\\right|\\;\\Pi\\_{\\vb{l}\\in I}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}}{kT}\\mathcal{\\hat N}\\_{\\vb{l}}}\\;\\left|n\\right>\\nonumber\\\\\n&=\\frac{1}{Z}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\delta\\_{\\vb{m},S}\\;n(\\vb{m})\\;\\Pi\\_{\\vb{l}\\in I}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}\\delta\\_{\\vb{l},S}n(\\vb{l})}{kT}}\\nonumber\\\\\n&=\\frac{1}{Z}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\delta\\_{\\vb{m},S}\\;n(\\vb{m})\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}.{\\label{eq: MeanOfNumberOperator1}}\\end{aligned}$$ \nThis is a rather complicated expression, but observe that\n \n$$\\begin{aligned}\n-\\frac{k\\;T}{\\hbar}\\frac{1}{Z}\\frac{\\partial\\;Z}{\\partial\\;\\omega\\_{\\vb{m}}}&=\\nonumber\\\\\n&=-\\frac{k\\;T}{\\hbar}\\frac{1}{Z}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\frac{\\partial}{\\partial\\_{\\omega\\_{\\vb{m}}}}\\left(\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}\\right)\\nonumber\\\\\n&=-\\frac{k\\;T}{\\hbar}\\frac{1}{Z}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\nonumber\\\\\n&\\left(-\\frac{\\hbar}{kT}\\right)\\delta\\_{\\vb{m},S}\\;n(\\vb{m})\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}\\nonumber\\\\\n&=\\frac{1}{Z}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\delta\\_{\\vb{m},S}\\;n(\\vb{m})\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}.{\\label{eq: MeanOfNumberOperator2}}\\end{aligned}$$ \nThus, we can conclude from ([eq: MeanOfNumberOperator1]) and ([eq: MeanOfNumberOperator2]) that\n \n$$\\begin{aligned}\nN\\_{\\vb{m}}\\equiv\\left<\\mathcal{\\hat N}\\_{\\vb{m}}\\right>=-\\frac{kT}{\\hbar}\\frac{\\partial\\;\\ln Z}{\\partial\\;\\omega\\_{\\vb{m}}}.{\\label{eq: MeanOfNumberOperator}}\\end{aligned}$$ \nThis thermodynamic formula is a special case of the general variational formula ([eq: QuantumFundamentalVariationalIdentity]) for the case at hand where both the observable \\hat H and the partition function *Z* depends on a countable set of parameters $\\{\\omega\\_{\\vb{m}}\\}\\_{m\\in I}$\n \n$$\\begin{aligned}\n\\frac{\\partial\\;\\ln Z}{\\partial\\;\\omega\\_{\\vb{m}}}&=-\\frac{1}{kT}\\left<\\frac{\\partial \\mathcal{\\hat H}}{\\partial\\;\\omega\\_{\\vb{m}}}\\right>,\\nonumber\\\\\n&\\Downarrow\\nonumber\\\\\n\\frac{\\partial\\;\\ln Z}{\\partial\\;\\omega\\_{\\vb{m}}}&=-\\frac{1}{kT}\\left<\\frac{\\partial }{\\partial\\;\\omega\\_{\\vb{m}}}\\left(\\sum\\_{\\vb{l}}\\;\\hbar\\;\\omega\\_{\\vb{l}}\\;\\mathcal{\\hat N}\\_{\\vb{l}}\\right)\\right>,\\nonumber\\\\\n&\\Downarrow\\nonumber\\\\\n\\frac{\\partial\\;\\ln Z }{\\partial\\;\\omega\\_{\\vb{m}}}&=-\\frac{\\hbar}{kT}\\left<\\;\\mathcal{\\hat N}\\_{\\vb{m}}\\;\\right>,\\nonumber\\\\\n&\\Updownarrow\\nonumber\\\\\n\\left<\\mathcal{\\hat N}\\_{\\vb{m}}\\right>&=-\\frac{kT}{\\hbar}\\frac{\\partial\\;\\ln Z}{\\partial\\;\\omega\\_{\\vb{m}}}.\\nonumber\\end{aligned}$$ \nUsing formula ([eq: MeanOfNumberOperator]) we get the following formula for the expected value of the total energy of the system\n \n$$\\begin{aligned}\nE&=\\left<\\mathcal{\\hat H}\\right>=\\sum\\_{\\vb{m}}\\hbar\\;\\omega\\_{\\vb{m}}\\left<\\mathcal{\\hat N}\\_{\\vb{m}}\\right>\\nonumber\\\\\n&=\\sum\\_{\\vb{m}}\\hbar\\;\\omega\\_{\\vb{m}}\\left(-\\frac{kT}{\\hbar}\\right)\\frac{\\partial\\;\\ln Z}{\\partial\\;\\omega\\_{\\vb{m}}}\\nonumber\\\\\n&=-kT\\;\\sum\\_{\\vb{m}}\\;\\omega\\_{\\vb{m}}\\frac{\\partial\\;\\ln Z}{\\partial\\;\\omega\\_{\\vb{m}}}.{\\label{eq: MeanOfEnergy}}\\end{aligned}$$ \nObserve that\n \n$$\\begin{aligned}\n\\frac{1}{Z}\\frac{\\partial\\; Z}{\\partial T}&=\\frac{1}{Z}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\partial\\_T\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}\\nonumber\\\\\n&=\\frac{1}{Z}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\sum\\_{\\vb{m}\\in I}\\delta\\_{\\vb{m},S}\\;n(\\vb{m})\\left(\\frac{\\hbar\\;\\omega\\_{\\vb{m}}}{kT^2}\\right)\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}\\nonumber\\\\\n&=\\frac{1}{Z}\\sum\\_{\\vb{m}\\in I}\\left(\\frac{\\hbar\\;\\omega\\_{\\vb{m}}}{kT^2}\\right)\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\delta\\_{\\vb{m},S}\\;n(\\vb{m})\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}},{\\label{eq: EnergyMean1}}\\end{aligned}$$ \nand from the ([eq: MeanOfEnergy]) we get\n \n$$\\begin{aligned}\nE&=-kT\\;\\sum\\_{\\vb{m}}\\hbar\\;\\omega\\_{\\vb{m}}\\frac{\\partial\\;\\ln Z}{\\partial\\;\\omega\\_{\\vb{m}}}\\nonumber\\\\\n&=-kT\\;\\frac{1}{Z}\\sum\\_{\\vb{m}}\\hbar\\;\\omega\\_{\\vb{m}}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\frac{\\partial}{\\partial\\_{\\omega\\_{\\vb{m}}}}\\left(\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}\\right)\\nonumber\\\\\n&=-kT\\;\\frac{1}{Z}\\sum\\_{\\vb{m}}\\hbar\\;\\omega\\_{\\vb{m}}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\n\\left(-\\frac{\\hbar}{kT}\\right)\\delta\\_{\\vb{m},S}\\;n(\\vb{m})\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}\\nonumber\\\\\n&=kT^2\\;\\frac{1}{Z}\\sum\\_{\\vb{m}\\in I}\\left(\\frac{\\hbar\\;\\omega\\_{\\vb{m}}}{kT^2}\\right)\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\delta\\_{\\vb{m},S}\\;n(\\vb{m})\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}.{\\label{eq: EnergyMean2}}\\end{aligned}$$ \nComparing ([eq: EnergyMean1]) and ([eq: EnergyMean2]) we get\n \n$$\\begin{aligned}\nE=kT^2\\frac{\\partial\\; \\ln Z}{\\partial T},\\nonumber\\end{aligned}$$ \nwhich is the formula we get from the thermodynamical formalism.\nThe partition function.\n-----------------------\nFor the simple situation of non-interacting bosons we can find an exact expression for the partition function. Starting with formula ([eq: PartitionFunctionFormula1]) we have\n \n$$\\begin{aligned}\nZ&=\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\Pi\\_{\\vb{l}\\in S}\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}n(\\vb{l})}{kT}}\\nonumber\\\\\n&=\\Pi\\_{\\vb{l}\\in I}\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}\\;\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}\\delta\\_{\\vb{l},S}n(\\vb{l})}{kT}}\\nonumber\\\\\n&=\\Pi\\_{\\vb{l}\\in I}\\sum\\_{N=0}^{\\infty}\n\\underbrace{\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}}\\_{n(\\vb{l})=N}\\;\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}\\delta\\_{\\vb{l},S}\\;N}{kT}}\\nonumber\\\\\n&=\\Pi\\_{\\vb{l}\\in I}\\sum\\_{N=0}^{\\infty}\\alpha\\_{N,\\vb{l}}\\;\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}\\;N}{kT}},{\\label{eq: PartitionFunctionFormula2}}\\end{aligned}$$ \nwhere we have defined\n \n$$\\begin{aligned}\n\\alpha\\_{N,\\vb{l}}&=\\underbrace{\\sum\\_{S\\in\\mathcal{P}(I)}\\sum\\_{n\\in\\mathcal{F}(S)}}\\_{n(\\vb{l})=N}\\;1.\\nonumber\\end{aligned}$$ \nAll the numbers $\\alpha\\_{N,\\vb{l}}$ are obviously infinite and they are equal to the number of elements in certain subsets of the countable index set *I*. They are therefore all countable and thus all equal to the transfinite Cardinal number ℵ0, the smallest number in Cantors infinite hierarchy of transfinite Cardinal numbers.\nUsing the arithmetic rules that apply to transfinite numbers we have\n \n$$\\begin{aligned}\nZ&=\\aleph\\_0\\;\\Pi\\_{\\vb{l}\\in I}\\sum\\_{N=0}^{\\infty}\\;\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}\\;N}{kT}}.\\nonumber\\end{aligned}$$ \nIn all applications of the partition function for computing means of observables, the same transfinite number ℵ0 will occur in the numerator and denominator and they will therefore, again using the rules of transfinite arithmetic, cancel exactly. We can therefore disregard the transfinite multiplier in the formula for the partition function and thus get\n \n$$\\begin{aligned}\nZ&=\\Pi\\_{\\vb{l}\\in I}\\sum\\_{N=0}^{\\infty}\\;\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}\\;N}{kT}}=\\Pi\\_{\\vb{l}\\in I}Z\\_{\\vb{l}}(T,\\omega\\_{\\vb{l}}),{\\label{eq: PartitionFunctionFormula3}}\\end{aligned}$$ \nwhere\n \n$$\\begin{aligned}\nZ\\_{\\vb{l}}(T,\\omega\\_{\\vb{l}})&=\\sum\\_{N=0}^{\\infty}\\;\\exp{-\\frac{\\hbar \\omega\\_{\\vb{l}}\\;N}{kT}}=\\frac{1}{1-\\exp{-\\frac{\\hbar\\;\\omega\\_{\\vb{l}}}{kT}}}.{\\label{eq: PartitionFunctionFormula4}}\\end{aligned}$$ \nFormulas for thermodynamical quantities pertaining to light fields in cavities.\n-------------------------------------------------------------------------------\nFrom the general thermodynamical formalism introduced in section two of these notes it is clear that the partition function *Z* = *Z*(*λ*1, ⋯, *λ**p* *α*1, ⋯, *α**q*) depends on the Lagrange multipliers {*λ**i*}, one for each observable, and possibly on a set of parameters {*α**j*}, one for each fixed constraint that define the thermodynamical system under investigation. Differentiation of ln*Z* with respect to the Lagrange multipliers determine all statistical moments of the observables, and differentiation with respect to the parameters determine other important physical quantities through the general thermodynamic identities ([eq: DParameterZ]), for the case of a countable number of parameters, and ([eq: FundamentalVariationalIdentity]) for the case of an uncountable number of parameters,for example a confining surface. We have in section ([eq: pressure]) seen how a formula like this for the pressure is derived from ([eq: FundamentalVariationalIdentity]). For the calculation of all other physical quantities pertaining to the system we must do it by taking traces over the bosonic density operator ([eq: QMaxEntDensityOperator]). We will see an example of this in the next section. In either case we end up with formulas that expresses all physical quantities pertaining to a thermodynamical system in terms of mode sums.\n### Formulas for arbitrary cavities expressed in terms of mode sums\nObserve that all quantities that are calculable directly from the partition function actually only depends on ln*Z*. From the previous section we have\n \n$$\\begin{aligned}\n\\ln Z&=\\ln\\left(\\Pi\\_{\\vb{m}\\in I}Z\\_{\\vb{m}}(T,\\omega\\_{\\vb{m}})\\right)=\\sum\\_{\\vb{m}}\\ln Z\\_{\\vb{m}}(T,\\omega\\_{\\vb{m}})\\nonumber\\\\\n&=-\\sum\\_{\\vb{m}}\\ln \\left(1-\\exp{-\\frac{\\hbar\\;\\omega\\_{\\vb{m}}}{kT}}\\right){\\label{eq: LogZ}}\\end{aligned}$$ \nUsing the expression ([eq: LogZ]) for ln*Z* we find the following mode sum for the mean of the total electromagnetic field energy in the cavity\n \n$$\\begin{aligned}\nE&=kT^2\\frac{\\partial\\; \\ln Z}{\\partial T}\\nonumber\\\\\n&=-kT^2\\frac{\\partial\\; }{\\partial T}\\left(\\sum\\_{\\vb{m}}\\ln \\left(1-\\exp{-\\frac{\\hbar\\;\\omega\\_{\\vb{m}}}{kT}}\\right)\\right)\\nonumber\\\\\n&=\\sum\\_{\\vb{m}}\\frac{\\hbar\\;\\omega\\_{\\vb{m}}}{\\exp{\\frac{\\hbar\\;\\omega\\_{\\vb{m}}}{kT}}-1}.\\end{aligned}$$ \nNote that this formula holds for *any* cavity whatsoever. For the mean photon number in the cavity we have\n \n$$\\begin{aligned}\nN&=\\left<\\mathcal{\\hat N}\\right>=\\sum\\_{\\vb{m}}\\left<\\mathcal{\\hat N\\_{\\vb{m}}}\\right>=-\\frac{kT}{\\hbar}\\sum\\_{\\vb{m}}\\frac{\\partial\\;\\ln Z}{\\partial\\;\\omega\\_{\\vb{m}}}\\nonumber\\\\\n&=\\sum\\_{\\vb{m}}\\frac{1}{\\exp{\\frac{\\hbar\\;\\omega\\_{\\vb{m}}}{kT}}-1}.{\\label{eq: MeanOfNumberOperator}}\\end{aligned}$$ \nWe can not use the general formula ([eq: FundamentalVariationalIdentity3.5])\n \n$$\\begin{aligned}\n\\left<p\\right>\\_{\\Gamma}= kT\\;\\frac{\\delta\\_\\Gamma\\ln Z}{d\\_{\\Gamma}V},{\\label{eq: VariationalFormulaForPressure}}\\end{aligned}$$ \nto express the pressure as a mode sum. The reason for this is that we only have the explicit dependence of the partition function on the mode frequencies, not it’s dependence on the closed surface Γ defining the cavity. Deforming the surface will or will not change the mode frequencies depending on the precise nature of the deformation. There are in fact for some cavities deformations that leads to no change in the mode frequencies at all. These deformations are called isospectral and there is a whole field of mathematics dedicated to investigating such deformations. Thus, there is in general no simple relation between the mode frequencies and deformations and therefore ([eq: VariationalFormulaForPressure]) is useless for determining a formula for the pressure for general cavities.\nThis means that for general cavities the pressure is one of the quantities that can only be computed by taking traces over the bosonic density operator. We now proceed to do this for the simplest case of an empty cavity. The approach we use for these simplest kind of cavities can also be used to treat the general type of cavities introduced in section three, but then delicate questions regarding which stress tensor is the correct one to use in the presence of material bodies must first be resolved. If a choice has been made the calculations will proceed in a similar way to the one below for empty cavities.\nThe correct quantum Maxwell stress tensor for an empty cavity is\n \n$$\\begin{aligned}\n\\hat T&=\\epsilon\\_0\\vb{\\hat E}\\vb{\\hat E}+\\mu\\_0\\vb{\\hat H}\\vb{\\hat H}-\\frac{1}{2}I\\left(\\epsilon\\_0\\vb{\\hat E}^2+\\mu\\_0\\vb{\\hat H}^2\\right){\\label{eq: MaxwellTensor1}}\\end{aligned}$$ \nRecall that the quantum electromagnetic field inside the cavity can be written in the form\n \n$$\\begin{aligned}\n \\vb{\\hat E}(\\vb{x},t)&=-\\sum\\_{\\vb{m}}\\frac{\\hat p\\_{\\vb{m}}(t)}{\\sqrt{\\epsilon\\_0}}\\;\\vb{u}\\_{\\vb{m}}(\\vb{x}),{\\label{eq: QEExpansion}}\\\\\n \\vb{\\hat H}(\\vb{x},t)&=\\sum\\_{\\vb{m}}\\frac{\\hat q\\_{\\vb{m}}(t)}{\\mu\\_0\\sqrt{\\epsilon\\_0}}\\;\\curl \\vb{u}\\_{\\vb{m}}(\\vb{x}),{\\label{eq: QHExpansion}}\n \\end{aligned}$$ \nwhere we have\n \n$$\\begin{aligned}\n\\hat q\\_{\\vb{m}}&=\\sqrt{\\"},"data_id":{"kind":"string","value":"arxiv_0182404"}}},{"rowIdx":45599,"cells":{"content":{"kind":"string","value":"The opposition effect in Saturn’s rings seen by Cassini/ISS:I. Morphology of phase curves\n=========================================================================================\n,\n,\n,\n, and\nNumber of pages: \nNumber of tables: [lasttable] \nNumber of figures: [lastfig] \nRunning head: The opposition effect in Saturn’s rings seen by Cassini/ISS 1. Direct editorial correspondence to : \nEstelle Déau \nCEA Saclay \nService d’Astrophysique \nAIM, UMR 7158 \nOrme des Merisiers Bat. 709 \n91191 Gif-sur-Yvette FRANCE \nphone: +33 1 69 08 80 56 \nfax: +33 1 69 08 65 77 \ne-mail : estelle.deau@cea.fr \nThe Cassini cameras have captured the opposition effect in Saturn’s rings with a high radial resolution at phase angles down to 0.01o in the entire main ring system. We derive phase functions from 0.01o to 25o of phase angle in the Wide-Angle Camera clear filters with a central wavelength *λ*cl=0.611 *μ*m and phase functions from 0.001o to 25o of phase angle in the Narrow-Angle and Wide-Angle Cameras color filters (from the blue, *λ*bl=0.451 *μ*m to the near infrared *λ*it=0.752 *μ*m). We characterize the morphology of the phase functions of different features in the main rings. We find that the shape of the phase function is accurately represented by a logarithmic model (Bobrov 1970, in Surfaces and Interiors of Planets and Satellites, Academic, edited by A. Dollfus). For practical purposes, we also parametrize the phase curves by a simple linear-by-part model (Lumme and Irvine 1976, Astronomical Journal, 81, p865), which provides three morphological parameters : the amplitude and the Half-Width at Half-Maximum (HWHM) of the surge, and the slope S of the linear-part of the phase function at larger phase angles. Our analysis demonstrates that all of these parameters show trends with the optical depth of the rings. These trends imply that the optical depth is a key-element determining the physical properties which act on the opposition effect. Wavelength variations of the morphological parameters of the surge show important trends with the optical depth in the green filter (*λ*gr=0.568 *μ*m), which implies that grain size effects are maximum in this wavelength.\nSaturn’s rings; phase curves; opposition effect; coherent backscattering; shadowing; shadow hiding\nIntroduction\n============\nWhen one views the rings of Saturn with the Sun directly behind the observer, a phenomenon called the *opposition effect* can be seen. The opposition effect, also known as the opposition surge, is a sudden, nonlinear rise in brightness with decreasing phase (Sun–ring–observer) angle that occurs as the phase angle approaches zero. The opposition effect, which was observed for most Solar System bodies and was discovered in the course of Müller’s long-term photometry of the Saturn system, beginning in 1878. inferred that the opposition effect was due to the rings, since Jupiter did not show a comparable opposition brightening.\nMost recent studies of the opposition effect in the Saturn’s rings were based on ground based and spatial data which resolved the main rings. Earth-based observations, though valuable, have a low to moderate spatial resolution. This indicates that the signal is averaged over large regions in the rings. Unfortunately, the rings are highly heterogeneous features that may present rapid spatial variations of their optical properties. So the interest of *spacecraft* observations is the ability to probe the signal in very narrow ring features ($\\thicksim$40 km in the present paper), which may have much more uniform optical properties. This allows an easier study of the opposition effect which is an already complex phenomenon. \nPioneer 11, Voyager 1 and Voyager 2 did not observe the rings at phase angles near zero. However, Cassini, which became an artificial satellite of Saturn in 2004, is the first spacecraft to observe the opposition effect in the rings, with several instruments, including the Narrow Angle (NAC) and Wide Angle (WAC) cameras of the Imaging Science Subsystem (ISS). For our data set (Table[tableoeinfoimages]), typically for the WAC 1 pixel=40 km and the FWHM is 1.8 pixels, for the NAC 1 pixel=5 km and the FWHM is 1.3 pixels. With Cassini images we are thus able to characterize how the surge varies throughout individual features in the ring system.\nThe opposition effect is now known to be the combined effect of the coherent backscatter (at small phase angles) which is a constructive interference of photons in a medium of using grains of wavelength size and the shadow hiding (at larger phase angles) which consists of shadows cast by particles on other particles that become invisible to the observer. Some analytical models have tried to combined these two effects, but recent laboratory measurements and data analysis show that models’ results could be compromised by some physical assumptions on the scatterer elements, the wavelength-dependence and the the angular size of the source. Two other effects, the external near-field effect and the single scattering internal-field coherence, were never been observed solely because they are intrinsically inseparable from the coherent backscatter effect and are for the moment only numerically simulated.\nBoth shadow-hiding and coherent backscattering are likely to play roles in determining the shape of the phase curves of Solar System bodies at low phase angles. Shadow-hiding probably dominates at phase angles greater than a few degrees, while coherent backscattering takes effect at the very smallest phase angles. The shadow-hiding effect gives clues about the three-dimensional structure of a layer of ring particles which, according to, could have a typical size of *r̄* = 15m. By contrast, the coherent backscattering component sheds light on the nature of the ring particle surfaces at a scale not much larger than optical wavelengths. Indeed, previous photometric studies which investigate the opposition effect in the Saturn’s rings determined ranges of typical size for coherent-backscattering effect of about *d* = 10 *μ*m. Thus, the shape of the opposition effect provides constraints on features vastly smaller than the camera’s resolution. Like, we refer to particles as macroscopic and individual objects in the rings and grains as microscopic objects at the surface of rings’ particles. Using this terminology, ring particles might be covered by regolithic grains.\nThis is the first of a series of papers dealing with the opposition effect in Saturn’s rings as seen by ISS/Cassini. In the present paper, we focus on the characterization of the morphological modeling of the shape of opposition effect in the rings. Since the recent theories have difficulties to link the behavior of the opposition effect with the physical properties of the surface material, we think it is necessary to have an experimental approach, consisting in (1) deriving the shape’s parameters and (2) trying to find correlations among themselves, and with the optical depth. In the second paper (Déau et al. in preparation), we will use recent analytical photometric models to derive some of the physical properties of the ring material. Indeed, to constrain completely the physical properties of the ring material, photometric and polarimetric phase curves are needed, however, Cassi/ISS did not captured the opposition spot with its polarized filters. In section[observeetreduit] of the present paper, we describe the ISS/Cassini data set and our procedure for extracting photometric data from images and fitting empirical models to the data. In section[results], we characterize the morphology of the opposition surge at different locations in the main rings and focus our attention on cross-correlations among the morphological parameters and correlations of these parameters with the optical depth and the wavelength. Finally, in section[discussion] we discuss these results and examine to which extents photometric models may explain the morphological trends derived in section[observeetreduit].\nObservations and reductions\n===========================\nThe Cassini Imaging Data Set\n----------------------------\nThe Cassini ISS instrument is composed of two cameras, a wide angle camera (WAC) and a narrow angle camera (NAC) equipped with 1024 × 1024 CCD matrices. Both use a set of about twenty filters ranging from near-infrared to ultraviolet. Here, we focus only on filters in the optical domain, with first the blue, green, red and near infrared filters, that will be called hereafter COLOR filters. According to, COLOR filters from NAC and WAC cameras do not have exactly the same central wavelength. For the blue filters, the NAC can have two different combinations of filters (CL1,BL1) or (BL1,CL2) which lead to two central wavelengths : $\\lambda\\_{\\textrm{bl}}^{\\textrm{\\textsc{nac}}}=0.440\\mu$m and $\\lambda\\_{\\textrm{bl}}^{\\textrm{\\textsc{nac}}}=0.451\\mu$m. In contrast, for the WAC, the blue (CL1,BL1) filter is characterized by a wavelength of about $\\lambda\\_{\\textrm{bl}}^{\\textrm{\\textsc{wac}}}=0.460\\mu$m. Because the spectral width of these filters is about ± 0.050*μ*m, we consider all images from blue filters of the NAC and the WAC as a consistent whole. The green filter (CL1,GRN) has almost the same spectral characteristics for the two cameras since the central wavelength of $\\lambda\\_{\\textrm{grn}}^{\\textrm{\\textsc{nac}}}=0.568\\mu$m for the NAC and $\\lambda\\_{\\textrm{grn}}^{\\textrm{\\textsc{wac}}}=0.567\\mu$m for the WAC. The red filter for the NAC corresponds to the filter (RED,CL2) with a central wavelength of $\\lambda\\_{\\textrm{red}}^{\\textrm{\\textsc{nac}}}=0.650\\mu$m. For the WAC, the combination (CL1,RED) is at a central wavelength of $\\lambda\\_{\\textrm{red}}^{\\textrm{\\textsc{wac}}}=0.649\\mu$m, which does not change to the that of the NAC at one nanometer. Finally, the filter (CL1,IR1) in the near infrared shows a difference of 10 nanometers between the central wavelength of the NAC ($\\lambda\\_{\\textrm{ir}}^{\\textrm{\\textsc{nac}}}=0.752\\mu$m) and that of the WAC ($\\lambda\\_{\\textrm{ir}}^{\\textrm{\\textsc{wac}}}=0.742\\mu$m). In summary, with this moderately high spectral resolution, the combination of images that are not coming exactly from the same filters are responsible for a shift of the central wavelength from 0.001 to 0.010 micron. Because each filter has a non negligible spectral range ( ± 0.050*μ*m), we consider without distinction the COLOR filters from the NAC and WAC cameras and in the rest of the paper, we take as a reference the following wavelengths (*λ*bl = 0.451*μ*m; *λ*grn = 0.568*μ*m; *λ*red = 0.650*μ*m and *λ*ir = 0.752*μ*m) to designate each COLOR phase curve. The other part of images discussed in the present paper has been taken in CLEAR mode of the WAC, designating the absence of filters resulting in a spectral bandwidth spanning from 0.20 to 1.10 *μ*m (the central wavelength is *λ*cl = 0.611*μ*m and the spectral range is ± 0.450*μ*m). For the present paper, the selected observation campaign of the opposition effect on the Saturn’s rings is divided in four sequences from May 2005 to July 2006 (Table[tableoeinfoimages]).\n**[Table[tableoeinfoimages]]**\nThese observations were conduced conjointly by the VIMS, CIRS and ISS instruments. Three of ISS sequences were obtained using the WAC only and the last one using the BOTSIM mode (using both cameras simultaneously). Depending on the sequence, different filters were used and different spatial and angular resolutions were achieved. In the present paper, the first of a series aimed at a detailed study of the opposition effect, we will focus on sequences obtained using the WAC camera in CLEAR filter mode and sequences using NAC and WAC cameras in COLOR filter mode, Table[tableoeinfoimages]. Images of the B and C ring are shown in Fig.[representoeimages] and [oehisurge1] to illustrate the quality of the data set.\n**[Fig.[representoeimages], Fig.[oehisurge1]]**\nIn the CLEAR filter mode, the wide angle camera captured rings at zero phase angle two times in 2005, on June 7 and June 26 (they will be designated as *June 7* and *June 26* sequences in the rest of the paper, for practical purpose). In the *June 7* sequence, the WAC images have an average resolution of $\\thicksim$ 44 km per pixel; the rings are observed in reflexion. In each individual image, the phase angle varies by about 3 degrees. In the majority of the set, the opposition point at zero phase angle is visible with a very good sampling : about 0.005 degree per pixel. This set covers the full ring system by tracking the opposition spot. Most of the set has an excellent photometric quality, however in dark regions the strong Saturnshine produces ghost images of the secondary mirror perturbing the signal significantly. The amplitude of this artifact is quite constant, about 0.05 in I/F, which does not significantly affect the photometry of the bright regions as A and B rings. However it is very troublesome for dark regions like the D ring, the C ring and the Cassini Division. This is why images from the *June 7* sequence have not been considered for these two regions. The same instrument artifact has been observed and discussed in. The *June 26* sequence has similar characteristics, see Table[tableoeinfoimages], with a better radial resolution of about 30 km per pixel and spans the B and C rings. The presence of the ghost signal in the C ring is not clear. A detailed photometric analysis in these images (cf. section[phasecurvesoe]) shows that the derived phase function is consistent with an unperturbed signal, very differently from images in which the ghost were clearly identified. It is why the images of the C ring taken on *June 26* have been considered for the C ring phase function. The *July 23* sequence has the best radial resolution ($\\thicksim$13 km.pixel− 1) and spans all A, B, C rings and the Cassini Division (the radial location of the opposition spot in each image is represented in the Fig.[oeimagescolorlocation]). In this set, the ghost artifact is absent owing to a much larger angular separation with the bright Saturn globe. Some images of this sequence provide also the larger phase angles.\n**[Fig.[oeimagescolorlocation]]**\nIn COLOR filters mode, the ISS instrument uses the BOTSIM mode in the *May 20* sequence, using both cameras simultaneously for a ride in A and B rings, Cassini Division and the outer C ring (Fig.[oeimagescolorlocation]). However, because the boresight of the NAC and the NAC are not exactly the same, NAC images did not represent a zoom of the opposition spot in WAC images. Thus, WAC and NAC images captured at the same time have opposition spots that are not exactly at the same location in the rings. The radial resolution in NAC images is 4.6 km.pixel− 1 and spans from 44 to 66 km.pixel− 1 for WAC images in COLOR filters. The other sequences (*Dec 31*, *Feb 20* and *Apr 25*), did not have the opposition spot but provided the larger phase angles.\nData reduction\n--------------\n### Calibration\nRaw images are calibrated first with a standard pipeline described in and called CISSCAL, using the 3.4 version. The DN values are converted into the I/F units. This ratio is dimensionless, with *I* is the specific intensity measured by Cassini, and *π**F* is the incident flux from the Sun. So it is a measure of the local reflectivity under the current observing geometry. Cassini images are not directly exploitable to study the photometric behavior of rings. In order to do this, the relevant data we need are the so-called *phase function* which is linked to the I/F ratio as a function of the phase angle, and corrected from effects of observation geometry. A full procedure has been designed to reconstruct the phase function from different images with different observation geometry and resolutions and is detailed below.\n### Extraction procedure\nThe first step is to reproject the images in a (Radius, Longitude) frame, in which features at a same radius from Saturn are horizontally aligned. This procedure critically depends on the quality on the navigation. When possible, the edges of the A, B, C rings as well as ring features reported in were used as fiducial references. Distances reported in were corrected according to. The resulting navigation error on WAC images varies from 1 to 1.5 pixels from the center to the edge of the image. The I/F in a ring at constant distance from Saturn is obtained by extracting the data on a line of constant radius (*i.e.* an horizontal line) in the reprojected map of calibrated brightness. Other geometrical parameters are extracted in the same way : phase angle *α*, cosine of incidence angle *μ*0, cosine of the emission angle *μ*, optical depth of the rings obtained from the PPS Voyager instrument (*λ* = 0.260 *μ*m), ; radius scale has been corrected with the procedure of. This procedure works very well for structures with width larger than the navigation accuracy. However radial structures are visible at all scales down to one pixel, (Fig.[oehisurge1]). For structures radially smaller than 5 pixels, navigation errors make false the extraction along a line of constant radius close to the edges of the image, inducing accidental extraction in nearby different features. To overcome this problem and to ensure that we always extract the same ring feature, we developed a *ring tracking* technique using a basic pattern recognition algorithm to follow a single feature. Extensive visual check of the result shows the method is reliable down to 1 pixel of radial width.\n### Construction of the phase function\nThe ultimate information we need is the phase function of individual ring particles to characterize their surface properties. Unfortunately, the signal from an individual particle is heavily altered because of the finite thickness of the rings. Also, the signal’s intensity depends on observation angle with respect to the ring’s normal. Inverting such complex collective photometric effects would require the use a detailed light scattering code with many assumptions concerning the photometric properties of particles. Such code has been developed by and. However, for our present purposes, they cannot used to derive the phase functions in hundreds of different regions as we wish to do here. Consequently, as a first approximation, we use in the present paper the classical approach of linking the I/F to the phase function with the following assumptions : homogeneous layer of particles and single scattering. This latter assumption may be justified for phase angles smaller than $\\thicksim$30o. In reflexion, the phase function $\\varpi\\_{0}\\cdot P(\\alpha)$ is derived from the solution to the radiative transfer equation (the designation *phase function* is not strictly accurate since what we really determine is the product of the single scattering albedo, $\\varpi\\_{0}$, times the particle phase function, *P*(*α*)) :\n \n$$\\varpi\\_{0}\\cdot\nP(\\alpha)=\\frac{I}{F}\\times\\frac{4(\\mu+\\mu\\_{0})}{\\mu\\_{0}}\\times\\left(\n1 - e^{-\\tau\\left(\\frac{1}{\\mu}+\\frac{1}{\\mu\\_{0}}\\right)} \\right)^{-1}\n\\label{eq\\_chandrasekhar}$$ \nwith *τ*, *μ*, *μ*0, *α* standing for : the normal optical depth, cosine of emission angle, cosine of incidence angle and phase angle respectively. In order to allow future comparisons with detailed numerical models, numerous analytical parametrization of the observations are provided by morphological model in section[describemodeloe]. As one can see, the value of the optical depth is necessary to derive the phase function. Preceding works has shown that the exponential factor can be neglected in first approximation for Earth-based observations. However, we noticed that we obtained much more coherent results when taking into account the exponential factor, when comparing results from different geometry of observations. Consequently, we keep the initial formula of, as previous photometric studies based on spacecraft observations of Saturn’s rings.\n*P*(*α*) obtained with Eq.[eqchandrasekhar] is the particle’s disk integrated phase function which determines the angular distribution of single scattered radiation from the body as a whole. The phase function is normalized over the solid angle Ω to the single scattering albedo : $\\varpi\\_{0}=\\frac{1}{4\\pi} \\int P(\\alpha)~\\textrm{d}\\Omega$. To derive the albedo, the full phase curve, from 0 to 180 degrees of phase angle, must be known. In this study, we have restricted our data (0<*α*<25 degrees), so that we may avoid separating the albedo from the phase function. The derivation of the albedo will be presented in the second paper which relates ISS data from 0 to 180 degrees.\n### Finite size of the Sun\nGenerally speaking, all phase curves present a bright opposition surge below 1o and a slope decreasing linearly for *α* > 1o. Whereas WAC images go down to $\\thicksim$0.01o of phase angle (Fig.[representoeimages]), NAC images taken in COLOR filters capture the opposition spot at better angular resolution (Fig.[oehisurge1]), the resulting phase functions from NAC images go down to $\\thicksim$0.001o (Fig.[representnacallrings]). This is the first time the opposition spot is imaged at such fine scale. Note that we define the phase angle as the angle between the vector pointing to the Sun’s center, and to the spacecraft, from the observed point. So from a strictly *mathematical point of view* there is no lower limit to the phase angle value despite the source’s finite size. For an extended illumination source, the phase curve should be the integral of a point source phase function over the Sun angular radius. Thus on a more *physical point of view* it is not possible to observe the phase function at angle below the Sun’s angular radius. Fig.[representnacallrings] demonstrates that the opposition surge flattens (in all rings and at all wavelengths) at phase angles below 0.029o, in good agreement with the Sun’s angular radius at the date of observations.\n**[Fig.[representnacallrings]]**\nData fit with Morphological models\n----------------------------------\nThe purpose of the present paper is to provide an accurate description of the morphological behavior of the observed phase curves. This is the very first step prior to any attempt of further modeling, which will be the subject of a forthcoming paper. As a consequence, special care has been given to parametrize the phase functions conveniently. In addition morphological parametrization is necessary to compare efficiently hundreds of phase curves at different locations in the rings and derive statistical trend as will be done in section[results]. Several morphological models have been used in the past to describe the shape of the phase functions : the logarithmic model of, the linear-by-part model of and the linear-exponential model of. The specific properties of these three models make them adapted for different and complementary purposes. The logarithmic model is interesting for direct comparisons with numerical models, the linear-by-part model is convenient to describe the shape in an intuitive way, and finally the linear-exponential model is adapted for comparison with other studies previously published.\n### The logarithmic model\nAs, and, we remark that a logarithmic model describes very well the solar phase curves of the Saturn’s rings. It depends on two parameters (*a*0 and *a*1). This first morphological model has the following form :\n \n$$\\varpi\\_{0}P(\\alpha)=a\\_{0} + a\\_{1}\\times\\log(\\alpha)$$ \nIn general, this model is the best morphological fit to the data. It is reasonably accurate down to 0.029o of phase angle. This is due to the finite angular size of the Sun which flattens data below 0.029o, whereas the logarithmic function continues increasing (see section[solarsizefinisat] and Fig.[representnacallrings]a). For large phase angles, the fit is satisfactory up to *α* ≃ 15 degrees, this was also noticed by who fitted temperature phase curves of the C ring with CIRS/Cassini. Whereas the meaning of *a*0 and *a*1 are not easily interpretable in term of shape, to allow an easy comparison with future numerical simulations the values of these two parameters are reported in Table 1 of Electronic supplement material.\n### The linear-exponential model\nThis model describes the shape of the phase function as a combination of an exponential peak and a linear part. Its main interest is that it has been used in previous work for the study of the backscattering of Solar System’s icy satellites and rings. We give the details of this model : the 4 parameters are the intensity of the peak *I**p*, the intensity of the background *I**b*, the absolute slope of the linear part *I**s* and the width of the exponential *w*, such that the phase function is represented by :\n \n$$\\varpi\\_{0}P(\\alpha)=I\\_{b} + I\\_{s}\\cdot\\alpha + I\\_{p}\\cdot\ne^{-\\frac{\\alpha}{2w}}$$ \nWith these 4 parameters, we characterize the shape of the phase function by introducing three morphological parameters : A, HWHM and S designating the amplitude of the surge, the half-width at half-maximum of the surge and the absolute slope at large phase angles respectively, so that :\n \n$$\\textrm{A}=\\frac{I\\_{p}+I\\_{b}}{I\\_{b}}~~~~~~~~\\textrm{HWHM}=2\\cdot\n\\ln2w~~~~\\textrm{and}~~~~\\textrm{S}=-I\\_{s}$$ \nAs, we noticed that this model did not fit well the phase curves, in particular, the derived model parameters appeared to depend substantially on the phase angle coverage (see section[incompletedatafits]), preventing a robust comparison of data. This is due to the fact that we did not use the converging procedure of but rather the common downhill minimization technique, as done by the previous users of the linear-exponential model. Moreover, it seems that the angular scale at which the phase function is observed may strongly influence the model parameters. So to avoid this problem, we used a much simpler morphological model that appeared much more robust for the comparison of heterogeneous data set : the linear-by-part model of, also we found a much better match of the data with the linear-by-part model of (see below). However, the linear-exponential was considered in the present study only to understand variations of A, HWHM and S between the phase curves and our phase curves (see section[incompletedatafits] and section[comparehstiss]).\n### The linear-by-part model\nFor an intuitive description of the main features of the phase curves, the linear-by-part model is the most convenient one. It is constituted of two linear functions fitting both the surge at small phase angle (*α* < *α*1) and the linear regime at higher phase angle (*α* > *α*2) :\n \n$$\\begin{aligned}\n\\varpi\\_{0}P(\\alpha<\\alpha\\_{1})=-A\\_{0}\\times\\alpha + B\\_{0} \\newline\n\\varpi\\_{0}P(\\alpha>\\alpha\\_{2})=-A\\_{1}\\times\\alpha + B\\_{1}\\end{aligned}$$ \n and use *α*1=0.27o and *α*2=1.5o. However, we encountered difficulties with the value of *α*1. By testing several values of *α*1, it appears that for our data set, values of *α*1 less than 0.3o yield a general overestimation of *A*0, especially in the C ring and values of *α*1 greater than 0.3o yield an underestimation of *A*0 only in the B ring. Consequently we found the our data were better reproduced using *α*1 = 0.3o which is now adopted in the rest of the paper. With this four outputs : the two absolute slopes *A*0 and *A*1 and the two y-intercepts *B*0 and *B*1, the shape of the curve is characterized by the morphological parameters A, HWHM and S are then defined by :\n \n$$\\textrm{A}=\\frac{B\\_{0}}{B\\_{1}}~~~~~~~~\\textrm{HWHM}=\\frac{(B\\_{0}-B\\_{1})}{2(A\\_{0}-A\\_{1})}\n~~~~\\textrm{and}~~~~\\textrm{S}=A\\_{1}$$ \nThe purpose of this model is not, of course, an accurate description of the data but rather a convenient description of the main trends of the phase curve.\n### Stability of the morphological parameters\nIn order to compare properly our results with those of and, for which the Saturn’s rings phase curves did not have the same phase angle coverage, we have tested the influence of the portion 0.05o-0.4o and 6o-25o on the converging solution. Indeed, phase curves of do not have data under 0.3o, and both studies of and do not have data over 6.5o. With two typical Saturn’s rings phase curves of ISS (Fig.[fig2]), we have removed data by section of 0.1o and fit the pseudo-incomplete phase curve with the linear-exponential model, which provides the new solution, designated by *A*remove, HWHMremove and *S*remove. The initial solution found for fuller phase function (0.01o-25o) is called *A*optimal, HWHMoptimal and *S*optimal. Both solutions are obtained with a downhill minimization technique, to reproduce the fitting method used by and. It is wise to recall that proposed a converging procedure more accurate, however because did not used it, and because we want to reproduce not the best converging solutions but the deviations lead by the fits driven by and, that is why we use the downhill method and not the probability distribution method of.\n**[Fig.[fig2], Fig.[fig3]]**\nIn Fig.[fig3], we represent the ratio of the morphological parameters of the incomplete phase curve over the morphological parameters of the fuller phase curve, called *A*remove/*A*optimal, HWHMremove/HWHMoptimal and *S*remove/*S*optimal. We observe a slight underestimation of A, a strong overestimation of HWHM and a moderate underestimation of S. The deviation of the optimal value is quite weak for A ($A\\_{\\textrm{remove}}/A\\_{\\textrm{optimal}}\\thicksim$0.96 at a cutoff of 0.3o) but its variation depends on the morphology of the surge. For the typical B ring phase curve (which has a narrower peak), we note a slight decrease of the ratio *A*remove/*A*optimal and for the C ring, we note a slight increase of the ratio. However the both ratios lead to an underestimation, which means that incomplete data fitted by the morphological model will have a smaller amplitude that data which cover the full surge. HWHM shows the strongest deviation of *A*remove. Indeed, HWHM is overestimated for the typical B ring phase curve (HWHMremove/HWHM$\\_{\\textrm{optimal}}\\thicksim$1.4 at a cutoff of 0.3o) but is strongly overestimated for the typical C ring (HWHMremove/HWHM$\\_{\\textrm{optimal}}\\thicksim$2.0 at a cutoff of 0.3o) which has a wider peak. For the slope, we notice an overestimation in the order of $S\\_{\\textrm{remove}}/S\\_{\\textrm{optimal}}\\thicksim$1.3 at a cutoff of 7o for the B ring phase curve and of $\\thicksim$2.5 for the C ring at the same cutoff. This means that the slope of the linear part is stabilized at roughly 15o. These comparisons are important when we compare morphological trends found by, in section[comparativeapproach] and, in section[comparehstiss].\n### Linking morphological parameters with the physical parameters of the models\nThe use of a simple morphological model is generally not adapted to derive the physical properties of the medium. However, the theories developed for the coherent backscattering and the shadow hiding effects deduce their properties by parameterizing the opposition phase curve. Recent laboratory experiments raise some doubts on the meaning of the physical parameters of these models and their correlation with the real physical properties of the medium. However, because there is no better modelization, we connect the morphological parameters A, HWHM and S with the physical characteristics of the medium derived from these models.\nA\nis the amplitude of the opposition peak and describes the behavior of the phase function at the smallest phase angles (*α*<2o). What we know about the opposition effect is that it occurs at the smallest phase angles and acts on the multiple scattered light in the regolith on the surface of the particles : the underlying phenomenon is the coherent backscattering effect. The coherent-backscattering effect increases in brightness by almost a factor two, while using grains size smaller than the wavelength of the incident light,. In contrast, the second phenomenon of the opposition effect, the shadow hiding effect, is known to produce a wide peak from 0 to 2 degrees, and to decrease the brightness up to 20 degrees. The combination of the two effects at very low phase angle is still a matter of debate and today two theories disagree in order to explain the peak of the opposition. The theory of assumes that the opposition peak is a pure coherent backscattering effect whereas the theory of shows that the opposition peak results from a coupling of coherent-backscattering and shadow hiding, even at low phase angles. This coupling should be due to the fact that the coherent backscattering could act on both multiple and single scattered light whereas the shadow hiding is a single scattered light effect. Thus, this theoretical model defines two amplitudes as output parameters : the coherent backscatter amplitude *B**C*0 and the shadow hiding amplitude *B**S*0. As a consequence, using this theory, it does not seem to be possible to ascribe the morphological parameter A solely to the coherent backscattering. For most laboratory measurements, the amplitude of the opposition peak is a function of grain size in such way that A decreases with increasing grain size. This anti-correlation finds a natural explanation by the fact that macroscopic irregularities (>*λ*) create less coherent effects than microscopic irregularities ( ≤ *λ*). Laboratory measurements of also confirmed that the opposition surge increases when irregularities are small. and underline the fact A is linked to the intensity of the background *I**b* which is a decreasing function of increasing absorption, thus A must increase with increasing absorption or decreasing albedo $\\varpi\\_{0}$, which was also confirmed by laboratory measurements of who found that the peak decreases with increasing sample albedo.\nHWHM\n, the half-width at half-maximum, is generally associated to the coherent backscatter effect. It can be related to the grain size, index of refraction and packing density of regolith. The HWHM is maximum for a effective grain size near *λ*/2 and increases when the regolith grains filling factor *f* increases. For high values of *f*, the HWHM shifts towards the greater grain size. However, as for the amplitude, the model of defines two HWHMs : the coherent backscatter angular width (*h**c*), which is defined similarly that in the model of, and the shadow hiding angular width (*h**s*). This reinforces the idea that the observed surge results from a coupling between the coherent backscattering and the shadow hiding.\nS\n, the slope of the linear part, seems to be due only to the shadow hiding effect: the interferences caused by coherent backscattering effect seem not to be very significant at larger phase angles. This means the shadow hiding light is not affected by the coherent backscattering at larger phase angles. Also, according to recent analytical and numerical models, the shadow hiding acts solely on the linear part of the phase function. We define here the absolute slope, whereas it is naturally a negative parameter, because the phase function always decreases in brightness from 10o to 40o of phase angles. The slope depends on the volume filling factor D and the optical depth of the slab. Two models has been developed for different regimes of the particle volume density D and predict opposite behavior of S as the function of D and *τ*. The *shadowing* model of and, also called *inter-particle shadow hiding* by, consists in shadows of macroscopic particles in a particulate medium, such as 8D ≪ 1. The smaller the volume density D, the steeper the phase function for increasing phase angle. Other refinements of this model exists but lead to the same results : the opposition peak due to the shadowing sharpens and the absolute slope increases with decreasing packing density (or filling factor). Another model exists, the *intra-particle shadow hiding*, which is valid for higher particle volume density according to. underlined the fact that the mutual shadowing among regolithic grains could be suited for understanding the textural properties of the regolith. As a consequence, this mechanism operates at the surface of ring material : e.g. in the regolith layer. However, other physical parameters need to be taken into account. In the analytical *inter-particle shadow hiding* model of, the slope can be linked to the angular width of the shadow hiding *h**s**h*. Thus a normalized and absolute slope would be $S=\\frac{1}{2h\\_{sh}}=\\frac{\\textrm{D}}{Q\\_{\\textrm{ext}}(\\lambda,\\bar{r})\\bar{r}\\ln(1-\\textrm{D})}$, with D the volume filling factor, *Q*ext(*λ*, *r̄*) the extinction coefficient and *r̄* the mean radius of particles. Then, the slope should depend on the wavelength and on the particle size. Interestingly, laboratory experiments of showed that the slope increases when the sample’s size increases. However, the microscopic and/or macroscopic roughness of the medium need also to be taken into account in the shadow hiding models, as underlined by and. For example, laboratory experiments of showed that the slope increases when the sample’s roughness increases. Then laboratory measurements and recent theoretical models could significantly increase the number of physical parameters on which S depends.\nFrom the above arguments, the HWHM and the amplitude A are governed by both coherent backscatter and shadow hiding effects whereas the slope S provides information of the shadow hiding effect solely.\nShape of the phase curves at opposition\n=======================================\nThe opposition effect in CLEAR filters\n--------------------------------------\nDue to the automation of extraction and fitting procedures (cf section[datareduction]) and due to the high images resolution, phase functions were extracted in as many as 211 different locations, in the D, C, B rings, Cassini Division and A ring (in increasing distance from Saturn, left column of Table 1 of Electronic supplementary material). In this section, we first present the typical behavior of some selected phase curves in different regions of the main rings (section[phasecurvesoe]), then we discuss similarities and differences and what are the general trends from sections[comparativeapproach] to [compareparametersoe].\n### Overview of the CLEAR phase curves\nExamples of phase curves in various ring regions are presented in Fig.[representoeallrings]. Each pair of graphs show on the left side a zoom from 0.01 to 2.5 degrees and on the right side, the fuller phase curve from 0 to 25 degrees. These curves were obtained by combining several WAC images with a large distribution of viewing geometries (Table[tableoeinfoimages]) each curve is built from the merging of 10 to 70 different images with various values of emission angle, incidence angle, phase angle etc. The dispersion of points is not due to the measurement uncertainty but reflects mainly the limits of the inversion formula that was used to extract the phase function from the measured values of I/F. It seems that some important physics may be missing (like multiple scattering), that could explain the scattering of points. Curves for the A and D rings (Fig.[representoeallrings]) are incomplete between 3 to 20o which is due to removal of images because of an artifact (cf section[dataset]).\n**[Fig.[representoeallrings]]**\nWhereas the general shape is similar from one ring to another (Fig.[representoeallrings]), some details in the shape may vary significantly. First pair of graphs in Fig.[representoeallrings] shows the phase curve derived in the D ring from images of *June 26*. Due to short exposure time (10 ms), the D ring ringlets are too faint to be detected. In images of *June 26*, a bright spot is visible from 67 000 km to the inner boundary of the C ring : this corresponds to the expected location of the background sheet of material constituting part of the D ring. However camera artifact may be visible in such dim regions of the image but could not be clearly identified here. So the fact that a strong increase of brightness at the expected location of the opposition and the coherent variation of the signal with observing geometry between different images suggest that we indeed see the opposition effect in the D ring. However some doubts still remain. From 0.5 to 2 degrees, the curve is similar to other rings. Below 0.5 degree an exponential surge and a flattening at zero degree distinguish this phase function from the other ones. Does it reflect optical properties of D ring dust ? Is it an artifact ? This plateau below 0.5 degree is much too large to be explained by the finite angular radius of the Sun (0.025 degree). Because of these uncertainties, at this point it is speculative to interpret this specific behavior as real.\nFor the C ring (Fig.[representoeallrings]), the shape of the phase function is well sampled below 2 degrees. It is comparatively wider than in dense A and B rings. More precisely, HWHM of the opposition surge is wider for the C ring (HWHM=0.26o) than for the A and B ring (HWHM$\\thicksim$0.20o). This could be also interpreted as a steeper slope of the linear regime of the phase function for *α* > 2o. Wavy features between 5 to 25 degrees can be attributed to images artifact. Their amplitude is about 15% of the total signal of the C ring.\nThe B ring opposition surge has the smallest amplitude of all rings (A=1.25 in Fig.[representoeallrings]). This was already underlined by the recent study of whereas previous observations of and give the exact opposite trend. However, their result could have a bias due to the lack of data below 0.3o of phase angle for and 0.1o for whereas had values much smaller. The B ring has also the steepest slope (S=0.105 $\\varpi\\_{0}P$.deg− 1) in the linear regime explaining why the opposition spot is so contrasty in the ISS images.\nThe Cassini Division has a similar amplitude and width (A$\\thicksim$1.47 and HWHM$\\thicksim$0.28o in Fig.[representoeallrings]) to the C ring (A$\\thicksim$1.45 and HWHM$\\thicksim$0.26o in Fig.[representoeallrings]). Their slope of the linear regime are also similar (S=0.033 $\\varpi\\_{0}P$.deg− 1 for the Cassini Division and S=0.030 $\\varpi\\_{0}P$.deg− 1 for the C ring). These similarities were first noticed by and are suggestive of a strong dependence of the opposition effect on the optical depth (we will come back to this in section[compareparametersoe]).\nAn example of the A ring phase function is given in the last pair of graphs in Fig.[representoeallrings]. At first sight, the signal appears much more disturbed than in other rings : specifically, pieces of phase functions extracted from different images show a wide dispersion in this graph, whereas signal from an individual image has a very low dispersion. The origin of this is not clear and may be due to the artifact reported in section [observeetreduit]. The dispersion of the data is about 15% of the signal whereas the camera artifact should represent at most 5% of the signal only (estimated on the background). It may be possible that the dispersion may be also due to an intrinsic photometric effect which is not corrected by the single scattering model (equation([eqchandrasekhar])). Indeed, the A ring has an intermediate optical depth $\\thicksim$0.5 so that it neither appears as a solid surface (like the B ring) or as a dilute system (like the C ring). Here, we are in an intermediate regime where many collective effects may influence strongly the apparent phase function (multiple scattering, gravitational wakes, density waves, etc.). A sophisticated model is required here to investigate such effect. However the general trends are quite clear and the A ring phase curve has a larger peak amplitude (A=1.39 in Fig.[representoeallrings]) than the B ring’s (A=1.25 in Fig.[representoeallrings]). We see in addition that the slope of the linear regime is shallower in the A ring (S=0.078 $\\varpi\\_{0}P$.deg− 1 in Fig.[representoeallrings]) than in the B ring (S=0.105 $\\varpi\\_{0}P$.deg− 1 in Fig.[representoeallrings]) but steeper than in less dense rings (S$\\thicksim$0.03 $\\varpi\\_{0}P$.deg− 1 in Fig.[representoeallrings]). Thus the phase curve at opposition in the A ring is somewhat intermediate between the B and C rings, strengthening the idea of a dependence on the optical depth.\nFinally, we conclude this section by remarking that the opposition effect is very diverse in Saturn’s rings, and could be the consequence of different properties of the surface ring particles in various ring regions. Some general trends can be underlined, as we see in the next section. In a first step, we check if some correlations exist between the morphological parameters which depend on the same portion of the curve (e.g. A and HWHM for the surge) and also if parameters describing different parts of the phase curve can be correlated (e.g. A, HWHM and S for, respectively, the surge and the linear part). This is the purpose of the next section.\n### Cross comparisons between morphological parameters\nIn order to constrain the morphology of the surge, we correlate the amplitude A with the angular width HWHM for the different main rings. We fit A and HWHM with a linear function and found correlation coefficients reported in Table[crossstudykaasalainenslopeahwhm].\n**[Table[crossstudykaasalainenslopeahwhm]]**\nIn Table[crossstudykaasalainenslopeahwhm] the Cassini Division, the amplitude A has the steepest function of the HWHM. A linear fit gives a slope of about 1.2 with a correlation coefficient of 47 %. This mean that narrow surges have a low amplitude and inversely wide surges have large amplitude. In the C ring, values of A and HWHM are generally greater than in the Cassini Division (but there are some exceptions). For the amplitude, this qualitative difference between the Cassini Division and the C ring is confirmed by previous study of and. For the angular width, although our trend of HWHM agrees with the results of, it contradicts previous work of, this is certainly due to the overestimation of HWHM when the smallest phase angles are missing (section[incompletedatafits] and Fig.[fig3]b). For the C ring, we find a slope for A=f(HWHM) of about 0.9 with a good correlation coefficient of 79 %. The A ring shows a similar variation of A=f(HWHM) as for the C ring (1.0) with a correlation coefficient of 56 %. Finally, A and HWHM in the B ring have values smaller than in the faint rings (C ring and Cassini Division). Data points of A and HWHM for the B ring are concentrated in a similar range as the A ring one, however with a much shallower slope (0.6 with a correlation coefficient weakly reliable of 29 %, see Table[crossstudykaasalainenslopeahwhm]). This means that the shape of opposition phase curves in the B ring may have various angular width with an almost constant value of the amplitude. To conclude, the amplitude of the surge seems correlated with the angular width, at least for the C ring, A ring and Cassini Division whereas the amplitude is independent of HWHM for the B ring. The slope of A=f(HWHM) seems to decrease from the Cassini Division to the B ring, passing by the C and A rings, suggesting that the slope is a decreasing function of the optical depth.\nWhereas the slope S on the one hand, A and HWHM on the other hand are thought to be related to different portion of the phase curve (linear part and surge respectively), it is interesting to note that they are somewhat correlated. We simply note that S is a decreasing linear function the angular width and is also correlated with the amplitude. Slopes and correlation coefficients of S=f(A) and S=f(HWHM) are reported in Table[crossstudykaasalainenslopeahwhm]. This could be due to the fact that we derive our slope from 1.5o to 25o whereas analytical model of, for example, describes the shadow hiding effect as a slope which fits the phase curve from 4.5o to larger phase angles, see.\n### Regional behavior\nWe now look for regional behavior inside each ring (Fig.[ahwhmsring]). The Fig.[ahwhmsring] displays A, HWHM and S as a function of the distance from Saturn by introducing a *ring type* nomenclature based on the regional behavior of the C ring, well studied by and decomposed into three *ring types* : inner ring, background and *plateaux*. We have modified and extended this nomenclature to five classes of ring features, then applicable to the entire main rings system :\n* inner regions characterized by low optical depth in all the rings (for example, the dark bands in the Cassini Division, see ),\n* background are morphological smooth regions without abrupt variation of *τ*,\n* bright regions (*plateaux* or *plateaus* in the C ring, density and bending waves in the A ring located by ) are the regions in each ring with the highest optical depth,\n* *ringlets* according to are thinner ring embedded in a less dense region or a gap,\n* outer regions (for example the so-called *ramp* for the C ring and the Cassini Division ) mark the transition at the boundaries of each ring.\n**[Fig.[ahwhmsring]]**\nIn Fig.[ahwhmsring]a and b, the amplitude A and the HWHM vary smoothly across the main ring system (C to A rings), with only little scattering around the main trend (appart from the Cassini Division), illustrating the stability of the linear-by-part model for comparing multiple observations in different ring regions. From the C ring to the middle of the B ring, a smooth decrease is observed. No sharp transition is observed between the C and B rings. The outer regions of the C ring which are rich in gaps, plateaux and ringlets, have a somewhat larger value of amplitude. From the middle of the B ring to the outer of the A ring, both A and HWHM increase again. The Cassini Division presents (1) larger values of A and HWHM as in the C ring and (2) strongly dispersed values which may be indeed real because no image artifact is visible in this Division. The slope has a significantly different behavior (Fig.[ahwhmsring]c) because strong jumps are observed at the boundaries of each ring. This reinforces differences of behavior of the surge and the linear part of the phase curve with the distance from Saturn.\nAs a result, the behavior of A, HWHM ans S did not show significant variations with ring type. noticed single scattering albedo of the C ring was dependent on the ring type classification. Maybe that the absence of correlation between morphological parameters and the ring type classification implies that the morphological parameters are independent of the single scattering albedo. Since the single scattering albedo is correlated with the optical depth,, we try now to correlate the morphological parameters with the optical depth in the next section.\n### Optical depth variations of the morphological parameters\nIn order to quantify differences in terms of morphological shape in our 211 phase curves in CLEAR filters (see section[phasecurvesoe]), we use the three parameters A, HWHM and S. They are represented in the Fig.[resultskaasalainentau] as a function of the normal optical depth of the rings.\n**[Fig.[resultskaasalainentau]]**\nBecause the overall correlations of A, and S HWHM with *τ* are not clear, although they seem to lead to an negative correlation for A and HWHM (Fig.[resultskaasalainentau]a,b) and a positive correlation for S (Fig.[resultskaasalainentau]c), we calculated with a linear fit the correlation coefficients for A=f(*τ*), HWHM=f(*τ*) and S=f(*τ*) for each main ring, see Table[correlationtauahwhms].\n**[Table[correlationtauahwhms]]**\nThe amplitude A of the surge (Fig.[resultskaasalainentau]a) is correlated with the optical depth of the rings. The following trends may be noted : first, the amplitudes of low optical depth (*τ*<0.5, typically the C ring and the Cassini Division) are positively correlated with the optical depth (Table[correlationtauahwhms]) and second, the amplitude at high optical depth (*τ*>1, typically the A and B rings) are negatively correlated with the optical depth (Table[correlationtauahwhms]). The second trend was quite clear by eye, whereas the first one is more difficult to distinguish in Fig.[resultskaasalainentau]a. This is due to the fact that the C ring and the Cassini Division have an optical depth restricted in the range 0.01<*τ*<0.5, which is quite compressed in the scale from 0 to 2.5 of Fig.[resultskaasalainentau]. The same behavior was noticed by for the amplitude of asteroids’ phase curves : for albedo<0.3 an increase of the amplitude with increasing albedo was noticed. We note also that for intermediate and high optical depth (0.5<*τ*<2.5, typically the A and B rings) the amplitude has a much smaller scattering and finally, the correlation coefficient are larger (-74% for the B ring). For the A ring, it is interesting to note that regions of lower optical depth (0.3<*τ*<0.5) connect well with data-points in the C ring both in terms in mean value and scattering (Fig.[resultskaasalainentau]a). A good continuity with the B ring is also observed (0.7<*τ*<1.1).\nThe angular half width of the peak at half maximum (Fig.[resultskaasalainentau]b) show similar correlation woth the optical depth than the amplitde of the surge : increase of HWHM when the optical depth increases for the C ring and the Cassini Division and decrease of HWHM when the optical depth increases for the A and B rings. The scattering of HWHM (HWHM$\\thicksim0.3\\pm0.2$) for the low optical depth regions behaves similarly as for the amplitude, and lead to small correlation coefficients (19%, see Table[correlationtauahwhms]). As for the amplitude, the scattering of HWHM is narrow for intermediate optical depth (0.5<*τ*<1.6). Again, the behavior of the A ring is clearly intermediate between the C and B rings. To summarize, the behavior of HWHM is a decreasing function of increasing optical depth, with important scattering at low *τ* that could be understood as an opposite behavior of HWHM (HWHM is an increasing function of increasing optical depth).\nThe general trend for the slope of the linear regime (Fig.[resultskaasalainentau]c) is a strong increase with increasing optical depth, with a uniform scattering and with central value well represented by S$\\thicksim0.07\\tau^{1/2}$. This is the first time that a correlation is established between the slope S and the optical depth. Previous morphological study on asteroids’ solar phase curves showed an exponential correlation between the slope of the phase function (the so-called phase coefficient *β* in mag.deg− 1) and the albedo. Our trend for the slope (the increase of S in $\\varpi\\_{0}P$.deg− 1 with increasing *τ*) is thus consistent with the slope of asteroids’ phase functions, since we assume that the optical depth is positively correlated with the albedo, as already noticed by. However, as for A and HWHM, low optical depth regions (as the C ring and the Cassini Division) have a distinct trend than the trend of the A and B rings.\nIn conclusion, whereas the slope S has a strong tendency with the optical depth (the average of the absolute correlation coefficients of S=f(*τ*, see values of Table[correlationtauahwhms]) is 53%), the first two parameters A and HWHM have a soft tendency with the optical depth (the average of the absolute correlation coefficients for A=f(*τ*) is 44% and for HWHM=f(*τ*) is 32%). Consequently, Fig.[resultskaasalainentau] which presents A, HWHM and S according to the optical depth, yields the following trends :\n* The morphological parameters of the surge (A and HWHM) have both similar behavior with *τ* in the C ring and the Cassini Division. Firstly, a positive correlation of A with *τ* and of HWHM with *τ*, with a strong scattering and become almost independent of the optical depth for *τ*>0.5. Secondly, we note a negative correlation of S with *τ*. These trends are reversed (negative correlations of A and HWHM with *τ* and a positive correlation of S with *τ*) for the moderate and high optical depth regions (typically the A and B rings) ;\n* the morphological parameter of the linear regime (S) is strongly positively correlated with the optical depth : negatively correlated in the C ring and the Cassini Division (where *τ*<0.5) and positively correlated in the A and B rings (where *τ*>0.5).\nWe conclude that the trends of all the morphological parameters are linked to the optical depth. Then the drastic differences between, on the one hand, the amplitude and the angular width of the surge and, on the other hand, the slope of the linear part across the main ring system suggests that these characteristics originate from different physical mechanisms, as predicted by physical models (see section[physmorph]).\nThe opposition effect in COLOR filters\n--------------------------------------\n### Overview of the COLOR phase curves\nFig.[representoeallringscolor] details examples of phase curves obtained for the C, B, A rings and the Cassini Division. A striking result is that the COLOR phase curves’ peaks are much narrower than the CLEAR phase curves’ peaks. Indeed, this is due to the fact that WAC images (exclusively used for the CLEAR phase curves) do not have the more peaked part of the phase function (*α*>0.01o) whereas NAC images have it (*α*>0.001o, see Fig.[representnacallrings]). To check this, we plot in the first column (labeled a) of Fig.[representoeallringscolor] the phase function obtained with the camera WAC and in the second (labeled b) we plot the fuller phase function obtained with all the NAC and WAC images combined. Indeed, Fig.[representoeallringscolor] demonstrates that the value of $\\varpi\\_{0}P(\\alpha)$ when *α* tend towards 0o is greater in the panel (b) than in the panel (a).\n**[Fig.[representoeallringscolor]]**\nThe fact that WAC images do not include the peakest part of the opposition surge explains also the difference between the peak’s intensity of curves in Fig.[oeimagescolorlocation] in CLEAR filters and COLOR filters because when we processed images in CLEAR filters, there was no NAC images. This explains also the shift in I/F at the minimum phase angle on NAC images and WAC images of the same filter (Fig.[oeimagescolorlocation]). We observe that the shift in I/F between NAC and WAC image is minimum in the blue filter (Fig.[oeimagescolorlocation]), which implies that the full surge is contained in WAC images for this wavelength.\nIn general, we noticed that the general shape of the curve is similar to that obtained previously : the C ring still has a fairly broad peak with a large amplitude (first pair of graphs at the top of Fig.[representoeallringscolor]). The B ring exhibits also narrow peaks as those of CLEAR filters comparatively to the other main rings (the second pair of graphs of Fig.[representoeallringscolor]). The Cassini division (third pair of graphs of Fig.[representoeallringscolor]) shows again a lot of scattering, which could be the consequence of the failure of the inversion when the optical depth is not well known. Finally, the A ring, yet very dense, shows also scattering for $\\varpi\\_{0}P(\\alpha)$, which could be due to the gravitational wakes, see the last pair of graphs in Fig.[representoeallringscolor]. Indeed, the inversion is very sensitive to the optical depth, and the wakes are known to modify locally the value of *τ*.\n### Results in COLOR filters and comparisons with the morphological behaviors of the CLEAR filters\nWe now consider the variation of morphological parameters A(*λ*), HWHM(*λ*) and S(*λ*) in CLEAR and COLOR filters with the distance from Saturn (see Fig.[ahwhmsringsclearcolor]).\n**[Fig.[ahwhmsringsclearcolor]]**\nThe amplitude of the opposition in the outer parts of the C and A rings has a much broader scattering in COLOR filters that in CLEAR filters (Fig.[ahwhmsringsclearcolor]a). There is, unfortunately, not enough radial coverage to generalize this effect across the C ring. The Cassini Division also shows similar behavior for the dataset at high and low spectral resolution. In the B ring, where there is a good radial coverage in COLOR filters, amplitudes of CLEAR filters are much lower than the smallest amplitudes in COLOR (typically in the blue at *λ*=0.451 *μ*m), also noticed for the amplitudes in CLEAR and COLOR filters in the A ring. This could be due to the exclusive use of the images of the WAC to resolve the CLEAR opposition surge (section[phasecurvesoecolor]) The half-width at half maximum of the peak shows a good agreement between the data in CLEAR filters and COLOR filters (Fig.[ahwhmsringsclearcolor]b). The values are the same order of magnitude in all the rings and the regional effect is the same, except for the A ring where HWHM at high spectral resolution (COLOR filters) decreases when the distance at Saturn increases while at low spectral resolution (CLEAR filters), HWHM seems to start to grow. Finally, for the slope *S* of the linear part of the phase function, we represent a normalized slope in order to compare data from the CLEAR filters with data from the COLOR filters. We follow the method of which divided the slope by the intensity of the background, thus the normalized slope is *S*/*B*1 for the linear-by-part model and now has the deg− 1 unit. In Fig.[ahwhmsringsclearcolor]c we have for *S* similar behavior as a function from Saturn in the inner A ring and the Cassini Division for data in CLEAR filters and in COLOR filters. Finally, the B ring shows regional effects very different at high and low spectral resolution, especially in the middle of the B ring, at 107 000 km (the most optically thick region) where the CLEAR filters’ slope is significantly overestimated compared to the slope in the blue filter (*λ*=0.451 *μ*m). In the same region, we also observed a strong scattering of HWHM (Fig.[ahwhmsringsclearcolor]b).\n### Comparisons with multi-wavelength HST observations\nWe now compare the Cassini data (Fig.[ahwhmsringsclearcolor]) with Earth-based data. For this purpose, the recent study of was chosen due to their small phase angles *α*>0.028o and one point below the minimum phase angle corresponding to the angular size of the Sun. The phase curves of were obtained in I/F for the main rings and adjusted with the linear-by-part model of which provided morphological parameters A, HWHM and S for different wavelengths ranging from the ultraviolet to infrared (see their figure 7).\nWe consider first the regional effects of morphological parameters derived by with the WFPC2/HST instrument. The dispersion observed by the HST for the three morphological parameters A, HWHM and S in the C ring and the Cassini Division is also very clear with ISS (Fig.[ahwhmsringsclearcolor]abc). However, it should be noted for the C ring scattering observed with our data (especially those in CLEAR filters) is highly localized (internal and external parts of the ring) and not present in the central regions, corresponding to the background according to the ring type classification of. In the B ring, noticed strong variations of A and S in the inner edge which are absent for A and are present for S in the ISS data (Fig.[ahwhmsringsclearcolor]a and c). For HWHM, HST did not have the raise of HWHM in the middle of the ring that we observed (Fig.[ahwhmsringsclearcolor]b). For the slope, we observe in both data set a decrease of S from the inner edge to the middle of the ring (Fig.[ahwhmsringsclearcolor]c). In the A ring, the regional trends for the amplitude A and the angular width HWHM are similar for HST in ISS (Fig.[ahwhmsringsclearcolor]a), but it is not the case for S. obtained an almost constant value of S. The absolute value of the slope of in all filters is roughly S$\\thicksim$0.04 deg− 1. With ISS/Cassini (Fig.[ahwhmsringsclearcolor]c), we observe two distincts trends. First, S at high and low spectral resolution have similar values. Second, the regional effect observed is the same : there is a decreasing trend with increasing distance from Saturn. At the inner edge, S$\\thicksim$0.025 deg− 1 and at the outer edge S$\\thicksim$0.015 deg− 1. There is, however, a different order of magnitude obtained by ISS/Cassini and WFPC2/HST which could be due to lack of data at large phase angles for Earth-based observation. Indeed, the factor 2 between the slope values is well explained by the overestimation of the slope when the phase angle coverage stops at 6 degrees (section[incompletedatafits] and Fig.[fig3]c).\nNow, we turn on the comparison on the variations of the morphological parameters with the wavelength, which leads to the following conclusions. First, found weak variations of A(*λ*) in the B ring. Indeed, figure 3 of shows weak wavelength-variations of the amplitude in the B ring, and also in the A ring. Only one notable difference occurs in the ultraviolet. Overall, observed a decrease of A(*λ*) from the ultraviolet to the green, and then an increase in the amplitude of the green to the infrared. This is not at all what is observed for the amplitudes of ISS where a notable increase from blue to green and a decrease in the infrared to the green (Fig.[ahwhmsringsclearcolor]a). The values of A(*λ*) for ISS/Cassini and WFPC2/HST are anti-correlated. For HWHM(*λ*), remarked a strong decrease of the half-width at half maximum when the wavelength increases, and in all the rings. No break in decreasing HWHM(*λ*) is observed by the HST. This decrease is very different of the behavior of HWHM(*λ*) with ISS/Cassini (Fig.[ahwhmsringsclearcolor]b). Thus the variations of HWHM(*λ*) presented here are also unprecedented. There is firstly not a decrease but an increase of HWHM with increasing wavelength. Second, the order of magnitude found is not the same. The angular half-width obtained with WFPC2/HST are generally between 0.05o and 0.2o while those of ISS are between 0.1o and 0.4o. This discrepancy could be the consequence of the relative thickness of our phase functions due to the use of several images at similar phase angles whereas did not have multiple recovered data because they obtained one point per phase angle. Finally, for S(*λ*), the wavelength behavior of are consistent with those of ISS/Cassini (S decreases from the blue to the red, and slightly increases in the infrared), Fig.[ahwhmsringsclearcolor]c. Strangely, this agreement shows a consistency of both data sets for variations of the slope with a wavelength whereas the regional effects of S observed by the two instruments are significantly different.\n### Variations of the opposition effect with the wavelength and the optical depth\nWe see that the morphological parameters depend on the optical depth (Fig.[resultskaasalainentau]) and also on the wavelength (Fig.[oemorphparamsclearcolortau]).\n**[Fig.[oemorphparamsclearcolortau]]**\nIt is appropriate now to quantify the variations of A(*λ*), HWHM(*λ*) and S(*λ*) by fitting them with a linear model. We thus obtain for each COLOR phase curve of the rings two linear functions (one from 0.451 to 0.568 *μ*m and another from 0.568 to 0.752 *μ*m) which fit the behavior of A(*λ*) and two linear functions HWHM(*λ*), with the same wavelength boundaries. For the slope S(*λ*), two linear functions are obtained (one from 0.451 *μ*m to 0.650 *μ*m and from 0.650 *μ*m to 0.752 *μ*m), see Fig.[linearbypartcolorrings] for an example.\n**[Fig.[linearbypartcolorrings]]**\nFor this study, is only kept the slope of each linear function (that we called the steepness), that we correlated with the optical depth of the rings (the linear functions and their correlation coefficients are given in Tables[ahwhmswavelenghttau] and [swavelenghttau]).\n**[Table[ahwhmswavelenghttau], Table[swavelenghttau]]**\nThe slopes of A(*λ*) and HWHM(*λ*) show both an increase with the optical depth from the blue to the green (Table[ahwhmswavelenghttau]). From the green to the infrared, we note a decrease of A(*λ*) with *τ*, but not for HWHM(*λ*) which can increase or decrease with the optical depth. However, the strongest trends are found for S(*λ*) which lead to an decrease with *τ* of S(*λ*) from the blue to the red and to an increase of S(*λ*) from the red to the infrared. Values are given in Table[swavelenghttau]. It is interesting to note the similar wavelength trends for A and HWHM, which are singularly distinct of the wavelength trends of S. This confirms the fact that the morphological parameters originate from different physical mechanisms, as predicted by the physical models (section[physmorph]).\nGeneral trends of the opposition phase curves’ morphology with the ISS data\n---------------------------------------------------------------------------\nTo conclude the section [results], the ISS data set provides several trends of the opposition effect in Saturn’s rings which concern the :\n* **regional behavior of A, HWHM and S across the main ring system** : indeed, the classification of ring type features defined in section [subregionaloe] shows that A and HWHM vary continuously across the ring system (thus quite independently of the ring type classification), whereas S varies abruptly at the boundaries of each ring (Fig.[ahwhmsring]) ;\n* **morphology of the rings’ phase curves** : in section [compareparametersoe], strong dependences on the optical depth of the rings of A, HWHM and S were established : anti-correlation of A and HWHM with *τ* and correlation of S with *τ* (Fig.[resultskaasalainentau]). A high scattering at low *τ* and high *τ* is noticed for HWHM in CLEAR and COLOR phase curves ;\n* **correlation between the morphological parameters** : we showed in the section [comparativeapproach] that the parameters of the surge A and HWHM are correlated (Table[crossstudykaasalainenslopeahwhm]), whereas there is only a weak dependence of HWHM and S on the one hand, and A and S on the other hand (Table[crossstudykaasalainenslopeahwhm]) ;\n* **variations of the morphological parameters in CLEAR and COLOR filters** : we remarked in section[resultsmorphclearcolor] that the CLEAR amplitude is smaller than the COLOR ones, because the CLEAR data set did not includes NAC images which generally capture the highest part of the surge (Fig.[ahwhmsringsclearcolor]). We showed also that the discrepancy in the slope of is due to their phase angle coverages which did not cover the larger phase angles. As a consequence, their slopes are overestimated, as predicted by our study on the influence of incomplete data (section[incompletedatafits] and Fig.[fig3]c) ;\n* **variations of the morphological parameters with the wavelength and the optical depth** : A(*λ*) increases from the blue to the green (0.451 *μ*m-0.568 *μ*m) and decreases from the green to the infrared (0.568 *μ*m-752 *μ*m), Fig.[oemorphparamsclearcolortau]. This increase is reinforced when the optical depth increases (Table[ahwhmswavelenghttau]) and is marked by a maximum value of A(*λ*) in the green regardless of the rings. HWHM(*λ*) also increases from the blue to the green, but no general trend is noticed from the green to the infrared (Table[ahwhmswavelenghttau]). S(*λ*) decreases from the blue to the red (0.451 *μ*m-650 *μ*m), then rises from the red to the near-infrared (0.650 *μ*m-752 *μ*m), Table[swavelenghttau]. The wavelength-variation of S is then quite distinct to that of the surge parameters (A and HWHM), then suggesting that they may originate from different physical mechanisms.\nDiscussion\n==========\nA : a combination of the coherent backscattering with the shadow hiding ?\n-------------------------------------------------------------------------\nWe showed in section[compareparametersoe] that the amplitude of the surge had a specific behavior according to the optical depth. With the model of, an estimation of the amplitude is possible but refers only to the coherent backscattering enhancement. derived the following expression for the amplitude of the surge : $A\\thicksim 1+e^{-d/L}/2$ where *d* is the effective radius of grains and *L* is the free mean path of photons in the regolithic medium. It turns out that the amplitude could not be greater than 1.5, which contradicts our morphological results (Fig.[ahwhmsringsclearcolor]a). This variation might be due to the shadow hiding effect which is not taken into account in this computation of the amplitude.\nVariations of S with the optical depth\n--------------------------------------\nIn shadow hiding numerical simulations of that take into account the optical depth and the filling factor, the absolute slope of the linear part has roughly the same value when the optical depth is greater than 1, whatever the filling factor value. This theoretical prediction is now confirmed by our results : we observed a saturation of S at *τ*>1 (Fig.[resultskaasalainentau]c). Also, we observe with the ISS data (Fig.[ahwhmsringsclearcolor]c) a wavelength-dependence of the slope S. This behavior was predicted by the model of because S depends on the extinction coefficient *Q*ext(*λ*, *r̄*) where *r̄* is the mean radius of particles (see section[incompletedatafits]).\nLinking photometric behaviors with dynamical concepts\n-----------------------------------------------------\nThe present study, thanks to the quality of its data (radial and angular resolution), allows us to highlight several observational facts never reported in the history of the observation of the opposition effect in the Saturn’s system. The slope, the angular width and the amplitude of the rings’ opposition phase curves are clearly correlated with the optical depth of the rings (Fig.[resultskaasalainentau]). Whereas a physical description of such a dependence would need a full new physical model, we provide here some arguments explaining how the optical depth may indeed influence these three parameters.\nThe optical depth is both a measure of the local volume filling factor of material, and the local collisional activity. Indeed, basic analytical computation shows that the filling factor of ring particle is proportional to *τ*/*H* (with *τ* standing for the optical depth and *H* standing for the vertical height of the rings). So one may expect regions of higher optical depth to have a much higher filling factor of particles. This is also in agreement with local simulation of ring dynamics, that shows that the vertical width of material increases with decrease optical depth, because of the lower efficiency of collisional damping. So one may expect the filling factor of particles to be an increasing function of the optical depth.\nHowever, in the case the slope of the phase function for phase angles >1o, analytical and numerical models predict that S should depend on the particle filling factor D and the vertical extension of the layer of particles in such way that the steeper is the slope, the lower is the filling factor. Our morphological trends (Fig.[resultskaasalainentau]c) seem unambiguously to contradict the theoretical trends of these models because our highest slopes are found in the high optical depth regions. Because high optical depth regions are known to have the highest filling factor, this implies that assumptions of diluted layers are not suited to the Saturn’s rings. As a consequence, the shadow hiding observed in the Saturn’s rings may result of *intra-particle shadow hiding*, as stated by. Another possibility is that the shadow hiding that operates in the rings are maintained in a “regime of transparency” of particles that dominates the shadowing effect due to packing density.\nWe now turn to the case of the HWHM and the amplitude. Whereas there is still a debate on what determine their value, authors agree to link them the coherent-backscattering process, which may be controlled by the regolith at the surface of ring particles. Like for planetary surfaces, the regolith is expected to be the result of space fractionation and erosional processes at the surface of the ring-particles, due (in particular) to the on-going collisional activity inside rings. Numerical studies of the dynamics of the ring-particles have shown that the optical depth is a key parameter controlling the collisional activity of rings. On the one hand, the number of collisions per orbit per particle is proportional to *τ*, on the other hand, the random velocity in a ring of thickness H is about *H* × Ω (with Ω standing for the local keplerian frequency). Since *H* is a decreasing function of *τ*, thus impact velocities are lower in regions of high optical depth. In short, particles in low-optical depth regions may suffer rare but violent collisions, conversely, in high-optical depth regions particles suffer frequent but gentle collisions. This may explain qualitatively why the HWHM and amplitude have different behavior in the data (Fig.[ahwhmsring] and [resultskaasalainentau]). However, impact velocities have a lower bound $\\thicksim 2r\\times\\Omega$ (with *r* standing for the particle’s radius) due to the keplerian shear across the diameter of a particle. This ``shear dominated limit\" is reached when the optical depth is high, typically for *τ*>1. In such high filling factor regime the dynamics of collisions is entirely controlled by the keplerian shear rather than by the random impact velocities. This may qualitatively explain why values of HWHM and Amplitude seem constant for *τ*>1 : in this regime, the collisional activity being about independent of optical depth, the physical properties of the regolith may be about constant which is indeed observed.\nIn conclusion, in the absence of self-consistent physical model of the opposition effect, these qualitative arguments show that there are good reasons to believe that the optical depth is a key factor determining the opposition effect in the ring through two different mechanisms :\n* the optical depth may influence the absolute slope S, assuming that shadow hiding is the preponderant mechanism at phase angles *α*>1o\n* the optical depth may controls the HWHM and the amplitude (at phase angles *α*<1o) if the structure of the regolith is influenced by the collisional activity.\nComparison of the opposition effect in optical light and infrared light\n-----------------------------------------------------------------------\nA comparison of the solar phase curves of ISS/Cassini and the thermal phase curves of CIRS/Cassini yields to the following trends :\n* found a prominent opposition surge for the thermal phase curves of the plateaux, well fitted by a logarithmic model. This is also the case of the solar phase curves of the plateaux observed by ISS (Fig.[ahwhmsring]a) ;\n* found that the thermal phase curves of the *background* (regions in the close environment of the plateaux) do not have opposition surge, whereas background has an opposition surge in the solar phase curves of ISS (Fig.[ahwhmsring]a) ;\nA priori, the emitted phase curves may not reflect the coherent backscatter effect because interferences of photons did not act on heat, and thus on infrared light. However, a pure shadow hiding model such as fails to reproduce the CIRS opposition surge of the plateaux. This could be the proof that the shadow hiding cannot produce solely the opposition surge in emitted light and that the coherent backscatter could act on the shadow hiding mechanism, by multiplying the single scattered light component at small phase angles, as underlined by. Interestingly, no similar surge were observed in the background, whereas both plateaus and background have an opposition surge in the solar phase curves of ISS. Because the background regions are more dim and reflective than the plateaus, this could explain why these regions did not have an opposition peak in emitted light.\nConclusion\n==========\nWe report here the main conclusions of this morphological study on the Saturn’s rings opposition effect seen by ISS/Cassini :\n* The amplitude A and the half angular width HWHM of the opposition surge decrease with increasing optical depth *τ*. A and HWHM may reflect both coherent backscattering and shadow hiding, because according to the morphological parameters of the surge are greater than their coherent backscatter counterparts ;\n* All the morphological parameters are linked together. We find correlations between A and HWHM, between HWHM and S, and also between A and S, which imply that S could be more or less affected by the coherent backscattering. This could be due to the fact that we derive our slope from 1.5o to 25o whereas analytical model of describes the shadow hiding effect as a slope which fits the phase curve from 4.5o to larger phase angles, see.\n* The absolute slope S of the linear part of the phase function increases with increasing optical depth *τ* (for optically thick rings) and shows distinct trends to the morphological surge parameters, which implies that this parameter is not totally affected by the coherent backscattering. As, we think that the Saturn’s rings could be in a regime of transparency of particles because the effect of packing density (decrease of S with increasing packing density) is not that expected for the Saturn’s rings (increase of S with optical depth, and if we assume that the optical depth and the packing density are correlated, increase of S with increasing packing density) ;\n* *τ*-dependence with the morphological parameters strengthen our assumptions saying that environmental effects are the key element determining the opposition effect because the optical depth is a direct measure of the collisional and dynamical activity in the surrounding of particles and is highly correlated with A, HWHM and S ;\n* Comparisons of ISS/Cassini solar phase curves and CIRS/Cassini thermal phase curves in the C ring show that the C ring’s plateaus can have a strong opposition surge both in solar and thermal phase curves whereas the C ring’s background has a strong opposition surge in the solar phase curve and no opposition surge in the thermal phase curve.\n* Wavelength variations of the amplitude A of the surge show a maximum in all the rings at *λ*=0.568 *μ*m. The increase of A from 0.451 to 0.568 *μ*m and the decrease of A from 0.568 to 0.752 *μ*m are reinforced with increasing *τ* ;\n* Wavelength variations of HWHM of the surge show also a maximum at *λ*=0.568 *μ*m but it is not systematic, HWHM can also increase from 0.568 *μ*m to 0.752 *μ*m. Moreover, there is no specific wavelength variations of HWHM with the optical depth ;\n* Wavelength variations of the slope S of the linear part imply that the shadow hiding depend on the wavelength, may be via the particle’s scattering cross-section of recent model. The decrease of S from 0.451 to 0.650 *μ*m and the increase of S from 0.650 to 0.752 *μ*m are reinforced with increasing *τ*.\nThe goal of this first paper was not to derive and quantify directly the physical properties obtained from the models. First, because there is a large set of models and it seemed more convenient to separate the morphological models to the more physical and sophisticated ones. Second, because recent physical models did not implement only the opposition effect but the main photometric effects which occur in the full phase function, from 0 to 180 degrees. Consequently, more investigations will be provided for this purpose by using full phase curves and photometric analytical models in the second paper. In the future, we hope to have the linear degree of polarization at the opposition to obtain more constrains on the ring particle’ textures.\n[lastpage]\nThe authors would like to thank F. Poulet, C. Ferrari, S. Rodriguez and the two referees for useful comments that improved the quality of the paper. This work was supported by the French *Conseil R/’egional de la Martinique*, the French *Centre National de Recherche Scientifique* (CNRS) and the Cassini Project.\n[!ht]\n| | | | | | | |\n| --- | --- | --- | --- | --- | --- | --- |\n| date | Nb im | *i* | *ε* | Radial res. | Azim. res. | Filters (Camera) |\n| | | (o) | (o) | | | |\n| *June 7* 2005 | 12 | 111.5 | 111.9 | 44.0 | 115.1 | CLEAR (WAC) |\n| *June 26* 2005 | 66 | 111.3 | 111.3 | 30.1 | 70.0 | CLEAR (WAC) |\n| *July 23* 2006 | 48 | 73.1 | 73.4 | 13.4 | 40.7 | CLEAR (WAC) |\n| *May 20* 2005 | 57 | 111.6 | 111.6 | 4.6 | 11.5 | COLOR (NAC) |\n| *May 20* 2005 | 59 | 111.5 | 111.9 | 44.0 | 115.1 | COLOR (WAC) |\n| *Dec 31* 2006 | 12 | 104.4 | 108.7 | 38.4 | 104.8 | COLOR (WAC) |\n| *Feb 20* 2007 | 20 | 103.7 | 122.7 | 66.3 | 72.5 | COLOR (WAC) |\n| *Apr 25* 2007 | 16 | 102.7 | 110.7 | 44.4 | 92.6 | COLOR (WAC) |\n[!ht]\n|l|r|c|r|c|r|c| & & & \n & function & correl. & function & correl. & function & correl. \nCass. Div. & 0.91 + 1.72*τ* & 59 % & 0.04 -0.03*τ* & -68 % & 0.049 -0.05*τ* & -35 % \nC ring & 1.20 + 0.91*τ* & 79 % & 0.04 -0.06*τ* & -46 % & 0.025 -0.04*τ* & -47 % \nA ring & 1.12 + 1.08*τ* & 56 % & 0.10 -0.14*τ* & -84 % & 0.079 -0.24*τ* & -81 % \nB ring & 1.18 + 0.58*τ* & 31 % & 0.15 -0.33*τ* & -79 % & 0.141 -0.81*τ* & -65 % \n[!ht]\n|l|c|r|c|r|c|r| & & & \n & function & correl. & function & correl. & function & correl. \nCass. Div. & 1.22 + 0.937 & 41 % & 0.20 + 0.175*τ* & 19 % & 0.040 - 0.030*τ* & -42 % \nC ring & 1.50 + 0.024 & 25 % & 0.33 + 0.077*τ* & 19 % & 0.024 - 0.026*τ* & -68 % \nA ring & 1.50 - 0.175 & -36 % & 0.31 - 0.106*τ* & -49 % & 0.024 + 0.025*τ* & 19 % \nB ring & 1.42 - 0.068 & -74 % & 0.24 - 0.016*τ* & -39 % & 0.035 + 0.031*τ* & 82 % \n[!ht]\n|l|c|r|c|r| & & \n & function & correl. & function & correl. \n0.451 *μ*m <*λ*< 0.568 *μ*m & 1.210 + 0.3*τ* & 55 % & 0.14 + 0.011*τ* & 38 % \n0.568 *μ*m <*λ*< 0.752 *μ*m & 0.001 - 0.5*τ* & -51 % & 0.04 - 0.037*τ* & -46 % \n[!ht]\n|l|r|r|r|r| & & \n& & \n & function & correl. & function & correl. \n0.451 *μ*m <*λ*< 0.650 *μ*m & -0.21 - 0.043*τ* & -73 % & -0.04 - 0.039*τ* & -49 % \n0.650 *μ*m <*λ*< 0.752 *μ*m & 0.04 + 0.018*τ* & 67 % & 0.02 + 0.054*τ* & 44 % \n[representoeimages] The opposition effect in the B ring. A typical image of the *26 June* sequence captured by the Wide Angle Camera (W1498453136.IMG). Concentric circles correspond to identical phase curves computed in the image (isophase). Numbers in the concentric circles represent the solar phase angles (in degrees).\n[oehisurge1] The opposition effect in the C ring. A typical image of the *20 may* sequence captured by the Narrow Angle Camera (N1595278165.IMG). The contrast is enhanced to make more visible the opposition spot in the C ring. Concentric circles correspond to identical phase curves computed in the image (isophase). Numbers in the concentric circles represent the solar phase angles (in degrees).\n[oeimagescolorlocation] Radial location of the opposition spot in the images taken in CLEAR and COLOR filters (filled symbols represent WAC images and empty symbols represent NAC images). We give in the *y*-axis the normalized brightness *I*/*F* of the minimum phase angle in the image, the *x*-axis is the corresponding distance from Saturn of this point. The vertical dotted lines correspond to ring boundaries. The optical depth of PPS/Voyager is plotted as a radial reference.\n \n[fig2] Typical phase curves of the B ring (a) and C ring (b) in CLEAR filters chosen for testing the stability of linear-exponential model of when the phase angle coverage is incomplete. Circles and squares correspond to the cutoff of the incomplete phase function and the solid curves correspond to the initial fit of the linear-exponential model of. The vertical dotted lines correspond to boundaries of the surge domain and the linear part domain that we have tested (see Fig.[fig3]).\n[fig3] Deviation of the linear-exponential model of for the morphological parameters A (a), HWHM (b) and S (c) when the phase angle coverage is incomplete. Circles and squares correspond to the cutoff of the incomplete phase functions of respectively the B ring and C ring typical phase curves (Fig.[fig2]).\n[representoeallrings] Representative CLEAR phase curves for the main rings with a zoom on the surge (a) and the full phase curve (b), fitted respectively with the logarithmic model of (a) and the linear-by-part model of (b).\n[ahwhmsring] Regional behavior of the morphological parameters : (a) the amplitude A, (b) the angular width HWHM in degrees and (c) the absolute slope S in $\\varpi\\_{0}P$.deg− 1 from the Linear-by-part model of for CLEAR phase curves using the ring type classification. The vertical dotted lines correspond to ring boundaries. The optical depth of PPS/Voyager is plotted in (c) as a radial reference.\n[resultskaasalainentau] Morphological parameters of CLEAR phase curves from the Linear-by-part model of : Amplitude A (a), Angular width HWHM (b) and absolute slope S (c) in $\\varpi\\_{0}P$.deg− 1.\n[representoeallringscolor] Representative COLOR phase curves for the main rings with a zoom on the surge with WAC images (a) and the full phase curve with NAC and WAC images (b). Full phase curves of (b) are fitted with the linear-exponential model of.\n[ahwhmsringsclearcolor] Regional behavior of morphological parameters : (a) the amplitude A, (b) the angular width HWHM and (c) unitless absolute slope *S*/*B*1 in deg− 1 from the Linear-by-part model of using CLEAR and COLOR phase curves. The vertical dotted lines correspond to ring boundaries. The optical depth of PPS/Voyager is plotted in (c) as a radial reference.\n[oemorphparamsclearcolortau] Morphological parameters of CLEAR and COLOR phase curves from the Linear-by-part model of : (a) the amplitude A, (b) the angular width HWHM and (c) the unitless absolute slope *S*/*B*1 in deg− 1.\n \n**Electronic supplementary material**\nTable 1: Outputs of the logarithmic model of for the CLEAR phase curves representing each ring types (*inner* corresponds to inner regions characterized by low optical depth, *background* are morphological smooth regions without abrupt variation of optical depth, *bright* corresponds to bright regions that have the highest optical depth in each ring, *ringlet* corresponds to a thinner ring embedded in a less dense region or a gap, and *outer* corresponds to outer regions that mark the transition at each ring boundary) of each main ring. Horizontal lines correspond to the ring boundaries. We give for each ring type, the normal optical depth of the Voyager/PPS instrument and the corresponding radius (the distance from Saturn’s center). f$\\_{\\varpi\\_{0}P\\rightarrow\\textrm{I/F}}$ is a conversion factor which corresponds to the mean level of I/F curves over the mean level of $\\varpi\\_{0}P$ curves. [reskaasalainentypicalrings]\n\nThe opposition effect in Saturn’s rings seen by Cassini/ISS:I. Morphology of phase curves\n=========================================================================================\n,\n,\n,\n, and\nNumber of pages: \nNumber of tables: [lasttable] \nNumber of figures: [lastfig] \nRunning head: The opposition effect in Saturn’s rings seen by Cassini/ISS 1. Direct editorial correspondence to : \nEstelle Déau \nCEA Saclay \nService d’Astrophysique \nAIM, UMR 7158 \nOrme des Merisiers Bat. 709 \n91191 Gif-sur-Yvette FRANCE \nphone: +33 1 69 08 80 56 \nfax: +33 1 69 08 65 77 \ne-mail : estelle.deau@cea.fr \nThe Cassini cameras have captured the opposition effect in Saturn’s rings with a high radial resolution at phase angles down to 0.01o in the entire main ring system. We derive phase functions from 0.01o to 25o of phase angle in the Wide-Angle Camera clear filters with a central wavelength *λ*cl=0.611 *μ*m and phase functions from 0.001o to 25o of phase angle in the Narrow-Angle and Wide-Angle Cameras color filters (from the blue, *λ*bl=0.451 *μ*m to the near infrared *λ*it=0.752 *μ*m). We characterize the morphology of the phase functions of different features in the main rings. We find that the shape of the phase function is accurately represented by a logarithmic model (Bobrov 1970, in Surfaces and Interiors of Planets and Satellites, Academic, edited by A. Dollfus). For practical purposes, we also parametrize the phase curves by a simple linear-by-part model (Lumme and Irvine 1976, Astronomical Journal, 81, p865), which provides three morphological parameters : the amplitude and the Half-Width at Half-Maximum (HWHM) of the surge, and the slope S of the linear-part of the phase function at larger phase angles. Our analysis demonstrates that all of these parameters show trends with the optical depth of the rings. These trends imply that the optical depth is a key-element determining the physical properties which act on the opposition effect. Wavelength variations of the morphological parameters of the surge show important trends with the optical depth in the green filter (*λ*gr=0.568 *μ*m), which implies that grain size effects are maximum in this wavelength.\nSaturn’s rings; phase curves; opposition effect; coherent backscattering; shadowing; shadow hiding\nIntroduction\n============\nWhen one views the rings of Saturn with the Sun directly behind the observer, a phenomenon called the *opposition effect* can be seen. The opposition effect, also known as the opposition surge, is a sudden, nonlinear rise in brightness with decreasing phase (Sun–ring–observer) angle that occurs as the phase angle approaches zero. The opposition effect, which was observed for most Solar System bodies and was discovered in the course of Müller’s long-term photometry of the Saturn system, beginning in 1878. inferred that the opposition effect was due to the rings, since Jupiter did not show a comparable opposition brightening.\nMost recent studies of the opposition effect in the Saturn’s rings were based on ground based and spatial data which resolved the main rings. Earth-based observations, though valuable, have a low to moderate spatial resolution. This indicates that the signal is averaged over large regions in the rings. Unfortunately, the rings are highly heterogeneous features that may present rapid spatial variations of their optical properties. So the interest of *spacecraft* observations is the ability to probe the signal in very narrow ring features ($\\thicksim$40 km in the present paper), which may have much more uniform optical properties. This allows an easier study of the opposition effect which is an already complex phenomenon. \nPioneer 11, Voyager 1 and Voyager 2 did not observe the rings at phase angles near zero. However, Cassini, which became an artificial satellite of Saturn in 2004, is the first spacecraft to observe the opposition effect in the rings, with several instruments, including the Narrow Angle (NAC) and Wide Angle (WAC) cameras of the Imaging Science Subsystem (ISS). For our data set (Table[tableoeinfoimages]), typically for the WAC 1 pixel=40 km and the FWHM is 1.8 pixels, for the NAC 1 pixel=5 km and the FWHM is 1.3 pixels. With Cassini images we are thus able to characterize how the surge varies throughout individual features in the ring system.\nThe opposition effect is now known to be the combined effect of the coherent backscatter (at small phase angles) which is a constructive interference of photons in a medium of using grains of wavelength size and the shadow hiding (at larger phase angles) which consists of shadows cast by particles on other particles that become invisible to the observer. Some analytical models have tried to combined these two effects, but recent laboratory measurements and data analysis show that models’ results could be compromised by some physical assumptions on the scatterer elements, the wavelength-dependence and the the angular size of the source. Two other effects, the external near-field effect and the single scattering internal-field coherence, were never been observed solely because they are intrinsically inseparable from the coherent backscatter effect and are for the moment only numerically simulated.\nBoth shadow-hiding and coherent backscattering are likely to play roles in determining the shape of the phase curves of Solar System bodies at low phase angles. Shadow-hiding probably dominates at phase angles greater than a few degrees, while coherent backscattering takes effect at the very smallest phase angles. The shadow-hiding effect gives clues about the three-dimensional structure of a layer of ring particles which, according to, could have a typical size of *r̄* = 15m. By contrast, the coherent backscattering component sheds light on the nature of the ring particle surfaces at a scale not much larger than optical wavelengths. Indeed, previous photometric studies which investigate the opposition effect in the Saturn’s rings determined ranges of typical size for coherent-backscattering effect of about *d* = 10 *μ*m. Thus, the shape of the opposition effect provides constraints on features vastly smaller than the camera’s resolution. Like, we refer to particles as macroscopic and individual objects in the rings and grains as microscopic objects at the surface of rings’ particles. Using this terminology, ring particles might be covered by regolithic grains.\nThis is the first of a series of papers dealing with the opposition effect in Saturn’s rings as seen by ISS/Cassini. In the present paper, we focus on the characterization of the morphological modeling of the shape of opposition effect in the rings. Since the recent theories have difficulties to link the behavior of the opposition effect with the physical properties of the surface material, we think it is necessary to have an experimental approach, consisting in (1) deriving the shape’s parameters and (2) trying to find correlations among themselves, and with the optical depth. In the second paper (Déau et al. in preparation), we will use recent analytical photometric models to derive some of the physical properties of the ring material. Indeed, to constrain completely the physical properties of the ring material, photometric and polarimetric phase curves are needed, however, Cassi/ISS did not captured the opposition spot with its polarized filters. In section[observeetreduit] of the present paper, we describe the ISS/Cassini data set and our procedure for extracting photometric data from images and fitting empirical models to the data. In section[results], we characterize the morphology of the opposition surge at different locations in the main rings and focus our attention on cross-correlations among the morphological parameters and correlations of these parameters with the optical depth and the wavelength. Finally, in section[discussion] we discuss these results and examine to which extents photometric models may explain the morphological trends derived in section[observeetreduit].\nObservations and reductions\n===========================\nThe Cassini Imaging Data Set\n----------------------------\nThe Cassini ISS instrument is composed of two cameras, a wide angle camera (WAC) and a narrow angle camera (NAC) equipped with 1024 × 1024 CCD matrices. Both use a set of about twenty filters ranging from near-infrared to ultraviolet. Here, we focus only on filters in the optical domain, with first the blue, green, red and near infrared filters, that will be called hereafter COLOR filters. According to, COLOR filters from NAC and WAC cameras do not have exactly the same central wavelength. For the blue filters, the NAC can have two different combinations of filters (CL1,BL1) or (BL1,CL2) which lead to two central wavelengths : $\\lambda\\_{\\textrm{bl}}^{\\textrm{\\textsc{nac}}}=0.440\\mu$m and $\\lambda\\_{\\textrm{bl}}^{\\textrm{\\textsc{nac}}}=0.451\\mu$m. In contrast, for the WAC, the blue (CL1,BL1) filter is characterized by a wavelength of about $\\lambda\\_{\\textrm{bl}}^{\\textrm{\\textsc{wac}}}=0.460\\mu$m. Because the spectral width of these filters is about ± 0.050*μ*m, we consider all images from blue filters of the NAC and the WAC as a consistent whole. The green filter (CL1,GRN) has almost the same spectral characteristics for the two cameras since the central wavelength of $\\lambda\\_{\\textrm{grn}}^{\\textrm{\\textsc{nac}}}=0.568\\mu$m for the NAC and $\\lambda\\_{\\textrm{grn}}^{\\textrm{\\textsc{wac}}}=0.567\\mu$m for the WAC. The red filter for the NAC corresponds to the filter (RED,CL2) with a central wavelength of $\\lambda\\_{\\textrm{red}}^{\\textrm{\\textsc{nac}}}=0.650\\mu$m. For the WAC, the combination (CL1,RED) is at a central wavelength of $\\lambda\\_{\\textrm{red}}^{\\textrm{\\textsc{wac}}}=0.649\\mu$m, which does not change to the that of the NAC at one nanometer. Finally, the filter (CL1,IR1) in the near infrared shows a difference of 10 nanometers between the central wavelength of the NAC ($\\lambda\\_{\\textrm{ir}}^{\\textrm{\\textsc{nac}}}=0.752\\mu$m) and that of the WAC ($\\lambda\\_{\\textrm{ir}}^{\\textrm{\\textsc{wac}}}=0.742\\mu$m). In summary, with this moderately high spectral resolution, the combination of images that are not coming exactly from the same filters are responsible for a shift of the central wavelength from 0.001 to 0.010 micron. Because each filter has a non negligible spectral range ( ± 0.050*μ*m), we consider without distinction the COLOR filters from the NAC and WAC cameras and in the rest of the paper, we take as a reference the following wavelengths (*λ*bl = 0.451*μ*m; *λ*grn = 0.568*μ*m; *λ*red = 0.650*μ*m and *λ*ir = 0.752*μ*m) to designate each COLOR phase curve. The other part of images discussed in the present paper has been taken in CLEAR mode of the WAC, designating the absence of filters resulting in a spectral bandwidth spanning from 0.20 to 1.10 *μ*m (the central wavelength is *λ*cl = 0.611*μ*m and the spectral range is ± 0.450*μ*m). For the present paper, the selected observation campaign of the opposition effect on the Saturn’s rings is divided in four sequences from May 2005 to July 2006 (Table[tableoeinfoimages]).\n**[Table[tableoeinfoimages]]**\nThese observations were conduced conjointly by the VIMS, CIRS and ISS instruments. Three of ISS sequences were obtained using the WAC only and the last one using the BOTSIM mode (using both cameras simultaneously). Depending on the sequence, different filters were used and different spatial and angular resolutions were achieved. In the present paper, the first of a series aimed at a detailed study of the opposition effect, we will focus on sequences obtained using the WAC camera in CLEAR filter mode and sequences using NAC and WAC cameras in COLOR filter mode, Table[tableoeinfoimages]. Images of the B and C ring are shown in Fig.[representoeimages] and [oehisurge1] to illustrate the quality of the data set.\n**[Fig.[representoeimages], Fig.[oehisurge1]]**\nIn the CLEAR filter mode, the wide angle camera captured rings at zero phase angle two times in 2005, on June 7 and June 26 (they will be designated as *June 7* and *June 26* sequences in the rest of the paper, for practical purpose). In the *June 7* sequence, the WAC images have an average resolution of $\\thicksim$ 44 km per pixel; the rings are observed in reflexion. In each individual image, the phase angle varies by about 3 degrees. In the majority of the set, the opposition point at zero phase angle is visible with a very good sampling : about 0.005 degree per pixel. This set covers the full ring system by tracking the opposition spot. Most of the set has an excellent photometric quality, however in dark regions the strong Saturnshine produces ghost images of the secondary mirror perturbing the signal significantly. The amplitude of this artifact is quite constant, about 0.05 in I/F, which does not significantly affect the photometry of the bright regions as A and B rings. However it is very troublesome for dark regions like the D ring, the C ring and the Cassini Division. This is why images from the *June 7* sequence have not been considered for these two regions. The same instrument artifact has been observed and discussed in. The *June 26* sequence has similar characteristics, see Table[tableoeinfoimages], with a better radial resolution of about 30 km per pixel and spans the B and C rings. The presence of the ghost signal in the C ring is not clear. A detailed photometric analysis in these images (cf. section[phasecurvesoe]) shows that the derived phase function is consistent with an unperturbed signal, very differently from images in which the ghost were clearly identified. It is why the images of the C ring taken on *June 26* have been considered for the C ring phase function. The *July 23* sequence has the best radial resolution ($\\thicksim$13 km.pixel− 1) and spans all A, B, C rings and the Cassini Division (the radial location of the opposition spot in each image is represented in the Fig.[oeimagescolorlocation]). In this set, the ghost artifact is absent owing to a much larger angular separation with the bright Saturn globe. Some images of this sequence provide also the larger phase angles.\n**[Fig.[oeimagescolorlocation]]**\nIn COLOR filters mode, the ISS instrument uses the BOTSIM mode in the *May 20* sequence, using both cameras simultaneously for a ride in A and B rings, Cassini Division and the outer C ring (Fig.[oeimagescolorlocation]). However, because the boresight of the NAC and the NAC are not exactly the same, NAC images did not represent a zoom of the opposition spot in WAC images. Thus, WAC and NAC images captured at the same time have opposition spots that are not exactly at the same location in the rings. The radial resolution in NAC images is 4.6 km.pixel− 1 and spans from 44 to 66 km.pixel− 1 for WAC images in COLOR filters. The other sequences (*Dec 31*, *Feb 20* and *Apr 25*), did not have the opposition spot but provided the larger phase angles.\nData reduction\n--------------\n### Calibration\nRaw images are calibrated first with a standard pipeline described in and called CISSCAL, using the 3.4 version. The DN values are converted into the I/F units. This ratio is dimensionless, with *I* is the specific intensity measured by Cassini, and *π**F* is the incident flux from the Sun. So it is a measure of the local reflectivity under the current observing geometry. Cassini images are not directly exploitable to study the photometric behavior of rings. In order to do this, the relevant data we need are the so-called *phase function* which is linked to the I/F ratio as a function of the phase angle, and corrected from effects of observation geometry. A full procedure has been designed to reconstruct the phase function from different images with different observation geometry and resolutions and is detailed below.\n### Extraction procedure\nThe first step is to reproject the images in a (Radius, Longitude) frame, in which features at a same radius from Saturn are horizontally aligned. This procedure critically depends on the quality on the navigation. When possible, the edges of the A, B, C rings as well as ring features reported in were used as fiducial references. Distances reported in were corrected according to. The resulting navigation error on WAC images varies from 1 to 1.5 pixels from the center to the edge of the image. The I/F in a ring at constant distance from Saturn is obtained by extracting the data on a line of constant radius (*i.e.* an horizontal line) in the reprojected map of calibrated brightness. Other geometrical parameters are extracted in the same way : phase angle *α*, cosine of incidence angle *μ*0, cosine of the emission angle *μ*, optical depth of the rings obtained from the PPS Voyager instrument (*λ* = 0.260 *μ*m), ; radius scale has been corrected with the procedure of. This procedure works very well for structures with width larger than the navigation accuracy. However radial structures are visible at all scales down to one pixel, (Fig.[oehisurge1]). For structures radially smaller than 5 pixels, navigation errors make false the extraction along a line of constant radius close to the edges of the image, inducing accidental extraction in nearby different features. To overcome this problem and to ensure that we always extract the same ring feature, we developed a *ring tracking* technique using a basic pattern recognition algorithm to follow a single feature. Extensive visual check of the result shows the method is reliable down to 1 pixel of radial width.\n### Construction of the phase function\nThe ultimate information we need is the phase function of individual ring particles to characterize their surface properties. Unfortunately, the signal from an individual particle is heavily altered because of the finite thickness of the rings. Also, the signal’s intensity depends on observation angle with respect to the ring’s normal. Inverting such complex collective photometric effects would require the use a detailed light scattering code with many assumptions concerning the photometric properties of particles. Such code has been developed by and. However, for our present purposes, they cannot used to derive the phase functions in hundreds of different regions as we wish to do here. Consequently, as a first approximation, we use in the present paper the classical approach of linking the I/F to the phase function with the following assumptions : homogeneous layer of particles and single scattering. This latter assumption may be justified for phase angles smaller than $\\thicksim$30o. In reflexion, the phase function $\\varpi\\_{0}\\cdot P(\\alpha)$ is derived from the solution to the radiative transfer equation (the designation *phase function* is not strictly accurate since what we really determine is the product of the single scattering albedo, $\\varpi\\_{0}$, times the particle phase function, *P*(*α*)) :\n \n$$\\varpi\\_{0}\\cdot\nP(\\alpha)=\\frac{I}{F}\\times\\frac{4(\\mu+\\mu\\_{0})}{\\mu\\_{0}}\\times\\left(\n1 - e^{-\\tau\\left(\\frac{1}{\\mu}+\\frac{1}{\\mu\\_{0}}\\right)} \\right)^{-1}\n\\label{eq\\_chandrasekhar}$$ \nwith *τ*, *μ*, *μ*0, *α* standing for : the normal optical depth, cosine of emission angle, cosine of incidence angle and phase angle respectively. In order to allow future comparisons with detailed numerical models, numerous analytical parametrization of the observations are provided by morphological model in section[describemodeloe]. As one can see, the value of the optical depth is necessary to derive the phase function. Preceding works has shown that the exponential factor can be neglected in first approximation for Earth-based observations. However, we noticed that we obtained much more coherent results when taking into account the exponential factor, when comparing results from different geometry of observations. Consequently, we keep the initial formula of, as previous photometric studies based on spacecraft observations of Saturn’s rings.\n*P*(*α*) obtained with Eq.[eqchandrasekhar] is the particle’s disk integrated phase function which determines the angular distribution of single scattered radiation from the body as a whole. The phase function is normalized over the solid angle Ω to the single scattering albedo : $\\varpi\\_{0}=\\frac{1}{4\\pi} \\int P(\\alpha)~\\textrm{d}\\Omega$. To derive the albedo, the full phase curve, from 0 to 180 degrees of phase angle, must be known. In this study, we have restricted our data (0<*α*<25 degrees), so that we may avoid separating the albedo from the phase function. The derivation of the albedo will be presented in the second paper which relates ISS data from 0 to 180 degrees.\n### Finite size of the Sun\nGenerally speaking, all phase curves present a bright opposition surge below 1o and a slope decreasing linearly for *α* > 1o. Whereas WAC images go down to $\\thicksim$0.01o of phase angle (Fig.[representoeimages]), NAC images taken in COLOR filters capture the opposition spot at better angular resolution (Fig.[oehisurge1]), the resulting phase functions from NAC images go down to $\\thicksim$0.001o (Fig.[representnacallrings]). This is the first time the opposition spot is imaged at such fine scale. Note that we define the phase angle as the angle between the vector pointing to the Sun’s center, and to the spacecraft, from the observed point. So from a strictly *mathematical point of view* there is no lower limit to the phase angle value despite the source’s finite size. For an extended illumination source, the phase curve should be the integral of a point source phase function over the Sun angular radius. Thus on a more *physical point of view* it is not possible to observe the phase function at angle below the Sun’s angular radius. Fig.[representnacallrings] demonstrates that the opposition surge flattens (in all rings and at all wavelengths) at phase angles below 0.029o, in good agreement with the Sun’s angular radius at the date of observations.\n**[Fig.[representnacallrings]]**\nData fit with Morphological models\n----------------------------------\nThe purpose of the present paper is to provide an accurate description of the morphological behavior of the observed phase curves. This is the very first step prior to any attempt of further modeling, which will be the subject of a forthcoming paper. As a consequence, special care has been given to parametrize the phase functions conveniently. In addition morphological parametrization is necessary to compare efficiently hundreds of phase curves at different locations in the rings and derive statistical trend as will be done in section[results]. Several morphological models have been used in the past to describe the shape of the phase functions : the logarithmic model of, the linear-by-part model of and the linear-exponential model of. The specific properties of these three models make them adapted for different and complementary purposes. The logarithmic model is interesting for direct comparisons with numerical models, the linear-by-part model is convenient to describe the shape in an intuitive way, and finally the linear-exponential model is adapted for comparison with other studies previously published.\n### The logarithmic model\nAs, and, we remark that a logarithmic model describes very well the solar phase curves of the Saturn’s rings. It depends on two parameters (*a*0 and *a*1). This first morphological model has the following form :\n \n$$\\varpi\\_{0}P(\\alpha)=a\\_{0} + a\\_{1}\\times\\log(\\alpha)$$ \nIn general, this model is the best morphological fit to the data. It is reasonably accurate down to 0.029o of phase angle. This is due to the finite angular size of the Sun which flattens data below 0.029o, whereas the logarithmic function continues increasing (see section[solarsizefinisat] and Fig.[representnacallrings]a). For large phase angles, the fit is satisfactory up to *α* ≃ 15 degrees, this was also noticed by who fitted temperature phase curves of the C ring with CIRS/Cassini. Whereas the meaning of *a*0 and *a*1 are not easily interpretable in term of shape, to allow an easy comparison with future numerical simulations the values of these two parameters are reported in Table 1 of Electronic supplement material.\n### The linear-exponential model\nThis model describes the shape of the phase function as a combination of an exponential peak and a linear part. Its main interest is that it has been used in previous work for the study of the backscattering of Solar System’s icy satellites and rings. We give the details of this model : the 4 parameters are the intensity of the peak *I**p*, the intensity of the background *I**b*, the absolute slope of the linear part *I**s* and the width of the exponential *w*, such that the phase function is represented by :\n \n$$\\varpi\\_{0}P(\\alpha)=I\\_{b} + I\\_{s}\\cdot\\alpha + I\\_{p}\\cdot\ne^{-\\frac{\\alpha}{2w}}$$ \nWith these 4 parameters, we characterize the shape of the phase function by introducing three morphological parameters : A, HWHM and S designating the amplitude of the surge, the half-width at half-maximum of the surge and the absolute slope at large phase angles respectively, so that :\n \n$$\\textrm{A}=\\frac{I\\_{p}+I\\_{b}}{I\\_{b}}~~~~~~~~\\textrm{HWHM}=2\\cdot\n\\ln2w~~~~\\textrm{and}~~~~\\textrm{S}=-I\\_{s}$$ \nAs, we noticed that this model did not fit well the phase curves, in particular, the derived model parameters appeared to depend substantially on the phase angle coverage (see section[incompletedatafits]), preventing a robust comparison of data. This is due to the fact that we did not use the converging procedure of but rather the common downhill minimization technique, as done by the previous users of the linear-exponential model. Moreover, it seems that the angular scale at which the phase function is observed may strongly influence the model parameters. So to avoid this problem, we used a much simpler morphological model that appeared much more robust for the comparison of heterogeneous data set : the linear-by-part model of, also we found a much better match of the data with the linear-by-part model of (see below). However, the linear-exponential was considered in the present study only to understand variations of A, HWHM and S between the phase curves and our phase curves (see section[incompletedatafits] and section[comparehstiss]).\n### The linear-by-part model\nFor an intuitive description of the main features of the phase curves, the linear-by-part model is the most convenient one. It is constituted of two linear functions fitting both the surge at small phase angle (*α* < *α*1) and the linear regime at higher phase angle (*α* > *α*2) :\n \n$$\\begin{aligned}\n\\varpi\\_{0}P(\\alpha<\\alpha\\_{1})=-A\\_{0}\\times\\alpha + B\\_{0} \\newline\n\\varpi\\_{0}P(\\alpha>\\alpha\\_{2})=-A\\_{1}\\times\\alpha + B\\_{1}\\end{aligned}$$ \n and use *α*1=0.27o and *α*2=1.5o. However, we encountered difficulties with the value of *α*1. By testing several values of *α*1, it appears that for our data set, values of *α*1 less than 0.3o yield a general overestimation of *A*0, especially in the C ring and values of *α*1 greater than 0.3o yield an underestimation of *A*0 only in the B ring. Consequently we found the our data were better reproduced using *α*1 = 0.3o which is now adopted in the rest of the paper. With this four outputs : the two absolute slopes *A*0 and *A*1 and the two y-intercepts *B*0 and *B*1, the shape of the curve is characterized by the morphological parameters A, HWHM and S are then defined by :\n \n$$\\textrm{A}=\\frac{B\\_{0}}{B\\_{1}}~~~~~~~~\\textrm{HWHM}=\\frac{(B\\_{0}-B\\_{1})}{2(A\\_{0}-A\\_{1})}\n~~~~\\textrm{and}~~~~\\textrm{S}=A\\_{1}$$ \nThe purpose of this model is not, of course, an accurate description of the data but rather a convenient description of the main trends of the phase curve.\n### Stability of the morphological parameters\nIn order to compare properly our results with those of and, for which the Saturn’s rings phase curves did not have the same phase angle coverage, we have tested the influence of the portion 0.05o-0.4o and 6o-25o on the converging solution. Indeed, phase curves of do not have data under 0.3o, and both studies of and do not have data over 6.5o. With two typical Saturn’s rings phase curves of ISS (Fig.[fig2]), we have removed data by section of 0.1o and fit the pseudo-incomplete phase curve with the linear-exponential model, which provides the new solution, designated by *A*remove, HWHMremove and *S*remove. The initial solution found for fuller phase function (0.01o-25o) is called *A*optimal, HWHMoptimal and *S*optimal. Both solutions are obtained with a downhill minimization technique, to reproduce the fitting method used by and. It is wise to recall that proposed a converging procedure more accurate, however because did not used it, and because we want to reproduce not the best converging solutions but the deviations lead by the fits driven by and, that is why we use the downhill method and not the probability distribution method of.\n**[Fig.[fig2], Fig.[fig3]]**\nIn Fig.[fig3], we represent the ratio of the morphological parameters of the incomplete phase curve over the morphological parameters of the fuller phase curve, called *A*remove/*A*optimal, HWHMremove/HWHMoptimal and *S*remove/*S*optimal. We observe a slight underestimation of A, a strong overestimation of HWHM and a moderate underestimation of S. The deviation of the optimal value is quite weak for A ($A\\_{\\textrm{remove}}/A\\_{\\textrm{optimal}}\\thicksim$0.96 at a cutoff of 0.3o) but its variation depends on the morphology of the surge. For the typical B ring phase curve (which has a narrower peak), we note a slight decrease of the ratio *A*remove/*A*optimal and for the C ring, we note a slight increase of the ratio. However the both ratios lead to an underestimation, which means that incomplete data fitted by the morphological model will have a smaller amplitude that data which cover the full surge. HWHM shows the strongest deviation of *A*remove. Indeed, HWHM is overestimated for the typical B ring phase curve (HWHMremove/HWHM$\\_{\\textrm{optimal}}\\thicksim$1.4 at a cutoff of 0.3o) but is strongly overestimated for the typical C ring (HWHMremove/HWHM$\\_{\\textrm{optimal}}\\thicksim$2.0 at a cutoff of 0.3o) which has a wider peak. For the slope, we notice an overestimation in the order of $S\\_{\\textrm{remove}}/S\\_{\\textrm{optimal}}\\thicksim$1.3 at a cutoff of 7o for the B ring phase curve and of $\\thicksim$2.5 for the C ring at the same cutoff. This means that the slope of the linear part is stabilized at roughly 15o. These comparisons are important when we compare morphological trends found by, in section[comparativeapproach] and, in section[comparehstiss].\n### Linking morphological parameters with the physical parameters of the models\nThe use of a simple morphological model is generally not adapted to derive the physical properties of the medium. However, the theories developed for the coherent backscattering and the shadow hiding effects deduce their properties by parameterizing the opposition phase curve. Recent laboratory experiments raise some doubts on the meaning of the physical parameters of these models and their correlation with the real physical properties of the medium. However, because there is no better modelization, we connect the morphological parameters A, HWHM and S with the physical characteristics of the medium derived from these models.\nA\nis the amplitude of the opposition peak and describes the behavior of the phase function at the smallest phase angles (*α*<2o). What we know about the opposition effect is that it occurs at the smallest phase angles and acts on the multiple scattered light in the regolith on the surface of the particles : the underlying phenomenon is the coherent backscattering effect. The coherent-backscattering effect increases in brightness by almost a factor two, while using grains size smaller than the wavelength of the incident light,. In contrast, the second phenomenon of the opposition effect, the shadow hiding effect, is known to produce a wide peak from 0 to 2 degrees, and to decrease the brightness up to 20 degrees. The combination of the two effects at very low phase angle is still a matter of debate and today two theories disagree in order to explain the peak of the opposition. The theory of assumes that the opposition peak is a pure coherent backscattering effect whereas the theory of shows that the opposition peak results from a coupling of coherent-backscattering and shadow hiding, even at low phase angles. This coupling should be due to the fact that the coherent backscattering could act on both multiple and single scattered light whereas the shadow hiding is a single scattered light effect. Thus, this theoretical model defines two amplitudes as output parameters : the coherent backscatter amplitude *B**C*0 and the shadow hiding amplitude *B**S*0. As a consequence, using this theory, it does not seem to be possible to ascribe the morphological parameter A solely to the coherent backscattering. For most laboratory measurements, the amplitude of the opposition peak is a function of grain size in such way that A decreases with increasing grain size. This anti-correlation finds a natural explanation by the fact that macroscopic irregularities (>*λ*) create less coherent effects than microscopic irregularities ( ≤ *λ*). Laboratory measurements of also confirmed that the opposition surge increases when irregularities are small. and underline the fact A is linked to the intensity of the background *I**b* which is a decreasing function of increasing absorption, thus A must increase with increasing absorption or decreasing albedo $\\varpi\\_{0}$, which was also confirmed by laboratory measurements of who found that the peak decreases with increasing sample albedo.\nHWHM\n, the half-width at half-maximum, is generally associated to the coherent backscatter effect. It can be related to the grain size, index of refraction and packing density of regolith. The HWHM is maximum for a effective grain size near *λ*/2 and increases when the regolith grains filling factor *f* increases. For high values of *f*, the HWHM shifts towards the greater grain size. However, as for the amplitude, the model of defines two HWHMs : the coherent backscatter angular width (*h**c*), which is defined similarly that in the model of, and the shadow hiding angular width (*h**s*). This reinforces the idea that the observed surge results from a coupling between the coherent backscattering and the shadow hiding.\nS\n, the slope of the linear part, seems to be due only to the shadow hiding effect: the interferences caused by coherent backscattering effect seem not to be very significant at larger phase angles. This means the shadow hiding light is not affected by the coherent backscattering at larger phase angles. Also, according to recent analytical and numerical models, the shadow hiding acts solely on the linear part of the phase function. We define here the absolute slope, whereas it is naturally a negative parameter, because the phase function always decreases in brightness from 10o to 40o of phase angles. The slope depends on the volume filling factor D and the optical depth of the slab. Two models has been developed for different regimes of the particle volume density D and predict opposite behavior of S as the function of D and *τ*. The *shadowing* model of and, also called *inter-particle shadow hiding* by, consists in shadows of macroscopic particles in a particulate medium, such as 8D ≪ 1. The smaller the volume density D, the steeper the phase function for increasing phase angle. Other refinements of this model exists but lead to the same results : the opposition peak due to the shadowing sharpens and the absolute slope increases with decreasing packing density (or filling factor). Another model exists, the *intra-particle shadow hiding*, which is valid for higher particle volume density according to. underlined the fact that the mutual shadowing among regolithic grains could be suited for understanding the textural properties of the regolith. As a consequence, this mechanism operates at the surface of ring material : e.g. in the regolith layer. However, other physical parameters need to be taken into account. In the analytical *inter-particle shadow hiding* model of, the slope can be linked to the angular width of the shadow hiding *h**s**h*. Thus a normalized and absolute slope would be $S=\\frac{1}{2h\\_{sh}}=\\frac{\\textrm{D}}{Q\\_{\\textrm{ext}}(\\lambda,\\bar{r})\\bar{r}\\ln(1-\\textrm{D})}$, with D the volume filling factor, *Q*ext(*λ*, *r̄*) the extinction coefficient and *r̄* the mean radius of particles. Then, the slope should depend on the wavelength and on the particle size. Interestingly, laboratory experiments of showed that the slope increases when the sample’s size increases. However, the microscopic and/or macroscopic roughness of the medium need also to be taken into account in the shadow hiding models, as underlined by and. For example, laboratory experiments of showed that the slope increases when the sample’s roughness increases. Then laboratory measurements and recent theoretical models could significantly increase the number of physical parameters on which S depends.\nFrom the above arguments, the HWHM and the amplitude A are governed by both coherent backscatter and shadow hiding effects whereas the slope S provides information of the shadow hiding effect solely.\nShape of the phase curves at opposition\n=======================================\nThe opposition effect in CLEAR filters\n--------------------------------------\nDue to the automation of extraction and fitting procedures (cf section[datareduction]) and due to the high images resolution, phase functions were extracted in as many as 211 different locations, in the D, C, B rings, Cassini Division and A ring (in increasing distance from Saturn, left column of Table 1 of Electronic supplementary material). In this section, we first present the typical behavior of some selected phase curves in different regions of the main rings (section[phasecurvesoe]), then we discuss similarities and differences and what are the general trends from sections[comparativeapproach] to [compareparametersoe].\n### Overview of the CLEAR phase curves\nExamples of phase curves in various ring regions are presented in Fig.[representoeallrings]. Each pair of graphs show on the left side a zoom from 0.01 to 2.5 degrees and on the right side, the fuller phase curve from 0 to 25 degrees. These curves were obtained by combining several WAC images with a large distribution of viewing geometries (Table[tableoeinfoimages]) each curve is built from the merging of 10 to 70 different images with various values of emission angle, incidence angle, phase angle etc. The dispersion of points is not due to the measurement uncertainty but reflects mainly the limits of the inversion formula that was used to extract the phase function from the measured values of I/F. It seems that some important physics may be missing (like multiple scattering), that could explain the scattering of points. Curves for the A and D rings (Fig.[representoeallrings]) are incomplete between 3 to 20o which is due to removal of images because of an artifact (cf section[dataset]).\n**[Fig.[representoeallrings]]**\nWhereas the general shape is similar from one ring to another (Fig.[representoeallrings]), some details in the shape may vary significantly. First pair of graphs in Fig.[representoeallrings] shows the phase curve derived in the D ring from images of *June 26*. Due to short exposure time (10 ms), the D ring ringlets are too faint to be detected. In images of *June 26*, a bright spot is visible from 67 000 km to the inner boundary of the C ring : this corresponds to the expected location of the background sheet of material constituting part of the D ring. However camera artifact may be visible in such dim regions of the image but could not be clearly identified here. So the fact that a strong increase of brightness at the expected location of the opposition and the coherent variation of the signal with observing geometry between different images suggest that we indeed see the opposition effect in the D ring. However some doubts still remain. From 0.5 to 2 degrees, the curve is similar to other rings. Below 0.5 degree an exponential surge and a flattening at zero degree distinguish this phase function from the other ones. Does it reflect optical properties of D ring dust ? Is it an artifact ? This plateau below 0.5 degree is much too large to be explained by the finite angular radius of the Sun (0.025 degree). Because of these uncertainties, at this point it is speculative to interpret this specific behavior as real.\nFor the C ring (Fig.[representoeallrings]), the shape of the phase function is well sampled below 2 degrees. It is comparatively wider than in dense A and B rings. More precisely, HWHM of the opposition surge is wider for the C ring (HWHM=0.26o) than for the A and B ring (HWHM$\\thicksim$0.20o). This could be also interpreted as a steeper slope of the linear regime of the phase function for *α* > 2o. Wavy features between 5 to 25 degrees can be attributed to images artifact. Their amplitude is about 15% of the total signal of the C ring.\nThe B ring opposition surge has the smallest amplitude of all rings (A=1.25 in Fig.[representoeallrings]). This was already underlined by the recent study of whereas previous observations of and give the exact opposite trend. However, their result could have a bias due to the lack of data below 0.3o of phase angle for and 0.1o for whereas had values much smaller. The B ring has also the steepest slope (S=0.105 $\\varpi\\_{0}P$.deg− 1) in the linear regime explaining why the opposition spot is so contrasty in the ISS images.\nThe Cassini Division has a similar amplitude and width (A$\\thicksim$1.47 and HWHM$\\thicksim$0.28o in Fig.[representoeallrings]) to the C ring (A$\\thicksim$1.45 and HWHM$\\thicksim$0.26o in Fig.[representoeallrings]). Their slope of the linear regime are also similar (S=0.033 $\\varpi\\_{0}P$.deg− 1 for the Cassini Division and S=0.030 $\\varpi\\_{0}P$.deg− 1 for the C ring). These similarities were first noticed by and are suggestive of a strong dependence of the opposition effect on the optical depth (we will come back to this in section[compareparametersoe]).\nAn example of the A ring phase function is given in the last pair of graphs in Fig.[representoeallrings]. At first sight, the signal appears much more disturbed than in other rings : specifically, pieces of phase functions extracted from different images show a wide dispersion in this graph, whereas signal from an individual image has a very low dispersion. The origin of this is not clear and may be due to the artifact reported in section [observeetreduit]. The dispersion of the data is about 15% of the signal whereas the camera artifact should represent at most 5% of the signal only (estimated on the background). It may be possible that the dispersion may be also due to an intrinsic photometric effect which is not corrected by the single scattering model (equation([eqchandrasekhar])). Indeed, the A ring has an intermediate optical depth $\\thicksim$0.5 so that it neither appears as a solid surface (like the B ring) or as a dilute system (like the C ring). Here, we are in an intermediate regime where many collective effects may influence strongly the apparent phase function (multiple scattering, gravitational wakes, density waves, etc.). A sophisticated model is required here to investigate such effect. However the general trends are quite clear and the A ring phase curve has a larger peak amplitude (A=1.39 in Fig.[representoeallrings]) than the B ring’s (A=1.25 in Fig.[representoeallrings]). We see in addition that the slope of the linear regime is shallower in the A ring (S=0.078 $\\varpi\\_{0}P$.deg− 1 in Fig.[representoeallrings]) than in the B ring (S=0.105 $\\varpi\\_{0}P$.deg− 1 in Fig.[representoeallrings]) but steeper than in less dense rings (S$\\thicksim$0.03 $\\varpi\\_{0}P$.deg− 1 in Fig.[representoeallrings]). Thus the phase curve at opposition in the A ring is somewhat intermediate between the B and C rings, strengthening the idea of a dependence on the optical depth.\nFinally, we conclude this section by remarking that the opposition effect is very diverse in Saturn’s rings, and could be the consequence of different properties of the surface ring particles in various ring regions. Some general trends can be underlined, as we see in the next section. In a first step, we check if some correlations exist between the morphological parameters which depend on the same portion of the curve (e.g. A and HWHM for the surge) and also if parameters describing different parts of the phase curve can be correlated (e.g. A, HWHM and S for, respectively, the surge and the linear part). This is the purpose of the next section.\n### Cross comparisons between morphological parameters\nIn order to constrain the morphology of the surge, we correlate the amplitude A with the angular width HWHM for the different main rings. We fit A and HWHM with a linear function and found correlation coefficients reported in Table[crossstudykaasalainenslopeahwhm].\n**[Table[crossstudykaasalainenslopeahwhm]]**\nIn Table[crossstudykaasalainenslopeahwhm] the Cassini Division, the amplitude A has the steepest function of the HWHM. A linear fit gives a slope of about 1.2 with a correlation coefficient of 47 %. This mean that narrow surges have a low amplitude and inversely wide surges have large amplitude. In the C ring, values of A and HWHM are generally greater than in the Cassini Division (but there are some exceptions). For the amplitude, this qualitative difference between the Cassini Division and the C ring is confirmed by previous study of and. For the angular width, although our trend of HWHM agrees with the results of, it contradicts previous work of, this is certainly due to the overestimation of HWHM when the smallest phase angles are missing (section[incompletedatafits] and Fig.[fig3]b). For the C ring, we find a slope for A=f(HWHM) of about 0.9 with a good correlation coefficient of 79 %. The A ring shows a similar variation of A=f(HWHM) as for the C ring (1.0) with a correlation coefficient of 56 %. Finally, A and HWHM in the B ring have values smaller than in the faint rings (C ring and Cassini Division). Data points of A and HWHM for the B ring are concentrated in a similar range as the A ring one, however with a much shallower slope (0.6 with a correlation coefficient weakly reliable of 29 %, see Table[crossstudykaasalainenslopeahwhm]). This means that the shape of opposition phase curves in the B ring may have various angular width with an almost constant value of the amplitude. To conclude, the amplitude of the surge seems correlated with the angular width, at least for the C ring, A ring and Cassini Division whereas the amplitude is independent of HWHM for the B ring. The slope of A=f(HWHM) seems to decrease from the Cassini Division to the B ring, passing by the C and A rings, suggesting that the slope is a decreasing function of the optical depth.\nWhereas the slope S on the one hand, A and HWHM on the"},"data_id":{"kind":"string","value":"arxiv_0182405"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":455,"numItemsPerPage":100,"numTotalItems":46503,"offset":45500,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6IjVHZDBvd0g5MTM2eDZjc1FvbE1zcktNWUZoRVFoUm5PVVVybEpjOUhaUEEifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4NjIyNTIyNCwianRpIjoiZmE3ZGQ4MWUtMWExNC00MTgzLWFkYTktOWJkN2QxYWM2YWYzIiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODYyMjg4MjQsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.9cVB34OtNm8LsSdpLgv6GSfqq7z4d7fkhi4jrb3hQuN_okl25aMB8XM3ORrCX5CKq_PPORIB-3ZRFXpzWpoSCA","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
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$, which is separable in *v*1 and *v*2. This implies the full-insurance property. In addition, the expected revenue is the second moment of the marginal distribution over the maximum valuation.
I show that this mechanism is a maxmin mechanism across all dominant-strategy mechanisms under certain regularity conditions... | arxiv_0182055 |
= Φ*v*(*δ*).
Our Contributions
-----------------
### Bounds for Deterministic Algorithms
In this section we require that the algorithm always output the correct answer. We show time space tradeoffs based on the vertex expansion properties of *G*. Our lower bounds are in the average case. Given a distribution *μ* over ... | arxiv_0182058 |
eclipsing binaries. IL Cnc was one of them that benefited from this mission in its rejuvenated stage - K2. The target was observed in the Kepler K2 mission (hereafter K2) with long cadence mode which took 30 minutes for each image. It was monitored during Kepler day 2307 to 2382 and 3419 to 3470 which corresponds to c... | arxiv_0182062 |
An Ephemerally Self-Exciting Point Process
==========================================
Introduction
============
*What’s past is prologue* – unavoidably, the present is shaped by what has already occurred. The current state of the world is indebted to our history. Our actions, behaviors, and decisions are both precur... | arxiv_0182065 |
still better than TCP+COPE, because when it makes probabilistic NC decision (when loss rate is less than 20%), it adds redundancy considering the loss rate over the overhearing link. This improves throughput, because adding redundancy using intra-session NC makes all packets equally beneficial to the receiver and the ... | arxiv_0182069 |
Patterns of traveling intrinsic localized modes in a driven electrical lattice., 77:066601, Jun 2008.
S. Flach, K. Kladko, and R. S. MacKay. Energy thresholds for discrete breathers in one-, two-, and three-dimensional lattices., 78:1207–1210, Feb 1997.
Sergej Flach and Andrey V. Gorbach. Discrete breathers advances i... | arxiv_0182075 |
regret bound scales at least $\Omega(\mathbb{E}\_{\bm{\theta}\sim\bm{P}}[W(\bm{\theta})])$. Note that in the definition of the regret, the performance of our algorithm is the total collected reward *after* adversarial corrupted, and the benchmark is the optimal policy *with* adversarial corruptions, denoted by *π*\*.
... | arxiv_0182081 |
Deep Learning for Time Series Anomaly Detection: A Survey
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Time series anomaly detection is important for a wide range of research fields and applications, including manufacturing and healthcare. The presence of anomalies can indicate novel or unexpected events, ... | arxiv_0182082 |
Mirror Neutron Stars
=====================
The fundamental nature of dark matter is entirely unknown. A compelling candidate is Twin Higgs mirror matter, invisible hidden-sector cousins of the Standard Model particles and forces. This predicts mirror neutron stars made entirely of mirror nuclear matter. We find their ... | arxiv_0182084 |
On generation of node distributions for meshless PDE discretizations [3](#fn3)
==============================================================================
In this paper we present an algorithm that is able to generate locally regular node layouts with spatially variable nodal density for interiors of arbitrary domai... | arxiv_0182086 |
MUSER: A MUlti-Step Evidence Retrieval Enhancement Framework for Fake News Detection
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The ease of spreading false information online enables individuals with malicious intent to manipulate public opinion and destabilize social stability... | arxiv_0182090 |
,$$
where $\frac{1}{p}+\frac{1}{q}=1$, for all functions *f* for which the right hand side is finite.
From Theorem [thm.carnot] we can claim that, often called *q*-*spectral gap*, holds for *q* ≤ 3/2. However, by Proposition 2.3 in, we immediately get that if is true for *q* then it is true for any *q*ʹ > *q* (and wi... | arxiv_0182095 |
The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme[1](#fn1)
==========================================================================================================
The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for... | arxiv_0182101 |
A Survey on Privacy-Preserving Caching at Network Edge: Classification, Solutions, and Challenges
=================================================================================================
Caching content at the network edge is a popular and effective technique widely deployed to alleviate the burden of network ... | arxiv_0182102 |
test, the BHM and NSB estimators clearly outperform the rest, even though their advantage is less significant as *N* increases.
Finally, for every particular *N*, we compute the mean squared error of the entropy estimators, Equation , as a function of *p*00 and *p*11. Its aggregated value
$$\label{eq:MSE}
\overline... | arxiv_0182104 |
Differences of Skew Schur Functions of Staircases with Transposed Foundations
Matthew Morin
University of British Columbia
mjmorin@math.ubc.ca
**Abstract**
We consider the skew diagram Δ*n*, which is the 180∘ rotation of the staircase diagram *δ**n* = (*n*, *n* − 1, *n* − 2, …, 2, 1). We create a *staircase with bad fo... | arxiv_0182106 |
in unitary gauge *ϕ**a* = *x**a* where we have [unitary] Sunitary &=&d-2 d x + m2 n=0d n*x*2 Un
&+& n n*x*2 L(n)Lovelock[g] ], with *x*2 = *x**a**x**b**η**a**b*. This explicit dependence on *x**a**x**a* implies that translation invariance is broken but not Lorentz invariance, at least around the preferred point *x**... | arxiv_0182110 |
Chemical modelling of dust-gas chemistry within AGB outflowsIII. Photoprocessing of the ice and return to the ISM
=================================================================================================================
### Accepted XXX. Received YYY; in original form ZZZ
[firstpage]
To explain the properties o... | arxiv_0182113 |
Normed symmetric monoidal categories
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We introduce categorical models of *N*∞ spaces, which we call normed symmetric monoidal categories (NSMCs). These are ordinary symmetric monoidal categories equipped with compatible families of norm maps, and when specialized to a particular clas... | arxiv_0182120 |
This paper studies distributed nonconvex optimization problems with stochastic gradients for a multi-agent system, in which each agent aims to minimize the sum of all agents’ cost functions by using local compressed information exchange. We propose a distributed stochastic gradient descent (SGD) algorithm, suitable for... | arxiv_0182124 |
The shapes of galaxies in the Sloan Digital Sky Survey
======================================================
### Accepted 2008 May 17. Received 2008 April 28; in original form 2008 February 6
[firstpage]
We determine the underlying shapes of spiral and elliptical galaxies in the Sloan Digital Sky Survey Data Release 6... | arxiv_0182125 |
0.0004—0.05. The obtained data are publicly available in electronic form at the CDS and also at `http://www.astro.uu.nl/~kruijs`. The results from our models are as follows.
* *The preferential loss of low-mass stars* slightly decreases the total disruption time of a cluster by a few percent. However, the most signifi... | arxiv_0182132 |
be observed: as the starting value *x* decreases to zero, *v* converges to zero for the unbounded variation case (**Cases 1**, **2**, and **3**), but not for the bounded variation case (**Cases 1’**, **2’**, and **3’**).
1.0
| | |
| --- | --- |
| image | image |
| Case 1 | Case 1’ |
| image | image |
| Case 2 | Case... | arxiv_0182139 |
New Constraints on the Escape of Ionizing Photons from Starburst Galaxies Using Ionization-Parameter Mapping
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The fate of ionizing radiation in starburst galaxies is key to understanding cosmic reionization. Howe... | arxiv_0182147 |
captured by the fact that H*S*1 is a *commutative Frobenius algebra*. This is a algebra equipped with an invertible bilinear form *η**i**j* called the Frobenius form. In the TQFT this is obtained from the generators
$$\begin{aligned}
\label{frob}
{\quad \vcenter{\hbox{\tikz{{ \begin{scope}[xshift=0,yshift=0]
\fill... | arxiv_0182152 |
Sensitivity of the Cherenkov Telescope Array for probing cosmology and fundamental physics with gamma-ray propagation
=====================================================================================================================
Gamma-ray propagation on cosmic scales
======================================
Over t... | arxiv_0182153 |
Induced nuclear activity in galaxy pairs with different morphologies (E+E), (E+S) and (S+S).
============================================================================================
[firstpage]
We analysed 385 galactic spectra from the Sloan Digital Sky Survey Data Release 7 (SDSS-DR7) that belong to the catalog of... | arxiv_0182160 |
Detection efficiency and photometry in supernova surveys
========================================================
– the Stockholm VIMOS Supernova Survey **I** [1](#fn1)
======================================================
### Received ; accepted
The aim of the work presented in this paper is to test and optimise supe... | arxiv_0182164 |
, chemically-active swimmers convert chemical energy stored internally or in their environment into interfacial flows along their surface, thus prescribing mechanical stresses on the surrounding fluid to achieve self-propulsion . Their development is currently driven by their various potential applications, including t... | arxiv_0182166 |
Tangle solutions for composite knots: application to Hin recombination
======================================================================
We extend the tangle model, originally developed by Ernst and Sumners, to include composite knots. We show that, for any prime tangle, there are no rational tangle attachments of... | arxiv_0182168 |
Archimedean Zeta Integrals for Unitary Groups
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We compute the archimedean zeta integrals for holomorphic discrete series of general vector weights arising in the doubling method for unitary groups, an integral representation of automorphic *L*-functions discovered by Garrett... | arxiv_0182169 |
, the probability of trapping is :
$$\Pi = \frac{1}{{2\pi |r|}}\left( {\frac{{\partial \Lambda }}{{\partial s}}\frac{{\partial S\_{res} }}{{\partial p}} - \frac{{\partial \Lambda }}{{\partial p}}\frac{{\partial S\_{res} }}{{\partial s}}} \right) =
\frac{{\left\{ {\Lambda,S\_{res} } \right\}\_{s,p} }}{{2\pi |r|}}
\la... | arxiv_0182171 |
an IC scenario, the cooling time scale for electrons with energy $E\_{\rm e}
\approx 5$ GeV that up-scatter 2500 K photons to ≈ 100 MeV is $t\_{\rm IC} \approx (3/4) \, m\_{\rm e}^2 \, c^3 \, [\sigma\_{\rm T} \,
E\_{\rm e} \, u\_{\rm IR}(R)]^{-1} \sim 3.1 \times 10^5$ s. Thus, $t/t\_{\rm IC}\sim 28\%$ of the electr... | arxiv_0182174 |
Black Box Galois Representations.
=================================
We develop methods to study 2-dimensional 2-adic Galois representations *ρ* of the absolute Galois group of a number field *K*, unramified outside a known finite set of primes *S* of *K*, which are presented as *Black Box* representations, where we onl... | arxiv_0182176 |
Long-Period Totally Eclipsing Binary Star at the Turnoff of the Open Cluster NGC 6819 Discovered with *Kepler*[1](#fn1)
=========================================================================================================================
We present the discovery of the totally eclipsing long-period (*P* = 771.8 d)... | arxiv_0182179 |
Latent Class Model with Application to Speaker Diarization
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In this paper, we apply a latent class model (LCM) to the task of speaker diarization. LCM is similar to Patrick Kenny’s variational Bayes (VB) method in that it uses soft information and avoids prematu... | arxiv_0182182 |
Delay-Optimal Probabilistic Scheduling withArbitrary Arrival and Adaptive Transmission
======================================================================================
In this paper, we aim to obtain the optimal delay-power tradeoff and the corresponding optimal scheduling policy for arbitrary i.i.d. arrival proc... | arxiv_0182183 |
-conserved EDE component with *c**s*2 = 1. See also for a similar study, in which the authors used a different binning approach, leaving also the EDE sound speed free in the MC; or for the effect of curvature and a late-time dark energy dynamics. However, we should bear in mind that the aforementioned measurements of... | arxiv_0182184 |
Lectures on controlled Reeb dynamics
====================================
These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for H... | arxiv_0182187 |
Searching for Trans Ethyl Methyl Ether in Orion KL [1](#fn1)[2](#fn2)
=====================================================================
### Received...; accepted...
Introduction
============
The spectral millimeter-wave survey of Orion KL carried out with the IRAM 30m radio telescope shows more than 15400 spectral... | arxiv_0182192 |
ALPs at Colliders
=================
Introduction
============
A long time has passed since the work of Peccei and Quinn , and yet no concrete hint of ALPs has been found. In this paper we discuss an alternative to the traditional ways of looking for ALPs using colliders. Colliders can both explore ALP masses beyond the... | arxiv_0182197 |
of the constructed automaton. Lastly, we report on a prototype implementation and compare it to the LTL monitor of Bauer et al. on a sample of examples.
Introduction
============
Runtime monitoring is nowadays routinely used to assess the satisfaction of properties of systems during their execution. To this end, a mon... | arxiv_0182206 |
On Adaptivity in Information-constrainedOnline Learning
=======================================================
We study how to adapt to smoothly-varying (‘easy’) environments in well-known online learning problems where acquiring information is expensive. For the problem of label efficient prediction, which is a budge... | arxiv_0182212 |
Freeway network design with exclusive lanes for automated vehicles under endogenous mobility demand
===================================================================================================
Automated vehicles (AV) have the potential to provide cost-effective mobility options along with overall system-level be... | arxiv_0182213 |
_{C \in \mathrm{6VDW}(n)}(a\_0 a\_1)^{\nu(C)}
(b\_0 b\_1)^{n(n-1)/2-\nu(C)-\mu(C)} c\_0^{\mu(C)}c\_1^{\mu(C)+n},
\label{comparisontwo}\end{aligned}$$
where *a*0 = 1 − *β**v*, *a*1 = *v*, *b*0 = 1 + *β**v*, *b*1 = *v*, *c*0 = 1, *c*1 = 2*v*.
Combining and and simplifying, one gets
$$\begin{aligned}
\sum\_{C \in \... | arxiv_0182215 |
Nonleptonic two-body decays of charmed mesons
=============================================
Nonleptonic decays of charmed mesons into two pseudoscalar mesons or one pseudoscalar meson and one vector meson are studied on the basis of a generalized factorization method considering the resonance effects in the pole model ... | arxiv_0182220 |
≈ 15 − 20%), but still result in a spherical core. We conclude that the ROI is suppressed in these systems simply because sufficient mass occupying radially anisotropic orbits is not retained during the collapse. We point out that not all systems that lose mass necessarily remain spherical. If ${\ensuremath{\mu\_{r,0}}... | arxiv_0182223 |
Characterizations of symmetric polyconvexity
============================================
Symmetric quasiconvexity plays a key role for energy minimization in geometrically linear elasticity theory. Due to the complexity of this notion, a common approach is to retreat to necessary and sufficient conditions that are eas... | arxiv_0182226 |
Black hole inner horizon evaporation in semiclassical gravity
=============================================================
December 2020
In this work we analyse the backreaction of a quantum field on a spherically symmetric black hole geometry with an inner horizon, i.e. an internal boundary of the trapped region. We ... | arxiv_0182230 |
The SILCC (SImulating the LifeCycle of molecular Clouds) project:I. Chemical evolution of the supernova-driven ISM
==================================================================================================================
### Accepted. Received 2014 December 8th; in original form
[firstpage]
The SILCC project (... | arxiv_0182231 |
On the chromatic number of graphons
===================================
We extend Bollobas’ classical result on the chromatic number of a binomial random graph to the exchangeable random graph model $\calG(n,W)$ defined by a graphon *W* : [0, 1]2 → [0, 1], which is a symmetric measurable function. In the case when *W* ... | arxiv_0182233 |
* (*Z**α*)2 to the data for *Z**α* *P*(*Z**α*) for small values of *Z* tabulated in Ref. .
Two-loop effects
----------------
For the calculation of two-loop contributions in the independent electron approximation, we use formulas from Refs. . These formulas assume a point-like nucleus and are perturbative in the nuclea... | arxiv_0182236 |
Disentangling protostellar evolutionary stages in clustered environments using *Spitzer*-IRS spectra and comprehensive SED modeling
===================================================================================================================================
When studying the evolutionary stages of protostars that... | arxiv_0182244 |
Some applications of Ricci flow to 3-manifolds
==============================================
Introduction
============
The purpose of this text is to describe some applications of Ricci flow to questions about the topology and geometry of 3-s. The most spectacular achievement in the recent years was the proof of W. Th... | arxiv_0182245 |
State discrimination with post-measurement information and incompatibility of quantum measurements
==================================================================================================
We discuss the following variant of the standard minimum error state discrimination problem: Alice picks the state she sen... | arxiv_0182253 |
Diagnosing Mass Flows Around Herbig Ae/Be Stars Using the He I *λ*10830 Line
============================================================================
We examine profile morphologies for a sample of 56 Herbig Ae/Be stars (HAEBES). We find significant differences between HAEBES and CTTSs in the statistics of both blu... | arxiv_0182259 |
Structure of 38Cl and the quest for a comprehensive shell model interaction.
============================================================================
The higher-spin structure of 38Cl (*N* = 21) was investigated following the 26Mg(14C, *p**n*) reaction at 30 and 37 MeV beam energies. The outgoing protons were detec... | arxiv_0182263 |
restrict our attention to stock *s* = 1, a low information stock if *L* > 0 in the information scenarios, and identical to the remaining stocks otherwise. We intially focus on the meta-model effects on MARE ratios Δ(*θ**s*) and complex aggregate $\overline\Delta(\theta)$ to interpret model configuration effects, and u... | arxiv_0182265 |
Entanglement transitions in the quantum Ising chain:A comparison between different unravelings of the same Lindbladian
======================================================================================================================
We study the dynamics of entanglement in the quantum Ising chain with dephasing di... | arxiv_0182267 |
The temporal logic of coalitional goal assignments in concurrent multi-player games
===================================================================================
We introduce and study a natural extension of the Alternating time temporal logic ATL, called *Temporal Logic of Coalitional Goal Assignments* (TLCGA). ... | arxiv_0182268 |
**Realistic Tunneling States for the Magnetic Effectsin Non-Metallic Real Glasses**
===================================================================================
The discovery of magnetic and compositional effects in the low temperature properties of multi-component glasses has prompted the need to extend the sta... | arxiv_0182272 |
, inflow, and outflow, have been obtained (e.g., ). And a few well-studied AGNs were measured multiple times for velocity-resolved results in different periods. For example, recovered velocity-delay maps of NGC 5548 from multi-year data and found that the kinematics of the BLR varied between inflow and virial. Then, ... | arxiv_0182276 |
The Radius–Luminosity Relationship for Active Galactic Nuclei:The Effect of Host-Galaxy Starlight on Luminosity Measurements II. The Full Sample of Reverberation-Mapped AGNs
==================================================================================================================================================... | arxiv_0182282 |
Multilevel domain decomposition-based architectures for physics-informed neural networks
========================================================================================
Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggl... | arxiv_0182286 |
Gepner-like boundary states on *T*4
===================================
Introduction and summary
========================
It is well known that D-branes in type II superstring are described (at weak string coupling) by superconformal boundary states. These satisfy the gluing conditions
[eq:glC]
$$\begin{aligned}
(L... | arxiv_0182292 |
L*2-norm with respect to the time parameter *k*. In Table [table1.1c], we have shown rate of convergence of the control variable in the energy norm with respect to the control discretization parameter $\sigma:=\sqrt{h^2+k^2}$.
[h!]
[table1.1a]
| *N* | *h* | ${\left\Vert \nabla(\bar{u}(\bar{q})-\bar{u}\_{kh}(\bar{q}\_{\... | arxiv_0182295 |
Tracer particle diffusion in a system with hardcore interacting particles
=========================================================================
In this study, inspired by the work of K. Nakazato and K. Kitahara [[*Prog. Theor. Phys.* **64**, 2261 (1980)](http://ptp.oxfordjournals.org/cgi/doi/10.1143/PTP.64.2261)], ... | arxiv_0182297 |
Two-weight dyadic Hardy’s inequalities
======================================
[1](#fn1)
[2](#fn2)
[3](#fn3)
[4](#fn4)
We present various results concerning the two-weight Hardy’s inequality on infinite trees. Our main scope is to survey known characterizations (and proofs) for trace measures, as well as to provide some... | arxiv_0182299 |
Homotopy Theory of Labelled Symmetric Precubical Sets
=====================================================
This paper is the third paper of a series devoted to higher dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition syst... | arxiv_0182301 |
Achievable Regions and Precoder Designs for the Multiple Access Wiretap Channels with Confidential and Open Messages
====================================================================================================================
This paper investigates the secrecy capacity region of multiple access wiretap (MAC-WT... | arxiv_0182302 |
140.4 | 6.52 × 10− 1 | 1.67 × 10− 2 | 1.64 × 10− 2 | 9.09 × 10− 2 | + 9.049 × 10− 2 − 7.662 × 10− 2 |
| | 234.6 | 5.40 × 10− 1 | 1.56 × 10− 2 | 1.52 × 10− 2 | 7.12 × 10− 2 | |
| | 325.2 | 4.24 × 10− 1 | 1.17 × 10− 2 | 1.14 × 10− 2 | 5.90 × 10− 2 | |
| | 14.5 | 3.99 × 10− 1 | 1.22 × 10− 1 | 9.72 × 10− 2 | 1.31 × 1... | arxiv_0182303 |
Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs
============================================================================
In his 1947 paper that inaugurated the probabilistic method, Erdös proved the existence of 2log*n*-Ramsey graphs on *n* vertices. Matching Erdös’ result with a const... | arxiv_0182312 |
7-1290, Nov. 1989.
J. Hamkins. “Optimal codes for the burst erasure channel,” *IPN Progress Report*, pp. 42-174, 2008.
F. Etezadi, A. Khisti and M. Trott, “Zero-delay sequential transmission of Markov sources over burst erasure channels,” *IEEE Trans. Inf. Theory*, vol. 60, pp. 4584-4613, Jan. 2014.
F. Etezadi, A. Khis... | arxiv_0182315 |
On strong Mal’cev conditions for congruence meet–semidistributivityin a locally finite variety
==============================================================================================
[1](#fn1) In this paper we examine four–element and five–element digraphs for existence of certain polymorphisms that imply congru... | arxiv_0182316 |
The reliability of observational measurements of column density probability distribution functions
==================================================================================================
### Received ; accepted
Using simulations, we show to what degree their determination is affected by noise, line-of-sight ... | arxiv_0182321 |
and also with the radial overlap between the orbits.
In Fig. [fig7] we show in a schematic way the positions of the neutron states 2*p*1/2, 2*p*3/2, and 1*f*5/2 using 1*f*7/2 as a reference state, calculated with the relativistic interaction NL3RHF0.5 which includes the one-pion-exchange tensor force with half the str... | arxiv_0182322 |
Bulk-boundary correspondence in (3+1)-dimensional topological phases
=====================================================================
[1](#fn1)
We discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level K, and its generalization. In particula... | arxiv_0182327 |
6). Figure 1 of Takahashi et al. (2009) shows an image where SMM 7 (MMS 6-NE) is spatially resolved using SMA compact configuration data with robust weighting. The sources, and were detected for the first time with our SMA 850 *μ*m continuum observations. For the other eight sources, this is the first time that they ha... | arxiv_0182328 |
*, *δ**B*∥) is never perfectly anti-correlated, reaching a minimum value of ≃ − 0.9. Third, as with the MHD plasma, the pure fast modes are perfectly correlated in the low *β**i* region and become drastically less correlated for *β**i* values above the mode conversion (see Appendix [app:branch]) at *β**i* ≃ 1.3.
Synt... | arxiv_0182335 |
A Bayesian view of doubly robust causal inference
=================================================
In causal inference confounding may be controlled either through regression adjustment in an outcome model, or through propensity score adjustment or inverse probability of treatment weighting, or both. The latter approa... | arxiv_0182345 |
We perform 3D numerical simulations to investigate the sedimentation of a single sphere in the absence and presence of a simple cross shear flow in a yield stress fluid with weak inertia. In our simulations, the settling flow is considered to be the primary flow, whereas the linear cross shear flow is a secondary flow ... | arxiv_0182349 |
Symmetries and Strings of Adjoint QCD2
======================================
Introduction and Summary
========================
Confinement in gauge theories has been a central subject of research in quantum field theory for decades. While the ultimate goal is to understand confinement in Yang-Mills (YM) theory and Qua... | arxiv_0182350 |
0) ↦ (Σ0(*η*, *α*0), Θ0(*η*, *α*0)). The styles of the sheading of the distinguished subregions and those of the characteristic curves and points are in correspondence in these two figures. An interesting feature of figure [figure3] is that, in virtue of the above argument in connection with *c**s*, min2, the slope of ... | arxiv_0182354 |
*A*Fe2As2 (*A* = Ca, Sr, Ba, Eu) and SrFe2 − *x**T**M**x*As2 (*T**M* = Mn, Co, Ni): crystal structure, charge doping, magnetism and superconductivity
=====================================================================================================================================================
The electronic struc... | arxiv_0182355 |
* = 0.82 which will serve as a comparison between linear analysis, experiments and direct numerical simulations. All the simulations are performed at the wave numbers *k**x* = 0.96, *k**z* = 5.16, where the most unstable mode appears in the *S**c* = ∞ case. The results are reported in figure [fig:Sceffects]. First we r... | arxiv_0182356 |
.
J. Shatah and C. Zeng. Local well-posedness for the fluid interface problem. 199 (2011), no. 2, 653-705.
M. Shinbrot. The initial value problem for surface waves under gravity. I. The simplest case. 25 (1976), no. 3, 281-300.
T. C. Sideris. Formation of singularities in three-dimensional compressible fluids. 101 (198... | arxiv_0182357 |
EFT of Interacting Spin-2 Fields
================================
Introduction
============
The search for all possible consistent field theories for various spins is a longstanding and ongoing research topic of fundamental importance. Although renormalizable theories only exist for *s* ≤ 1, effective field theories (E... | arxiv_0182363 |
A proof of the instability of AdSfor the Einstein–null dust system with an inner mirror
=======================================================================================
In 2006, Dafermos and Holzegel formulated the so-called AdS instability conjecture, stating that there exist *arbitrarily small* perturbations ... | arxiv_0182364 |
believe that this is due to enhanced turbulence diffusion caused by the strong turbulence motions at the small driving scale. Owing to the smaller linear growth rate, the system reaches a final saturation stage later. Figure [f:weakB](b) shows the magnetic energy densities are all similar during the saturation stage i... | arxiv_0182365 |
model involving shooting rays from the sites in various directions. It can be thought of as being a “polar model” or a “spherical model” or a “constant orientation model”. The advantage of this model is that a significant part of the difficulty in the computation is reduced to a one dimensional setting (the ray on whi... | arxiv_0182368 |
sign of tidal or other processes that could be connected to their outstanding chemical peculiarity. In contrast, the transition between circular and eccentric orbits seems to occur at periods of ∼ 4 − 800 days for binaries with former AGB companions.
Further information on the presently unseen companions can be deriv... | arxiv_0182371 |
Hitting Sets and Reconstruction for Dense Orbits in VPe and ΣΠΣ Circuits
========================================================================
In this paper we study polynomials in VPe (polynomial-sized formulas) and in ΣΠΣ (polynomial-size depth-3 circuits) whose orbits, under the action of the affine group GL*n*af... | arxiv_0182375 |
Achieving Budget-optimality with Adaptive Schemes in Crowdsourcing
===================================================================
Crowdsourcing platforms provide marketplaces where task requesters can pay to get labels on their data. Such markets have emerged recently as popular venues for collecting annotations ... | arxiv_0182382 |
**Supplementary Materials: Resource requirements for efficient quantum communication using all-photonic graph states generated from a few matter qubits**
=========================================================================================================================================================
Quantum comm... | arxiv_0182388 |
0,2); at (0,0) [label=below:5] ; at (-1,0) [label=below:3] ; at (-2,0) [label=below:1] ; at (0,1) [label=left:3] ; at (0,2) [label=left:1] ; at (1,0) [label=below:3] ; at (2,0) [label=below:2] ; at (3,0) [label=below:1] ;
$\scriptsize{(5)}$& & 1 & 5 & 30 & (, ) &
[x=.7cm,y=.7cm] (-3,0)–(4,0); (0,0)–(0,1); at (0,0) [... | arxiv_0182394 |
Quantum Tanner codes
====================
Tanner codes are long error correcting codes obtained from short codes and a graph, with bits on the edges and parity-check constraints from the short codes enforced at the vertices of the graph. Combining good short codes together with a spectral expander graph yields the cele... | arxiv_0182398 |
, we outline the background for this survey in Section 2. Preliminaries of DFL are reviewed in Section 3. Section 4 studies possible threats to DFL. Section 5 discusses the potential defense mechanisms in DFL. Section 6 elaborates on the verifiability of DFL. Future research directions are discussed in Section 7. Final... | arxiv_0182401 |
On Security Research TowardsFuture Mobile Network Generations
=============================================================
Over the last decades, numerous security and privacy issues in all three active mobile network generations have been revealed that threaten users as well as network providers. In view of the newes... | arxiv_0182403 |
Thermodynamics of light
=======================
Introduction
============
These notes where written as a running documentation of a project whose goal was to explore the thermodynamical properties of light in a cavity. The notes does not make claims of originality with respect to any the topics that is discussed, perha... | arxiv_0182404 |
The opposition effect in Saturn’s rings seen by Cassini/ISS:I. Morphology of phase curves
=========================================================================================
,
,
,
, and
Number of pages:
Number of tables: [lasttable]
Number of figures: [lastfig]
Running head: The opposition effect in Saturn’... | arxiv_0182405 |
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