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samples/texts/1693838/page_1.md

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+ON MODULI OF INSTANTON BUNDLES ON $\mathbb{P}^{2n+1}$
+
+VINCENZO ANCONA AND G. OTTAVIANI
+
+Let $M\mathcal{I}_{\mathbb{P}^{2n+1}}(k)$ be the moduli space of stable instanton bundles on $\mathbb{P}^{2n+1}$ with $c_2 = k$. We prove that $M\mathcal{I}_{\mathbb{P}^{2n+1}}(2)$ is smooth, irreducible, unirational and has zero Euler-Poincaré characteristic, as it happens for $\mathbb{P}^3$. We find instead that $M\mathcal{I}_{\mathbb{P}^5}(3)$ and $M\mathcal{I}_{\mathbb{P}^5}(4)$ are singular.
+
+# 1. Definition and preliminaries.
+
+Instanton bundles on a projective space $\mathbb{P}^{2n+1}(\mathbb{C})$ were introduced in [OS] and [ST]. In [AO] we studied their stability, proving in particular that special symplectic instanton bundles on $\mathbb{P}^{2n+1}$ are stable, and that on $\mathbb{P}^5$ every instanton bundle is stable.
+
+In this paper we study some moduli spaces $M\mathcal{I}_{\mathbb{P}^{2n+1}}(k)$ of stable instanton bundles on $\mathbb{P}^{2n+1}$ with $c_2 = k$. For $k=2$ we prove that $M\mathcal{I}_{\mathbb{P}^{2n+1}}(2)$ is smooth, irreducible, unirational and has zero Euler-Poincaré characteristic (Theor. 3.2), just as in the case of $\mathbb{P}^3$ [Har].
+
+We find instead that $M\mathcal{I}_{\mathbb{P}^5}(k)$ is singular for $k=3,4$ (theor. 3.3), which is not analogous with the case of $\mathbb{P}^3$ [ES], [P]. To be more precise, all points corresponding to symplectic instanton bundles are singular. Theor. 3.3 gives, to the best of our knowledge, the first example of a singular moduli space of stable bundles on a projective space. The proof of Theorem 3.3 needs help from a personal computer in order to calculate the dimensions of some cohomology group [BaS].
+
+We recall from [OS], [ST] and [AO] the definition of instanton bundle on $\mathbb{P}^{2n+1}(\mathbb{C})$.
+
+**Definition 1.1.** A vector bundle $E$ of rank $2n$ on $\mathbb{P}^{2n+1}$ is called an instanton bundle of quantum number $k$ if
+
+(i) The Chern polynomial is $c_t(E) = (1-t^2)^{-k} = 1 + kt^2 + \binom{k+1}{2}t^2 + \dots$
+
+(ii) $E(q)$ has natural cohomology in the range $-2n - 1 \le q \le 0$ (that is $h^i(E(q)) \ne 0$ for at most one $i = i(q)$)
+
+(iii) $E|_{\tau} \simeq \mathcal{O}_{\tau}^{2n}$ for a general line $\tau$.
+
+Every instanton bundle is simple [AO]. There is the following characterization:

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+**Theorem 1.2 ([ST], [AO]).** A vector bundle $E$ of rank $2n$ on $\mathbb{P}^{2n+1}$ satisfies the properties (i) and (ii) if and only if $E$ is the cohomology of a monad
+
+$$ (1.1) \qquad \mathcal{O}(-1)^k \xrightarrow{\Lambda} \mathcal{O}^{2n+2k} \xrightarrow{\mathcal{B}} \mathcal{O}(1)^k. $$
+
+With respect to a fixed system of homogeneous coordinates the morphism $A$ (resp. $B$) of the monad can be identified with a $k \times (2n + 2k)$ (resp. $(2n + 2k) \times k$) matrix whose entries are homogeneous polynomials of degree 1. Then the conditions that (1.1) is a monad are equivalent to:
+
+> $A, B$ have rank $k$ at every point $x \in \mathbb{P}^{2n+1}$, $A \cdot B = 0$.
+
+**Definition 1.3.** A bundle $S$ appearing in an exact sequence:
+
+$$ (1.2) \qquad 0 \to S^* \to \mathcal{O}^d \xrightarrow{\mathcal{B}} \mathcal{O}(1)^c \to 0 $$
+
+is called a Schwarzenberger type bundle (*STB*).
+
+The kernel bundle *Ker* *B* in the monad (1.1) is the dual of a STB.
+
+**Definition 1.4.** An instanton bundle is called special if it arises from a monad (1.1) where the morphism *B* is defined in some system of homogeneous coordinates $(x_0, \dots, x_n, y_0, \dots, y_n)$ on $\mathbb{P}^{2n+1}$ by the matrix
+
+$$ B = \begin{bmatrix} x_0 & & & \\ & \ddots & & \\ x_n & x_0 & & \\ & & \ddots & \\ y_0 & & & \\ & \ddots & & \\ y_n & y_0 & & \\ & & \ddots & \\ y_n \end{bmatrix} $$

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+**Example 1.5.** Take
+
+$$
+A = \begin{bmatrix}
+y_n & \cdots & y_0 & & & -x_n & \cdots & -x_0 \\
+& \ddots & & \ddots & & & & \\
+& & \ddots & & \ddots & & & \\
+& & & \ddots & \ddots & & & \\
+\multicolumn{2}{c}{\begin{bmatrix} y_n & \cdots & y_0 \\ & \ddots & \\ & & \ddots \\ & & & \ddots \\ & & & & \ddots \\ & & & & & \ddots \\ & & & & & & \vdots \end{bmatrix}} &
+\multicolumn{2}{c}{-x_n \cdots -x_0}
+\end{bmatrix}
+$$
+
+$$
+B = \begin{bmatrix}
+[x_0] \\
+[ x_n \quad x_0 ] \\
+[ x_n ] \\
+[ y_0 ] \\
+[ y_n \quad y_0 ] \\
+[ \vdots ] \\
+[ y_n ]
+\end{bmatrix}
+$$
+
+$E = \text{Ker } B / \text{Im } A$ is a special instanton bundle.
+
+Property (iii) of the definition 1.1 can be checked by the following:
+
+**Theorem 1.6 [OS].** Let $E = \text{Ker } B / \text{Im } A$ as in (1.1). Let $r$ be the line joining two distinct points $P, Q \in \mathbb{P}^{2n+1}$. Then
+
+$$
+E|_{r} \cong \mathcal{O}_{r}^{2n} \Leftrightarrow A(P) \cdot B(Q) \text{ is an invertible matrix.}
+$$
+
+**Example 1.7.** Consider the special instanton bundle *E* of the example
+1.5. Let *P* = (1, 0, ..., 0, ..., 0), *Q* = (0, ..., 0, ..., 1). Then
+
+$$
+A(P) = \begin{bmatrix}
+& -1 \\
+& \cdot \\
+-1 & & \\
+& \cdot \\
+& & \ddots
+\end{bmatrix}
+$$
+
+$$
+B(Q) = \begin{bmatrix}
+1 \\
+\vdots \\
+\cdot \\
+\vdots \\
+1
+\end{bmatrix}
+$$
+
+and $A(P) \cdot B(Q) = \begin{bmatrix} -1 \\ \cdot \\ -1 \end{bmatrix}$ is invertible. Hence $E$ is trivial on the line $\{x_1 = \dots = x_n = y_0 = \dots = y_{n-1} = 0\}$.
+
+**Proposition 1.8.** Let $E$ be an instanton bundle as in (1.1). Then
+
+$$
+H^2(E \otimes E^*) = H^2[(\mathrm{Ker}\,B) \otimes (\mathrm{Ker}\,A^t)]
+$$

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+*Proof.* See [AO] Theorem 3.13 and Remark 2.22.
+
+**Remark 1.9.** If $E \simeq E^*$, then
+
+$$H^2(E \otimes E^*) = H^2[(\ker A^t) \otimes (\ker A^t)] = H^2[(\ker B) \otimes (\ker B)].$$
+
+**Remark 1.10.** The single complex associated with the double complex obtained by tensoring the two sequences
+
+$$0 \to \ker A^t \to \mathcal{O}^{2n+2k} \xrightarrow{\Lambda^t} \mathcal{O}(1)^k \to 0$$
+
+$$0 \to \ker B^t \to \mathcal{O}^{2n+2k} \xrightarrow{\mathcal{B}^t} \mathcal{O}(1)^k \to 0$$
+
+gives the resolution
+
+$$0 \to (\ker A^t) \otimes (\ker B) \to \mathcal{O}^{2n+2k} \otimes \mathcal{O}^{2n+2k} \\ \to \mathcal{O}^{2n+2k} \otimes \mathcal{O}(1)^k \oplus \mathcal{O}(1)^k \otimes \mathcal{O}^{2n+2k} \xrightarrow{\alpha} \mathcal{O}(1)^k \otimes \mathcal{O}(1)^k \to 0$$
+
+where $\alpha = (A^t \otimes \text{id}, \text{id} \otimes B)$.
+
+Hence
+
+$$H^2(E \otimes E^*) = \operatorname{Coker} H^0(\alpha)$$
+
+and its dimension can be computed using [BaS]. For the convenience of the reader we sketch the steps needed in the computations.
+
+$A, B^t$ are given by $k \times (2n + 2k)$ matrices whose entries are linear homogeneous polynomials.
+
+$$A \otimes \mathrm{Id}_k = (a_1, \dots, a_{k(2n+2k)})$$
+
+and
+
+$$\mathrm{Id}_k \otimes B^t = (b_1, \dots, b_{k(2n+2k)})$$
+
+are both $k^2 \times (2n + 2k)k$ matrices. Let
+
+$$C = (a_1, \dots, a_{k(2n+2k)}, b_1, \dots, b_{k(2n+2k)}).$$
+
+We will denote by $\text{syz}_m C$ the dimension of the space of the syzygies of $C$ of degree $m$. Then
+
+$$h^2(E \otimes E^*) = h^0(\mathcal{O}(2)^{k^2}) - (4n + 4k)h^0(\mathcal{O}(1)^k) + \text{syz}_1 C \\
+= k(n + 1)[k(2n - 5) - 8n] + \text{syz}_1 C \\
+h^1(E \otimes E^*) = h^2(E \otimes E^*) + 1 - k^2 + 8n^2k - 4n^2 + 3nk^2 - 2n^2k^2 \\
+= 1 - 6k^2 - 8kn - 4n^2 + \text{syz}_1 C.$$

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+Note also that $h^0(E(1)) = \text{syz}_1 B^t - k$ and $h^0(E^*(1)) = \text{syz}_1 A - k$.
+
+**Remark 1.11.** In the same way we obtain
+
+$$h^1(E \otimes E^*(-1)) = \text{syz}_0 C$$
+
+$$h^2(E \otimes E^*(-1)) = 2k(nk - 2n - k) + \text{syz}_0 C.$$
+
+## 2. Example on $\mathbb{P}^5$.
+
+Let $(a, b, c, d, e, f)$ be homogeneous coordinates in $\mathbb{P}^5$.
+
+**Example 2.1.** ($k=3$) Let
+
+$$B^t = \begin{bmatrix}
+a & b & c & d & e & f \\
+a & b & c & d & e & f \\
+a & b & c & d & e & f
+\end{bmatrix}$$
+
+$$A = \begin{bmatrix}
+f & e & d & -c-b-a \\
+f & e & d & -c-b-a \\
+f & e & d & -c-b-a
+\end{bmatrix}$$
+
+The corresponding monad gives a special symplectic instanton bundle on $\mathbb{P}^5$ with $k=3$. With the notation of remark 1.10, using [BaS] we can compute $\text{syz}_0 C = 14$, $\text{syz}_1 C = 174$. Hence $h^2(E \otimes E^*) = 3$ from the formulas of Remark 1.10. Moreover $h^0(E(1)) = 4$.
+
+**Example 2.2.** ($k=3$) Let $B^t$ as in the Example 2.1 and
+
+$$A = \begin{bmatrix}
+f & e & d & -c-b-a \\
+c & d & 2f-b-a & -2c \\
+d & f & e & -c-b
+\end{bmatrix}$$
+
+We have $\text{syz}_0 C = 10$, $\text{syz}_1 C = 171$. Hence $h^2(E \otimes E^*) = 0$. We can compute also the syzygies of $B^t$ and $A$ and we get $h^0(E(1)) = 4$, $h^0(E^*(1)) = 3$, hence $E$ is not self-dual.
+
+**Example 2.3.** ($k=4$) Let
+
+$$B^t = \begin{bmatrix}
+a & b & c & d & e & f \\
+a & b & c & d & e & f \\
+a & b & c & d & e & f \\
+a & b & c & d & e & f
+\end{bmatrix}$$
+
+$$A = \begin{bmatrix}
+f & e & d & -c-b-a \\
+f & e & d & -c-b-a \\
+f & e & d & -c-b-a \\
+f & e & d & -c-b-a
+\end{bmatrix}$$

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+$E$ is a special symplectic instanton bundle with $k = 4$. We compute
+
+$$h^2(E \otimes E^*) = 12.$$
+
+**Example 2.4.** ($k=4$) Let $B^t$ as in the Example 2.3. Let
+
+$$A = \begin{bmatrix}
+f & e & d & & -c & -b & -a \\
+& e & d & 2f & -b & -a & -2c \\
+3d & f & e & -3a & & -c & -b \\
+& f & e & d & -c & -b & -a
+\end{bmatrix}$$
+
+In this case $h^2(E \otimes E^*) = 6$, $h^0(E(1)) = 4$, $h^0(E^*(1)) = 3$.
+
+**Example 2.5.** ($k=4$) Let $B^t$ as in the Example 2.3. Let
+
+$$A = \begin{bmatrix}
+f & e & d & & -c & -b & -a \\
+e & d & 2f & -b & -a & -2c \\
+3d & f & e & -3a & & -c & -b \\
+5d & f & e & -5a & -c & -b - a - c & -b
+\end{bmatrix}$$
+
+Now $H^2(E \otimes E^*) = 0$, $h^0(E(1)) = 4$, $h^0(E^*(1)) = 2$.
+
+### 3. On the singularities of moduli spaces.
+
+The stable Schwarzenberger type bundles on $\mathbb{P}^m$ (see (1.2)) form a Zariski open subset of the moduli space of stable bundles. Let $N_{\mathbb{P}^m}(k, q)$ be the moduli space of stable STB whose first Chern class is $k$ and whose rank is $q$. The following proposition is easy and well known:
+
+**Proposition 3.1.** The space $N_{\mathbb{P}^m}(k,q)$ is smooth, irreducible of dimension $1-k^2-(q+k)^2+k(q+k)(m+1)$.
+
+We denote by $\text{MI}_{\mathbb{P}^{2n+1}}(k)$ the moduli space of stable instanton bundles with quantum number $k$. It is an open subset of the moduli space of stable $2n$-bundles on $\mathbb{P}^{2n+1}$ with Chern polynomial $(1-t^2)^{-k}$.
+
+On $\mathbb{P}^5$ (as on $\mathbb{P}^3$) all instanton bundles are stable by [AO], Theorem 3.6. $\text{MI}_{\mathbb{P}^{2n+1}}(2)$ is smooth ([AO] Theorem 3.14), unirational of dimension $4n^2 + 12n - 3$ and has zero Euler-Poincaré characteristic ([BE], [K]).
+
+**Theorem 3.2.** The space $\text{MI}_{\mathbb{P}^{2n+1}}(2)$ is irreducible.
+
+*Proof.* The moduli space $N = N_{\mathbb{P}^{2n+1}}(2, n+2)$ of stable STB of rank $2n+2$ and $c_1 = 2$ is irreducible of dimension $4n^2 + 8n - 3$ by Prop. 3.1

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+For a given instanton bundle $E$ there is a STB $S$ associated with $E$, which is stable ([AO], Theorem 2.8) and unique (ibid., Prop. 2.17). It is easy to prove that the map $\pi: M \to N$ defined by $\pi([E]) = [S]$ is algebraic, moreover $\pi$ is dominant by [ST]. If $m = [E] \in M$, the fiber $\pi^{-1}(\pi(m))$ is a Zariski open subset of the grassmannian of planes in the vector space $H^0(\mathbb{P}^{2n+1}, S^*(1))$, where $\pi(m) = [S]$; by the Theorem 3.14 of [AO], $h^0(\mathbb{P}^{2n+1}, S^*(1)) = 2n+2$, hence $\dim \pi^{-1}(\pi(m)) = 4n$.
+
+In order to prove that $M$ is irreducible, we suppose by contradiction that there are at least two irreducible components $M_0$ and $M_1$ of $M$. Then $M_0 \cap M_1 = \emptyset$ ($M$ is smooth), $\pi(M_0)$ and $\pi(M_1)$ are constructible subset of $N$ by Chevalley's theorem. Looking at the dimensions of $M_0, M_1, N$ and the fibers of $\pi$ we conclude that both $\pi(M_0)$ and $\pi(M_1)$ must contain an open subset of $N$, which implies $\pi(M_0) \cap \pi(M_1) \neq \emptyset$ by the irreducibility of $N$. This is a contradiction because the fibers of $\pi$ are connected. □
+
+For $n \ge 2$ and $k \ge 3$, it is no longer true that $\mathrm{MI}_{p^{2n+1}}(k)$ is smooth. In fact on $\mathbb{P}^5$ we have:
+
+**Theorem 3.3.** The space $\mathrm{MI}_{\mathbb{P}^5}(k)$ is singular for $k=3,4$. To be more precise, the irreducible component $M_0(k)$ of $\mathrm{MI}_{\mathbb{P}^5}(k)$ containing the special instanton bundles is generically reduced of dimension $54(k=3)$ or $65(k=4)$, and $\mathrm{MI}_{\mathbb{P}^5}(k)$ is singular at the points corresponding to special symplectic instanton bundles.
+
+*Proof.* Let $E_0$ be the special instanton bundle on $\mathbb{P}^5$ of the Example 2.2($k=3$) or of the Example 2.5($k=4$). Then $h^2(E_0 \otimes E_0^*) = 0$ and $M_0(k)$ is smooth at the point corresponding to $E_0$, of dimension $h^1(E_0 \otimes E_0^*) = 54(k=3)$ or $65(k=4)$. In particular, $M_0(k)$ is generically reduced. If $E_1$ is a special symplectic instanton bundle on $\mathbb{P}^5$, the computations in 2.1 and 2.3 show that $h^2(E_1 \otimes E_1^*) = 3(k=3)$ or $12(k=4)$, and $h^1(E_1 \otimes E_1^*) = 57$ or $77$ respectively. Hence $\mathrm{MI}_{\mathbb{P}^5}(k)$ is singular at $E_1$ for $k=3$ and $4$. □
+
+**Remark 3.4.** It is natural to conjecture that $\mathrm{MI}_{\mathbb{P}^{2n+1}}(k)$ is singular for all $n \ge 2$ and $k \ge 3$.
+
+**Theorem 3.5.** Let $E$ be an instanton bundle on $\mathbb{P}^{2n+1}$ with $c_2(E) = k$. Then
+
+$$h^1(E(t)) = 0 \text{ for } t \le -2 \text{ and } k-1 \le t.$$
+
+*Proof.* The result is obvious for $t \le -2$. It is sufficient to prove $h^1(S^*(t)) = 0$ for $t \ge k-1$. We have
+
+$$S^*(t) = \bigwedge^{2n+k-1} S(t-k).$$

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+Taking wedge products of (1.2) we have the exact sequence
+
+$$
+\begin{align*}
+0 \to \mathcal{O}(t+1-2n-2k)^{\alpha_0} \to \dots \to \mathcal{O}(t-k-1)^{\alpha_{2n+k-2}} \\
+\qquad \to \mathcal{O}(t-k)^{\alpha_{2n+k-1}} \to \bigwedge^{2n+k-1} S(t-k) \to 0
+\end{align*}
+$$
+
+for suitable $\alpha_i \in \mathbb{N}$ and from this sequence we can conclude.
+
+Ellia proves Theorem 3.5 in the case of $\mathbb{P}^3$ ([E], Prop. IV.1). He also remarks that the given bound is sharp. This holds on $\mathbb{P}^{2n+1}$ as it is shown by the following theorem, which points out that the special symplectic instanton bundles are the “furthest” from having natural cohomology. $\square$
+
+**Theorem 3.6.** Let $E$ be a special symplectic instanton bundle on $\mathbb{P}^{2n+1}$ with $c_2 = k$. Then
+
+$$h^1(E(t)) \neq 0 \text{ for } -1 \le t \le k-2.$$
+
+*Proof*. For $n=1$ the thesis is immediate from the exact sequence
+
+$$0 \rightarrow \mathcal{O}(t-1) \rightarrow E(t) \rightarrow \mathcal{J}_C(t+1) \rightarrow 0$$
+
+where $C$ is the union of $k+1$ disjoint lines in a smooth quadric surface. Then the result follows by induction on $n$ by considering the sequence
+
+$$0 \rightarrow E(t-2) \rightarrow E(t-1)^2 \rightarrow E(t) \rightarrow E(t)|_{\mathbb{P}^{2n-1}} \rightarrow 0$$
+
+and the fact that, for a particular choice of the subspace $\mathbb{P}^{2n-1}$, the restriction $E|_{\mathbb{P}^{2n-1}}$ splits as the direct sum of a rank-2 trivial bundle and a special symplectic instanton bundle on $\mathbb{P}^{2n-1}$([ST] 5.9). $\square$
+
+**Remark 3.7.** In [OT] it is proved that if $E_k$ is a special symplectic instanton bundle on $\mathbb{P}^5$ with $c_2 = k$ then $h^1(\mathrm{End}\,E_k) = 20k - 3$.
+
+In the following table we summarize what we know about the component $M_0(k) \subset \mathrm{MI}_{\mathbb{P}^5}(k)$ containing $E_k$.
+
+**Table 3.10**
+
+<table>
+  <thead>
+    <tr>
+      <th></th>
+      <th>h<sup>1</sup>(E<sub>k</sub> &otimes; E<sub>k</sub><sup>*</sup>)</th>
+      <th>h<sup>2</sup>(E<sub>k</sub> &otimes; E<sub>k</sub><sup>*</sup>)</th>
+      <th>dim M<sub>0</sub>(k)</th>
+      <th>MI<sub>P<sup>5</sup></sub>(k)</th>
+    </tr>
+  </thead>
+  <tbody>
+    <tr>
+      <td>k = 1</td>
+      <td>14</td>
+      <td>0</td>
+      <td>14</td>
+      <td>open subset of P<sup>14</sup></td>
+    </tr>
+    <tr>
+      <td>k = 2</td>
+      <td>37</td>
+      <td>0</td>
+      <td>37</td>
+      <td>smooth, irreduc., unirat.</td>
+    </tr>
+    <tr>
+      <td>k = 3</td>
+      <td>57</td>
+      <td>3</td>
+      <td>54</td>
+      <td>singular</td>
+    </tr>
+    <tr>
+      <td>k = 4</td>
+      <td>77</td>
+      <td>12</td>
+      <td>65</td>
+      <td>singular</td>
+    </tr>
+    <tr>
+      <td>k &ge; 2</td>
+      <td>20k - 3</td>
+      <td>3(k - 2)<sup>2</sup></td>
+      <td>?</td>
+      <td>?</td>
+    </tr>
+  </tbody>
+</table>

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+References
+
+[AO] V. Ancona and G. Ottaviani, *On the stability of special instanton bundles on P<sup>2n+1</sup>*,
+Trans. Amer. Math. Soc., **341** (1994), 677-693.
+
+[BE] J. Bertin and G. Elencwajg, *Symétries des fibrés vectoriels sur P<sup>n</sup> et nombre d'Euler*, Duke Math. J., **49** (1982), 807-831.
+
+[BaS] D. Bayer and M. Stillman, *Macaulay, a computer algebra system for algebraic geometry*.
+
+[BoS] G. Bohnhorst and H. Spindler, *The stability of certain vector bundles on P<sup>n</sup>*,
+Proc. Bayreuth Conference "Complex Algebraic Varieties", LNM **1507**, Springer Berlin
+(1992), 39-50.
+
+[E] Ph. Ellia, *Some vanishings for the cohomology of stable rank two vector bundles on P<sup>3</sup>*,
+J. reine angew. Math., **451** (1994), 1-14.
+
+[ES] G. Ellingsrud and S.A. Stromme, *Stable rank-2 bundles on P<sup>3</sup> with c<sub>1</sub> = 0 and c<sub>2</sub> = 3*, Math. Ann., **255** (1981), 123-137.
+
+[Har] R. Hartshorne, *Stable vector bundles of rank 2 on P<sup>3</sup>*,
+Math. Ann., **238** (1978), 229-280.
+
+[K] T. Kaneyama, *Torus-equivariant vector bundles on projective spaces*, Nagoya Math.
+J., **111** (1988), 25-40.
+
+[M] M. Maruyama, *Moduli of stable sheaves*, II, J. Math. Kyoto Univ., **18** (1978),
+557-614.
+
+[OS] C. Okonek and H. Spindler, *Mathematical instanton bundles on P<sup>2n+1</sup>*,
+Journal reine angew. Math., **364** (1986), 35-50.
+
+[OT] G. Ottaviani and G. Trautmann, *The tangent space at a special symplectic instanton bundle on P<sup>2n+1</sup>*,
+Manuscr. Math., **85** (1994), 97-107.
+
+[P] J. LePotier, *Sur l'espace de modules des fibrés de Yang et Mills*, in Mathématique et Physique, Sém. E.N.S. 1979-1982, Basel-Stuttgart-Boston 1983.
+
+[S] R.L.E. Schwarzenberger, *Vector bundles on the projective plane*, Proc. London Math. Soc., **11** (1961), 623-640.
+
+[ST] H. Spindler and G. Trautmann, *Special instanton bundles on P<sup>2n+1</sup>, their geometry and their moduli*, Math. Ann., **286** (1990), 559-592.
+
+Received January 5, 1993. Both authors were supported by MURST and by GNSAGA of CNR.
+
+DIPARTIMENTO DI MATEMATICA
+VIALE MORGAGNI 67 A
+I-50134 FIRENZE
+
+E-mail address: ancona@udininw.math.unifi.it
+
+AND
+
+DIPARTIMENTO DI MATEMATICA
+VIA VETOIO, COPPITO
+I-67010 L'AQUILA
+
+E-mail address: ottaviani@vxscaq.aquila.infn.it
+
+Added in proof. After this paper has been written we received a preprint of R. Miró-Reig and J. Orus-Lacort where they prove that the conjecture stated in the Remark 3.4 is true.

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+New (and Old) Proof Systems for Lattice Problems
+
+Navid Alamati*
+
+Chris Peikert†
+
+Noah Stephens-Davidowitz‡
+
+December 19, 2017
+
+Abstract
+
+We continue the study of statistical zero-knowledge (SZK) proofs, both interactive and noninteractive, for computational problems on point lattices. We are particularly interested in the problem GapSPP of approximating the $\epsilon$-smoothing parameter (for some $\epsilon < 1/2$) of an $n$-dimensional lattice. The smoothing parameter is a key quantity in the study of lattices, and GapSPP has been emerging as a core problem in lattice-based cryptography, e.g., in worst-case to average-case reductions.
+
+We show that GapSPP admits SZK proofs for remarkably low approximation factors, improving on prior work by up to roughly $\sqrt{n}$. Specifically:
+
+* There is a noninteractive SZK proof for $O(\log(n)\sqrt{\log(1/\epsilon)})$-approximate GapSPP. Moreover, for any negligible $\epsilon$ and a larger approximation factor $\tilde{O}(\sqrt{n\log(1/\epsilon)})$, there is such a proof with an efficient prover.
+
+* There is an (interactive) SZK proof with an efficient prover for $O(\log n + \sqrt{\log(1/\epsilon)/\log n})$-approximate coGapSPP. We show this by proving that $O(\log n)$-approximate GapSPP is in coNP.
+
+In addition, we give an (interactive) SZK proof with an efficient prover for approximating the lattice covering radius to within an $O(\sqrt{n})$ factor, improving upon the prior best factor of $\omega(\sqrt{n \log n})$.
+
+*Computer Science and Engineering, University of Michigan. Email: alamati@umich.edu.
+
+†Computer Science and Engineering, University of Michigan. Email: cpeikert@umich.edu. This material is based upon work supported by the National Science Foundation under CAREER Award CCF-1054495 and CNS-1606362, the Alfred P. Sloan Foundation, and by a Google Research Award. The views expressed are those of the authors and do not necessarily reflect the official policy or position of the National Science Foundation, the Sloan Foundation, or Google.
+
+‡Courant Institute of Mathematical Sciences, New York University. Email: noahsd@gmail.com. Supported by the National Science Foundation (NSF) under Grant No. CCF-1320188, and the Defense Advanced Research Projects Agency (DARPA) and Army Research Office (ARO) under Contract No. W911NF-15-C-0236. Part of this work was done while visiting the second author at the University of Michigan.

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+**Theorem 2.11.** There is an efficient algorithm that takes as input a basis $\mathbf{B} \in \mathbb{Q}^{n \times n}$ and any parameter $s \ge \|\tilde{\mathbf{B}}\|_{\sqrt{\log n}}$ and outputs a sample from $D_{\mathcal{L},s}$, where $\mathcal{L} \subset \mathbb{R}^n$ is the lattice generated by $\mathbf{B}$.
+
+**Corollary 2.12.** There is an efficient algorithm that takes as input a (basis for a) lattice $\mathcal{L} \subset \mathbb{Q}^n$ and parameter $s \ge 2^n \eta_\varepsilon(\mathcal{L})$ and outputs a sample from $D_{\mathcal{L},s}$.
+
+*Proof.* Combine the above with the celebrated LLL algorithm [LLL82], which in particular allows us to find a basis for $\mathcal{L}$ with $\|\tilde{\mathbf{B}}\| \le 2^{n/2}\eta_\varepsilon(\mathcal{L})$. $\square$
+
+We also need the following result, which is implicit in [Ban93]. See, e.g., [DR16] for a proof.
+
+**Lemma 2.13.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ and $\varepsilon \in (0, 1/2)$,
+
+$$ \lambda_n(\mathcal{L}) \le 2\mu(\mathcal{L}) \le \sqrt{n} \cdot \eta_\varepsilon(\mathcal{L}). $$
+
+In particular, there exists a basis $\mathbf{B}$ of $\mathcal{L}$ with $\|\tilde{\mathbf{B}}\| \le \lambda_n(\mathcal{L}) \le \sqrt{n} \cdot \eta_{1/2}(\mathcal{L})$.
+
+**Corollary 2.14.** For any lattice $\mathcal{L} \subset \mathbb{Q}^n$ with basis $\mathbf{B}$, there exists preprocessing $P$ whose size is polynomial in the bit length of $\mathbf{B}$ and an efficient algorithm that, on input $P$ and $s \ge \sqrt{n \log n} \cdot \eta_{1/2}(\mathcal{L})$ outputs a sample from $D_{\mathcal{L},s}$.
+
+*Proof.* By Lemma 2.13, there exists a basis $\mathbf{B}'$ with $\|\tilde{\mathbf{B}}'\| \le \sqrt{n} \cdot \eta_{1/2}(\mathcal{L})$. By Lemma 2.10, the bit length of $\mathbf{B}'$ is polynomial in the bit length of $\mathbf{B}$. We use this as our preprocessing $P$. The result then follows by Theorem 2.11. $\square$
+
+## 2.5 Computational Problems
+
+Here we define two promise problems that will be considered in this paper.
+
+**Definition 2.15 (Covering Radius Problem).** For any approximation factor $\gamma = \gamma(n) \ge 1$, an instance of $\gamma$-GapCRP is a (basis for a) lattice $\mathcal{L} \subset \mathbb{Q}^n$. It is a YES instance if $\mu(\mathcal{L}) \le 1$ and a NO instance if $\mu(\mathcal{L}) > \gamma$.
+
+**Definition 2.16 (Smoothing Parameter Problem).** For any approximation factor $\gamma = \gamma(n) \ge 1$ and $\varepsilon = \varepsilon(n) > 0$, an instance of $\gamma$-GapSPP$_\varepsilon$ is a (basis for a) lattice $\mathcal{L} \subset \mathbb{Q}^n$. It is a YES instance if $\eta_\varepsilon(\mathcal{L}) \le 1$ and a NO instance if $\eta_\varepsilon(\mathcal{L}) > \gamma$.
+
+We will need the following result from [CDLP13].
+
+**Theorem 2.17.** For any $\varepsilon \in (0, 1/2)$, $\gamma$-GapSPP$_\varepsilon$ is in SZK for $\gamma = O(1 + \sqrt{\log(1/\varepsilon)/\log(n)}橋$.
+
+## 2.6 Noninteractive Proof Systems
+
+**Definition 2.18 (Noninteractive Proof System).** A pair $(P, V)$ is a noninteractive proof system for a promise problem $\Pi = (\Pi^{\text{YES}}, \Pi^{\text{NO}})$ if $P$ is a (possibly unbounded) algorithm and $V$ is a polynomial-time algorithm such that
+
+* Completeness: for every $x \in \Pi_n^{\text{YES}}$, Pr[$V(x, r, P(x, r))$ accepts] $\ge 1 - \varepsilon$; and
+
+⁴In [CDLP13], this result is proven only for $\varepsilon < 1/3$. However, it is immediate from, e.g., Lemma 2.9 that the result can be extended to any $\varepsilon < 1/2$.

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+* Soundness: for every $x \in \Pi_n^{\text{NO}}$, $\Pr[\exists \pi : V(x, r, \pi) \text{ accepts}] \le \varepsilon$,
+
+where $n$ is the input length, $\varepsilon = \varepsilon(n) \le \text{negl}(n)$, and the probabilities are taken over $r$, which is sampled uniformly at random from $\{0, 1\}^{\text{poly}(n)}$.
+
+A noninteractive proof system $(P, V)$ for a promise problem $\Pi = (\Pi^{\text{YES}}, \Pi^{\text{NO}})$ is statistical zero knowledge if there exists a probabilistic polynomial-time algorithm $S$ (called a *simulator*) such that for all $x \in \Pi^{\text{YES}}$, the statistical distance between $S(x)$ and $(r, P(x, r))$ is negligible in $n$. The class of promise problems having noninteractive statistical zero-knowledge proof systems is denoted NISZK.
+
+## 2.7 Probability
+
+The entropy of a random variable $X$ over a countable set $S$ is given by
+
+$$H(X) := \sum_{a \in S} \Pr[X = a] \cdot \log_2(1/\Pr[X = a]) .$$
+
+We will also need the Chernoff-Hoeffding bound [Hoe63].
+
+**Lemma 2.19 (Chernoff-Hoeffding bound).** Let $X_1, \dots, X_m \in [0, 1]$ be independent and identically distributed random variables with $\bar{X} := \mathbb{E}[X_i]$. Then, for any $s > 0$,
+
+$$\Pr[m\bar{X} - \sum X_i \ge s] \le \exp(-s^2/(2m)) .$$
+
+Finally, we will need a minor variant of the above inequality.
+
+**Lemma 2.20.** Let $X_1, \dots, X_m \in \mathbb{R}$ be independent (but not necessarily identically distributed) random variables. Suppose that there exists an $\alpha \ge 0$ and $s > 0$ such that for any $r > 0$,
+
+$$\Pr[|X_i| \ge r] \le \alpha \exp(-r^2/s^2) .$$
+
+Then, for any $r > 0$,
+
+$$\Pr\left[\sum X_i^2 \ge r\right] \le (1+\alpha)^m \exp(-r/(2s^2)).$$
+
+*Proof.* For any index $i$, we have
+
+$$ 
+\begin{align*}
+\mathbb{E}[\exp(X_i^2/(2s^2))] &= 1 + \frac{1}{s^2} \cdot \int_0^\infty r \exp(r^2/(2s^2)) \Pr[|X_i| \ge r] dr \\
+&\le 1 + \frac{\alpha}{s^2} \cdot \int_0^\infty r \exp(-r^2/(2s^2)) dr \\
+&= 1 + \alpha.
+\end{align*}
+$$
+
+Since the $X_i$ are independent, it follows that
+
+$$\mathbb{E}\left[\exp\left(\sum X_i^2/(2s^2)\right)\right] = \mathbb{E}\left[\prod_i \exp(X_i^2/(2s^2))\right] \le (1+\alpha)^m .$$
+
+The result then follows by Markov's inequality. $\square$

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+**3 Two NISZK Proofs for GapSPP**
+
+Recall the definition
+
+$$
+\eta_{\det}(\mathcal{L}) := \max_{\pi} \det(\pi(\mathcal{L}))^{1/\mathrm{rank}(\pi(\mathcal{L}))}.
+$$
+
+We will also need the following definition from [DR16],
+
+$$
+C_{\eta}(n) := \sup_{\mathcal{L}} \frac{\eta_{1/2}(\mathcal{L})}{\eta_{\det}(\mathcal{L})},
+$$
+
+where the supremum is taken over all lattices $\mathcal{L} \subset \mathbb{R}^n$. In this notation, Theorem 1.6 is equivalent to the
+inequality
+
+$$
+C_{\eta}(n) \leq 10(\log n + 2).
+$$
+
+We note that the true value of $C_\eta(n)$ is still not known. (In particular, the best lower bound is $C_\eta(n) \ge \sqrt{\log(n)/\pi} + o(1)$, which follows from the fact that $\eta_{1/2}(\mathbb{Z}^n) = \sqrt{\log(n)/\pi} + o(1)$.) We therefore state our results in terms of $C_\eta(n)$.
+
+**3.1 An Explicit Proof System**
+
+We first consider the NISZK proof system for $\sqrt{n}$-coGapSVP due to [PV08], shown in Figure 1. We show that this is actually also a NISZK proof system for $O(\sqrt{\log(1/\epsilon)} \cdot \log n)$-GapSPP$_\epsilon$ for negligible $\epsilon$. (In Section 3.2, we show a different proof system that works for all $\epsilon \in (0, 1/2)$, also with an approximation factor of $O(\log(n)\sqrt{\log(1/\epsilon)})$.)
+
+NISZK proof system for GapSPP.
+
+**Common Input:** A basis $\mathbf{B}$ for a lattice $\mathcal{L} \subset \mathbb{Q}^n$.
+
+**Random Input :** $m$ vectors $t_1, \dots, t_m \in \mathcal{P}(\mathbf{B})$, sampled uniformly at random.
+
+**Prover P:** Sample $m$ vectors $e_1, \dots, e_m \in \mathbb{R}^n$ independently from $D_{\mathcal{L}+t_i}$, and output them as the
+proof.
+
+**Verifier V:** Accept if and only if $e_i \equiv t_i \bmod \mathcal{L}$ for all $i$ and $\left\| \sum e_i e_i^T \right\| \le 3m$.
+
+Figure 1: The non-interactive zero-knowledge proof system for GapSPP, where $m := 100n$.
+
+**Theorem 3.1.** For any $\varepsilon \le \text{negl}(n)$, $\gamma$-GapSPP$_\varepsilon$ is in NISZK for
+
+$$
+\gamma := O(C_{\eta}(n) \sqrt{\log(1/\varepsilon)}) \le O(\log(n) \sqrt{\log(1/\varepsilon)})
+$$
+
+via the proof system shown in Figure 1.
+
+We will prove in turn that the proof system is statistical zero knowledge, complete, and sound. In fact,
+the proofs of statistical zero knowledge and completeness are nearly identical to the corresponding proofs
+in [PV08].

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+To prove the zero-knowledge property of the proof system, we consider the simulator that behaves as follows. Let $\mathbf{e}_1, \dots, \mathbf{e}_m \in \mathbb{R}^m$ be sampled independently from the continuous Gaussian centered at 0. Let $\mathbf{t}_1, \dots, \mathbf{t}_m \in \mathcal{P}(\mathbf{B})$ such that $\mathbf{e}_i \equiv \mathbf{t}_i \bmod \mathcal{L}$. The simulator then outputs $\mathbf{t}_1, \dots, \mathbf{t}_m$ as the random input and $\mathbf{e}_1, \dots, \mathbf{e}_m$ as the proof.
+
+**Lemma 3.2 (Statistical zero knowledge).** For any $\varepsilon \in (0, 1)$ and lattice $\mathcal{L} \subset \mathbb{Q}^n$ with $\eta_\varepsilon(\mathcal{L}) \le 1$, the output of the simulator described above is within statistical distance $\varepsilon m$ of honestly generated random input and an honestly generated proof as in Figure 1. In particular, the proof system in Figure 1 is statistical zero knowledge for negligible $\varepsilon$.
+
+*Proof.* Notice that, conditioned on the random input $\mathbf{t}_i$, the distribution of $\mathbf{e}_i$ is exactly $D_{\mathcal{L}+\mathbf{t}_i,s}$. So, we only need to show that the random input $\mathbf{t}_1, \dots, \mathbf{t}_m \in \mathcal{P}(\mathbf{B})$ chosen by the simulator is within statistical distance $\varepsilon m$ of uniform. Indeed, this follows from Lemma 2.8 and the union bound. $\square$
+
+The proof of completeness is a bit tedious and nearly identical to proofs of similar statements in [AR04, PV08, DR16]. We include a proof in Appendix A.
+
+**Lemma 3.3 (Completeness).** For any lattice $\mathcal{L} \subset \mathbb{Q}^n$ with $\eta_{1/2}(\mathcal{L}) \le 1$, the proof given in Figure 1 will be accepted except with negligible probability. I.e., the proof system is complete.
+
+**Soundness.** We now show the soundness of the proof system shown in Figure 1, using Theorem 1.6. We note that [DR16] contains an implicit proof of a very similar result in a different context. (Dadush and Regev conjectured a form of Theorem 1.6 and showed a number of implications [DR16]. In particular, they showed that with non-negligible probability over a single uniformly random shift $\mathbf{t} \in \mathbb{R}^n/\mathcal{L}$, there is no list of vectors $\mathbf{e}_1, \dots, \mathbf{e}_m \in \mathcal{L} + \mathbf{t}$ with small covariance.)
+
+**Theorem 3.4.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ with basis **B** satisfying $\eta_{1/2}(\mathcal{L}) \ge 100C_\eta(n)$ (and in particular any lattice with $\eta_{1/2}(\mathcal{L}) \ge 1000(\log(n) + 2)$), if $\mathbf{t}_1, \dots, \mathbf{t}_m$ are sampled uniformly from $\mathbb{R}^n/\mathcal{L}$, then the probability that there exists any proof $\mathbf{e}_1, \dots, \mathbf{e}_m$ with $\mathbf{e}_i \equiv \mathbf{t}_i \bmod \mathcal{L}$ and
+
+$$
+\left\| \sum_i \mathbf{e}_i \mathbf{e}_i^T \right\| \le 3m
+$$
+
+is at most $\exp(-\Omega(m^2))$.
+
+*Proof.* By the definition of $C_\eta(n)$ there is a lattice projection $\pi$ such that $\det(\pi(\mathcal{L})) \ge 100^k$, where $k := \text{rank}(\pi(\mathcal{L}))$. For any $\mathbf{e}_1, \dots, \mathbf{e}_m$ with $\mathbf{e}_i \equiv \mathbf{t}_i \bmod \mathcal{L}$, we have
+
+$$
+\begin{align*}
+\left\| \sum_i \mathbf{e}_i \mathbf{e}_i^T \right\| &\geq \left\| \sum_i \pi(\mathbf{e}_i) \pi(\mathbf{e}_i)^T \right\| \\
+&\geq \frac{1}{k} \operatorname{Tr} \left( \sum_i \pi(\mathbf{e}_i) \pi(\mathbf{e}_i)^T \right) \\
+&= \frac{1}{k} \sum_i \| \pi(\mathbf{e}_i) \|_2^2 \\
+&\geq \frac{1}{k} \sum_i \operatorname{dist}(\pi(\mathbf{t}_i), \pi(\mathcal{L}))^2,
+\end{align*}
+$$
+
+where the first inequality on the spectral norms follows from the fact that $\langle \mathbf{u}, \pi(\mathbf{e}_i) \rangle = \langle \pi(\mathbf{u}), \pi(\mathbf{e}_i) \rangle$ and
+$\|\pi(\mathbf{u})\| \leq \|\mathbf{u}\|$; the second inequality follows from the fact that the spectral norm is the largest eigenvalue
+and the trace is the sum of the $k$ eigenvalues; and the equality is by definition of trace.

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+Now by Claim 2.4, $\pi(\mathbf{t}_i)$ is uniformly distributed mod $\pi(\mathcal{L})$, and therefore by Lemma 2.3,
+
+$$ \mathbb{E}[\mathrm{dist}(\pi(\mathbf{t}_i), \pi(\mathcal{L}))^2] \geq \mu(\pi(\mathcal{L}))^2/4. $$
+
+Furthermore, since the $\mathbf{t}_i$ are independent and identically distributed with $\mathrm{dist}(\pi(\mathbf{t}_i), \pi(\mathcal{L})) \leq \mu(\pi(\mathcal{L}))$, we
+can apply the Chernoff-Hoeffding bound (Lemma 2.19) to get
+
+$$ \mathrm{Pr} \left[ \sum \mathrm{dist}(\pi(\mathbf{t}_i), \pi(\mathcal{L}))^2 \leq m\mu(\pi(\mathcal{L}))^2/5 \right] \leq \exp(-Cm^2). $$
+
+The result follows by noting that $\mu(\pi(\mathcal{L}))^2/(5k) \ge 3$ by Claim 2.1, together with the fact that $\det(\pi(\mathcal{L})) \ge 100^k$. $\square$
+
+**Corollary 3.5 (Soundness).** For any $\varepsilon \in (0, 1/2)$ and lattice $\mathcal{L} \subset \mathbb{R}^n$ with basis **B** satisfying $n \ge 2$ and $\eta_\varepsilon(\mathcal{L}) \ge 100C_\eta(n)\sqrt{\log(1/\varepsilon)}$ (and in particular any lattice with $\eta_\varepsilon(\mathcal{L}) \ge 1000(\log(n)+2)\sqrt{\log(1/\varepsilon)})$, if $\mathbf{t}_1, \dots, \mathbf{t}_m$ are sampled uniformly from $\mathcal{P}(\mathbf{B})$, then the probability that there exists a proof $\mathbf{e}_1, \dots, \mathbf{e}_m$ with $\mathbf{e}_i \equiv \mathbf{t} \bmod \mathcal{L}$ and
+
+$$ \left\| \sum \mathbf{e}_i \mathbf{e}_i^T \right\| \le 3m $$
+
+is at most $\exp(-\Omega(m^2))$. In other words, the proof system in Figure 1 is $\exp(-\Omega(m^2))$-statistically sound.
+
+*Proof.* By Lemma 2.9, we have $\eta_{1/2} \ge 100C_\eta(n)$, and the result follows from Theorem 3.4. $\square$
+
+**Making the prover efficient.** Finally, following [PV08] we observe that the prover in the proof system shown in Figure 1 can be made efficient if we relax the approximation factor. In particular, if $\eta_\epsilon(\mathcal{L}) \le 1/\sqrt{n\log n}$, then by Corollary 2.14, there is in fact an efficient prover. Theorem 1.2 then follows immediately from the above analysis.
+
+## 3.2 A Proof via Entropy Approximation
+
+We recall from Goldreich, Sahai, and Vadhan [GSV99] the Entropy Approximation problem, which asks us to approximate the entropy of the distribution obtained by calling some input circuit $C$ on the uniform distribution over its input space. In particular, we recall that [GSV99] proved that this problem is NISZK-complete. (Formally, we only need the fact that Entropy Approximation is in NISZK.)
+
+**Definition 3.6.** An instance of the Entropy Approximation problem is a circuit $C$ and an integer $k$. It is a YES instance if $H(C(U)) > k+1$ and a NO instance if $H(C(U)) < k-1$, where $U$ is the uniform distribution on the input space of $C$.
+
+**Theorem 3.7 ([GSV99]).** *Entropy Approximation is NISZK-complete.*
+
+In the rest of this section, we show a Karp reduction from $O(\log(n)\sqrt{\log(1/\varepsilon)})$-GapSPP$_\varepsilon$ to Entropy Approximation. I.e., we give an efficient algorithm that takes as input a basis for a lattice $\mathcal{L}$ and outputs a circuit $C_\mathcal{L}$ such that (1) if $\eta_\varepsilon(\mathcal{L}) \le 1$, then $H(C_\mathcal{L}(U))$ is large; but (2) if $\eta_\varepsilon(\mathcal{L}) \ge C \log(n)\sqrt{\log(1/\varepsilon)}$, then $H(C_\mathcal{L}(U))$ is small.
+
+Intuitively, we want to use a circuit that samples from the continuous Gaussian with parameter one modulo the lattice $\mathcal{L}$. Then, by Claim 2.7, if $\eta_\varepsilon(\mathcal{L}) \le 1$, the resulting distribution will be nearly uniform over $\mathbb{R}^n/\mathcal{L}$. On the other hand, we know that, with high probability, the continuous Gaussian lies in a set of

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+volume roughly one. And, by definition, if $\eta_\epsilon(\mathcal{L}) \ge \Omega(C_\eta(n)\sqrt{\log(1/\epsilon)})$, then there exists a projection $\pi$ such that, say, $\text{vol}(\pi(\mathbb{R}^n/\mathcal{L})) = \det(\pi(\mathcal{L})) \ge 100$. Therefore, the projected Gaussian lies in a small fraction of $\pi(\mathbb{R}^n/\mathcal{L})$ with high probability.
+
+To make this precise, we must discretize $\mathbb{R}^n/\mathcal{L}$ appropriately to, say, $(\mathcal{L}/q)/\mathcal{L}$ for some large integer $q > 1$ and sample from a discretized version of the continuous Gaussian. Naturally, we choose $D_{\mathcal{L}/q}$. The following theorem shows that $D_{\mathcal{L}/q} \bmod \mathcal{L}$ lies in a small subset of $(\mathcal{L}/q)/\mathcal{L}$ when $\eta_{1/2}(\mathcal{L})$ is large.
+
+**Theorem 3.8.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ with sufficiently large $n$ and integer $q \ge 2^n (\eta_{2-n}(\mathcal{L}) + \mu(\mathcal{L}))$, if $\eta_{1/2}(\mathcal{L}) \ge 1000C_\eta(n)$ (and in particular if $\eta_{1/2}(\mathcal{L}) \ge 10^4(\log(n)+2)$), then there is a subset $S \subset (\mathcal{L}/q)/\mathcal{L}$ with $|S| \le q^n/200$ such that
+
+$$ \mathbf{x} \sim D_{\mathcal{L}/q} \bmod \mathcal{L} [\mathbf{X} \in S] \ge \frac{9}{10}. $$
+
+*Proof.* It is easy to see that $D_{\mathcal{L}/q}$ is statistically close to the distribution obtained by sampling from a continuous Gaussian with parameter one and rounding to the closest vector in $\mathcal{L}/q$. (One must simply recall from Lemma 2.6 that nearly all of the mass of $D_{\mathcal{L}/q}$ lies in a ball of radius $\sqrt{n}$ and notice that for such short points, shifts of size $\mu(\mathcal{L}/q) < 2^{-n}$ have little effect on the Gaussian mass.) It therefore suffices to show that the above probability is at least 19/20 when $\mathbf{X}$ is sampled from this new distribution. We write $CVP(\mathbf{t})$ for the closest vector in $\mathcal{L}/q$ to $\mathbf{t}$.
+
+By assumption, there is a lattice projection $\pi$ onto a $k$-dimensional subspace such that $\det(\pi(\mathcal{L})) \ge 1000^k$. Notice that $\|\pi(CVP(\mathbf{t}))\| \le \|\pi(\mathbf{t})\| + \mu(\mathcal{L})/q \le \|\mathbf{t}\| + 2^{-n}$ for any $\mathbf{t} \in \mathbb{R}^n$. In particular, if $\mathbf{X}$ is sampled from a continuous Gaussian with parameter one,
+
+$$ \Pr \left[ \| \pi(CVP(\mathbf{X})) \| \ge \sqrt{k} \right] \le \Pr \left[ \| \pi(\mathbf{X}) \| \ge \sqrt{k} - 2^{-n} \right] \le \frac{1}{20}, $$
+
+where we have applied Lemma 2.20. But, by Lemma 2.2, there are at most $(q/200)^k$ points $y \in (\pi(\mathcal{L})/q)/\pi(\mathcal{L}) \cap \sqrt{k}B_2^k$. Therefore, there are at most $q^n/200^k \le q^n/200$ points $y \in (\mathcal{L}/q)/\mathcal{L}$ with 19/20 of the mass, as needed. $\square$
+
+**Corollary 3.9.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ with $n \ge 2$, $\varepsilon \in (0, 1/2)$, and integer $q \ge 2$, let $\mathbf{X} \sim D_{\mathcal{L}/q} \bmod \mathcal{L}$. Then,
+
+1. if $\eta_\varepsilon(\mathcal{L}) \le 1$, then $H(\mathbf{X}) > n \log_2 q - 2$; but
+
+2. if $\eta_\varepsilon(\mathcal{L}) \ge 1000C_\eta(n) \cdot \sqrt{\log(1/\varepsilon)}$ (and in particular if $\eta_\varepsilon(\mathcal{L}) \ge 10^4 \log(n)\sqrt{\log(1/\varepsilon)})$ and $q \ge 2^n(\eta_{2-n}(\mathcal{L}) + \mu(\mathcal{L}))$, then $H(\mathbf{X}) < n \log_2 q - 6$.
+
+*Proof.* Suppose that $\eta_\varepsilon(\mathcal{L}) \le 1$. Then, by Claim 2.7, for any $y \in (\mathcal{L}/q)/\mathcal{L}$,
+
+$$ \Pr_{\mathbf{x} \sim D_{\mathcal{L}/q} \bmod \mathcal{L}} [\mathbf{X} = y] = \frac{\rho(\mathcal{L} + y)}{\rho(\mathcal{L}/q)} \le \frac{1+\varepsilon}{1-\varepsilon} \cdot \frac{1}{q^n}. $$
+
+It follows that
+
+$$ H(D_{\mathcal{L}/q} \bmod \mathcal{L}) \ge n \log_2 q + \log_2(1-\varepsilon) - \log_2(1+\varepsilon) > n \log_2 q - 2, $$
+
+as needed.

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+Suppose, on the other hand, that $\eta_{\varepsilon}(\mathcal{L}) \ge 1000C_{\eta}(n) \cdot \sqrt{\log(1/\varepsilon)}$ and $q \ge 2^n(\eta_{2-n}(\mathcal{L}) + \mu(\mathcal{L}))$. By Lemma 2.9, $\eta_{1/2}(\mathcal{L}) \ge 1000C_{\eta}(n)$, so that by Theorem 3.8, there is a set $S$ of size $|S| = q^n/200$ with at least 9/10 of the mass of $D_{\mathcal{L}/q} \bmod \mathcal{L}$. Therefore,
+
+$$H(D_{\mathcal{L}/q} \bmod \mathcal{L}) \le \frac{9}{10} \cdot \log_2 |S| + \frac{1}{10} \cdot n \log_2 q < n \log_2 q - 6,$$
+
+as needed. $\square$
+
+Corollary 3.9 shows that, in order to reduce $O(\log(n)\sqrt{\log(1/\varepsilon)})$-GapSPP$_{\varepsilon}$ to Entropy Approximation, it suffices to construct a circuit that samples from $D_{\mathcal{L}/q} \bmod \mathcal{L}$. The main result of this section follows immediately from Corollary 2.12.
+
+**Theorem 3.10.** There is an efficient Karp reduction from $\gamma$-GapSPP$_{\varepsilon}$ to Entropy Approximation for
+
+$$\gamma := O(C_{\eta}(n)\sqrt{\log(1/\varepsilon)}) \le O(\log(n)\sqrt{\log(1/\varepsilon)}).$$
+
+and any $\varepsilon \in (0, 1/2)$. I.e., $\gamma$-GapSPP$_{\varepsilon}$ is in NISZK.
+
+*Proof.* The reduction behaves as follows on input $\mathcal{L} \subset \mathbb{Q}^n$. By Lemma 2.10, we can find an integer $q \ge 2$ with polynomial bit length that satisfies $q \ge 2^n(\eta_{2-n}(\mathcal{L}) + \mu(\mathcal{L}))$. The reduction constructs the circuit $C_{\mathcal{L}/q}$ from Corollary 2.12 and outputs the modified circuit $C_{(\mathcal{L}/q)/\mathcal{L}}$ that takes the output from $C_{\mathcal{L}/q}$ and reduces it modulo $\mathcal{L}$. It then outputs the Entropy Approximation instance $(C_{(\mathcal{L}/q)/\mathcal{L}}, k := n \log_2 q - 4)$.
+
+The running time is clear. Suppose that $\eta_{\varepsilon}(\mathcal{L}) \le 1$. Then, by Corollary 3.9,
+
+$$H(D_{\mathcal{L}/q} \bmod \mathcal{L}) > n \log_2 q - 2.$$
+
+Since the output of $C_{(\mathcal{L}/q)/\mathcal{L}}$ is statistically close to $D_{\mathcal{L}/q} \bmod \mathcal{L}$, it follows that $H(C_{(\mathcal{L}/q)/\mathcal{L}}(U)) > n \log_2 q - 3$, as needed.
+
+If, on the other hand, $\eta_{\varepsilon}(\mathcal{L}) \ge \Omega(C_{\eta}(n) \cdot \sqrt{\log(1/\varepsilon)})$, then by Corollary 3.9,
+
+$$H(D_{\mathcal{L}/q} \bmod \mathcal{L}) < n \log_2 q - 6.$$
+
+Since the output of $C_{(\mathcal{L}/q)/\mathcal{L}}$ is statistically close to $D_{\mathcal{L}/q} \bmod \mathcal{L}$, it follows that $H(C_{(\mathcal{L}/q)/\mathcal{L}}(U)) < n \log_2 q - 5$. $\square$
+
+# 4 A coNP Proof for $O(\log n)$-GapSPP
+
+We will need the following result from [RS17], which extends Theorem 1.6 to smaller $\varepsilon$ by noting that $\rho_{1/s}(\mathcal{L}^* \setminus \{\mathbf{0}\})$ decays at least as quickly as $\rho_{1/s}(\lambda_1(\mathcal{L}^*))$.
+
+**Theorem 4.1.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ and any $\varepsilon \in (0, 1/2)$,
+
+$$\eta_{\varepsilon}(\mathcal{L})^2 \le C_{\eta}(n)^2 \eta_{\det}(\mathcal{L})^2 + \frac{\log(1/\varepsilon)}{\pi \lambda_1(\mathcal{L}^*)^2} \le 100(\log n + 2)^2 \eta_{\det}(\mathcal{L})^2 + \frac{\log(1/\varepsilon)}{\pi \lambda_1(\mathcal{L}^*)^2}.$$

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+*Proof.* We may assume without loss of generality that $\eta_{\det}(\mathcal{L}) = 1$. Then, by definition, $\rho_{1/C_\eta(n)}(\mathcal{L}^*\setminus\{\mathbf{0}\}) \le 1/2$. Therefore, for any $s \ge C_\eta(n)$,
+
+$$
+\begin{align*}
+\rho_{1/s}(\mathcal{L}^*) &= 1 + \sum_{\mathbf{w} \in \mathcal{L}^* \setminus \{\mathbf{0}\}} \exp(-\pi(s^2 - C_\eta(n)^2) \|w\|^2) \rho_{1/C_\eta(n)}(\mathbf{w}) \\
+&\le 1 + \sum_{\mathbf{w} \in \mathcal{L}^* \setminus \{\mathbf{0}\}} \exp(-\pi(s^2 - C_\eta(n)^2) \lambda_1(\mathcal{L}^*)^2) \rho_{1/C_\eta(n)}(\mathbf{w}) \\
+&\le 1 + \exp(-\pi(s^2 - C_\eta(n)^2) \lambda_1(\mathcal{L}^*)^2)/2,
+\end{align*}
+$$
+
+and the result follows.
+
+Next, we prove an easy lower bound with a similar form (by taking the average of two trivial lower bounds).
+
+**Lemma 4.2.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ and any $\varepsilon \in (0, 1/2)$,
+
+$$
+\eta_{\varepsilon}(\mathcal{L})^2 \geq \eta_{\det}(\mathcal{L})^2/8 + \frac{\log(2/\varepsilon)}{2\pi\lambda_1(\mathcal{L}^*)^2}.
+$$
+
+*Proof.* First, note that $\rho_{1/s}(\mathcal{L}^* \setminus \{\mathbf{0}\}) \ge 2\rho_{1/s}(\lambda_1(\mathcal{L}^*))$. Rearranging, we see that
+
+$$
+\eta_{\epsilon}(\mathcal{L})^2 \geq \frac{\log(2/\epsilon)}{\pi \lambda_1 (\mathcal{L}^*)^2}.
+$$
+
+On the other hand, recall that for any lattice projection $\pi$ onto a subspace $W$, $\det(\mathcal{L}^* \cap W) = 1/\det(\pi(\mathcal{L}))$. I.e., $\eta_{\det}(\mathcal{L}) = \max_{\mathcal{L}' \subseteq \mathcal{L}^*} \det(\mathcal{L}')^{-1/\rank(\mathcal{L}')}$. So, suppose $s \le \eta_{\det}(\mathcal{L})/2$. Then, by Lemma 2.5,
+
+$$
+\rho_{1/s}(\mathcal{L}^*) = \max_{\mathcal{L}' \subseteq \mathcal{L}^*} \rho_{1/s}(\mathcal{L}') \geq \max_{\mathcal{L}' \subseteq \mathcal{L}^*} s^{-\operatorname{rank}(\mathcal{L}') / \det(\mathcal{L}')} \geq 2.
+$$
+
+So, $\eta_\epsilon(\mathcal{L})^2 \geq \eta_1(\mathcal{L})^2 \geq \eta_{\det}(\mathcal{L})^2/4$. The result follows by taking the average of the two bounds. $\square$
+
+The main theorem of this section now follows immediately.
+
+**Theorem 4.3.** For any $\varepsilon \in (0, 1/2)$, $\gamma$-GapSPP$_\varepsilon$ is in coNP for $\gamma = O(C_\eta(n)) \le O(\log n)$.
+
+*Proof.* Let $\gamma := 2\sqrt{2C_\eta(n)}$. On input a lattice $\mathcal{L} \subset \mathbb{R}^n$, the prover simply sends a lattice projection $\pi$ with $\det(\pi(\mathcal{L}))^{1/\rank(\pi(\mathcal{L}))} = \eta_{\det}(\mathcal{L})$ and a vector $\mathbf{w} \in \mathcal{L}^*$ with $\|\mathbf{w}\| = \lambda_1(\mathcal{L}^*)$. The verifier checks that $\pi$ is indeed a lattice projection and that $\mathbf{w} \in \mathcal{L}^* \setminus \{\mathbf{0}\}$. It then answers NO if and only if
+
+$$
+\gamma^2 \det(\pi(\mathcal{L}))^{2/\rank(\pi(\mathcal{L}))}/8 + \frac{\log(1/\varepsilon)}{\pi \|\mathbf{w}\|^2} > \gamma^2. \quad (4.1)
+$$
+
+To prove completeness, suppose that $\eta_\epsilon(\mathcal{L}) > \gamma$. Then, by Theorem 4.1,
+
+$$
+\gamma^2 \eta_{\det}(\mathcal{L})^2 / 8 + \frac{\log(1/\epsilon)}{\pi \lambda_1 (\mathcal{L}^*)^2} \geq \eta_\epsilon(\mathcal{L})^2 > \gamma^2.
+$$
+
+I.e., there exists a valid proof, as needed.

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+To prove soundness, suppose that $\eta_\varepsilon(\mathcal{L}) \le 1$. Then, by Lemma 4.2,
+
+$$
+\eta_{\det}(\mathcal{L})^2/8 + \frac{\log(1/\varepsilon)}{2\pi\lambda_1(\mathcal{L}^{*})^2} \leq \eta_{\varepsilon}(\mathcal{L})^2 \leq 1.
+$$
+
+Therefore,
+
+$$
+\begin{align*}
+\gamma^2 \eta_{\det}(\mathcal{L})^2 / 8 + \frac{\log(1/\varepsilon)}{\pi \lambda_1 (\mathcal{L}^*)^2} &\le \frac{\gamma^2 \eta_{\det}(\mathcal{L})^2 / 8 + \frac{\log(1/\varepsilon)}{\pi \lambda_1 (\mathcal{L}^*)^2}}{\eta_{\det}(\mathcal{L})^2 / 8 + \frac{\log(1/\varepsilon)}{2\pi \lambda_1 (\mathcal{L}^*)^2}} \\
+&\le \max\{\gamma^2, 2\} \\
+&\le \gamma^2.
+\end{align*}
+$$
+
+In other words, Equation (4.1) cannot hold for any pair $w \in \mathcal{L}^* \setminus \{0\}$ and lattice projection $\pi$. I.e., the
+verifier will always answer YES, as needed. $\square$
+
+Finally, we derive the following corollary.
+
+**Corollary 4.4.** For any $\epsilon \in (0, 1/2)$, $\gamma$-coGapSPP$_\epsilon$ has an SZK proof system with an efficient prover for
+
+$$
+\gamma := O(C_{\eta}(n) + \sqrt{\log(1/\epsilon)/\log n}) \le O(\log n + \sqrt{\log(1/\epsilon)/\log n}).
+$$
+
+*Proof.* By Theorem 2.17, $\gamma$-GapSPP$_\epsilon$ is in SZK. Since SZK is closed under complements [SV97, Oka96], $\gamma$-coGapSPP$_\epsilon$ is in SZK as well. By Theorem 4.3, $\gamma$-coGapSPP$_\epsilon$ is in NP. The result then follows by the fact that any language in SZK $\cap$ NP has an SZK proof system with an efficient prover [NV06]. $\square$
+
+5 An SZK Proof for $O(\sqrt{n})$-GapCRP
+
+In this section we prove that $O(\sqrt{n})$-GapCRP is in SZK, which improves the previous known result by a
+$\omega(\sqrt{\log n})$ factor [PV08]. First we need the following result from [CDLP13].
+
+**Lemma 5.1.** For any lattice $\mathcal{L}$ and parameter $s > 0$,
+
+$$
+\rho_s(\mathcal{L}) \cdot \gamma_s(\mathcal{V}(\mathcal{L})) \leq 1.
+$$
+
+Here we prove an upper bound on the smoothing parameter of a lattice in terms of its covering radius. This
+bound is implicit in [DR16].
+
+**Lemma 5.2.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ and $\epsilon > 0$, we have
+
+$$
+\eta_{\epsilon}(\mathcal{L}) \leq \sqrt{\frac{\pi}{\log(1+\epsilon)}} \cdot \mu(\mathcal{L}).
+$$
+
+In particular, $\eta_\epsilon(\mathcal{L}) \le O(\mu(\mathcal{L}))$ for any $\epsilon \ge \Omega(1)$.

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+*Proof.*
+
+$$
+\begin{align*}
+\rho_{1/s}(\mathcal{L}^*) &= s^{-n} \cdot \det(\mathcal{L}) \cdot \rho_s(\mathcal{L}) && (\text{Lemma 2.5}) \\
+&\le \frac{s^{-n} \cdot \det(\mathcal{L})}{\gamma_s(\nu(\mathcal{L}))} && (\text{Lemma 5.1}) \\
+&\le \frac{s^{-n} \cdot \det(\mathcal{L})}{\int_{\nu(\mathcal{L})} s^{-n} \exp(-\pi \mathbf{x}^2/s^2) \, d\mathbf{x}} \\
+&\le \frac{\det(\mathcal{L})}{\int_{\nu(\mathcal{L})} \exp(-\pi \mu(\mathcal{L})^2/s^2) \, d\mathbf{x}} \\
+&\le \exp(\pi \mu(\mathcal{L})^2/s^2),
+\end{align*}
+$$
+
+where we used the fact that $\|\mathbf{x}\| \le \mu(\mathcal{L})$ for any $\mathbf{x} \in \nu(\mathcal{L})$. By setting $s = \sqrt{\frac{\pi}{\log(1+\epsilon)}} \cdot \mu(\mathcal{L})$ we have the desired result. $\square$
+
+**Theorem 5.3.** The problem $O(\sqrt{n})$-GapCRP has an SZK proof system with an efficient prover, as does $O(\sqrt{n})$-coGapCRP.
+
+*Proof.* Fix some some constant $\epsilon \in (0, 1/2)$. By Lemma 2.13 and Lemma 5.2, we know that there exist $C_1$ and $C_2$ such that
+
+$$ C_1\eta_\varepsilon(\mathcal{L}) \leq \mu(\mathcal{L}) \leq C_2\sqrt{n} \cdot \eta_\varepsilon(\mathcal{L}), $$
+
+and hence there is a simple reduction from $O(\sqrt{n})$-GapCRP to $O(1)$-GapSPP$_\varepsilon$. It follows from Theorem 2.17 that $O(\sqrt{n})$-GapCRP is in SZK. To see that the prover can be made efficient, we recall from [GMR04] that $O(\sqrt{n})$-GapCRP is in NP $\cap$ coNP. The result then follows by the fact that any language in SZK $\cap$ NP has an SZK proof system with an efficient prover [NV06]. $\square$
+
+## References
+
+[ADRS15] D. Aggarwal, D. Dadush, O. Regev, and N. Stephens-Davidowitz. Solving the shortest vector problem in $2^n$ time using discrete Gaussian sampling. In *STOC*, pages 733–742. 2015.
+
+[ADS15] D. Aggarwal, D. Dadush, and N. Stephens-Davidowitz. Solving the closest vector problem in $2^n$ time - the discrete Gaussian strikes again! In *FOCS*, pages 563–582. 2015.
+
+[Ajt96] M. Ajtai. Generating hard instances of lattice problems. *Quaderni di Matematica*, 13:1–32, 2004. Preliminary version in STOC 1996.
+
+[AR04] D. Aharonov and O. Regev. Lattice problems in NP $\cap$ coNP. *J. ACM*, 52(5):749–765, 2005. Preliminary version in FOCS 2004.
+
+[Bab85a] L. Babai. Trading group theory for randomness. In *STOC*, pages 421–429. 1985.
+
+[Bab85b] L. Babai. On Lovász' lattice reduction and the nearest lattice point problem. *Combinatorica*, 6(1):1–13, 1986. Preliminary version in STACS 1985.
+
+[Ban93] W. Banaszczyk. New bounds in some transference theorems in the geometry of numbers. *Mathematische Annalen*, 296(4):625–635, 1993.

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+# 1 Introduction
+
+Informally, a *proof system* [GMR85, Bab85a] is a protocol that allows a (possibly unbounded and malicious) prover to convince a skeptical verifier of the truth of some statement. A proof system is *zero knowledge* if the verifier “learns nothing more” from the interaction, other than the statement’s veracity. The system is said to be *statistical zero knowledge* if the revealed information is negligible, even to an unbounded verifier; the class of problems having such proof systems is called SZK. Since their introduction, proof systems and zero-knowledge have found innumerable applications in cryptography and complexity theory. As a few examples, they have been used in constructions of secure multiparty computation [GMW87], digital signatures [BG89], actively secure public-key encryption [NY90], and “ZAPs” [DN00]. And if a problem has an SZK (or even coAM) proof, it is not NP-hard unless the polynomial-time hierarchy collapses [BHZ87], so interactive proofs have been used as evidence against NP-hardness; see, e.g., [GMR85, GMW91, GG98, HR14].
+
+A proof system is *noninteractive* [BDMP88, GSV99] if it consists of just one message from the prover, assuming both it and the verifier have access to a truly random string. Noninteractive statistical zero-knowledge (NISZK) proof systems are especially powerful cryptographic primitives: they have minimal message complexity; they are concurrently and even “universally” composable [Can01]; and their security holds against unbounded malicious provers and verifiers, without any computational assumptions. However, we do not understand the class NISZK of problems that have noninteractive statistical zero-knowledge proof systems nearly as well as SZK. In particular, while NISZK is known to have complete problems, it is not known whether it is closed under complement or disjunction [GSV99], unlike SZK [SV97, Oka96].
+
+**Lattices and proofs.** An $n$-dimensional lattice is a (full-rank) discrete additive subgroup of $\mathbb{R}^n$, and consists of all integer linear combinations of some linearly independent vectors $\mathbf{B} = \{\mathbf{b}_1, \dots, \mathbf{b}_n\}$, called a *basis* of the lattice. Lattices have been extensively studied in computer science, and lend themselves to many natural computational problems. Perhaps the most well-known of these are the *Shortest Vector Problem* (SVP), which is to find a shortest nonzero vector in a given lattice, and the *Closest Vector Problem* (CVP), which is to find a lattice point that is closest to a given vector in $\mathbb{R}^n$. Algorithms for these problems and their approximation versions have many applications in computer science; see, e.g., [LLL82, Len83, Kan83, Odl90, JS98, NS01, DPV11]. In addition, many cryptographic primitives, ranging from public-key encryption and signatures to fully homomorphic encryption, are known to be secure assuming the (worst-case) hardness of certain lattice problems (see, e.g., [MR04, Reg05, GPV08, Pei09, BV11, BGV12]).
+
+Due to the importance of lattices in cryptography, proof systems and zero-knowledge protocols for lattice problems have received a good deal of attention. Early on, Goldreich and Goldwasser [GG98] showed that for $\gamma = O(\sqrt{n}/\log n)$, the $\gamma$-approximate Shortest and Closest Vector Problems, respectively denoted $\gamma$-GapSVP and $\gamma$-GapCVP, have SZK proof systems; this was later improved to coNP for $\gamma = O(\sqrt{n})$ factors [AR04].¹ Subsequently, Micciancio and Vadhan [MV03] gave different SZK proofs for the same problems, where the provers are *efficient* when given appropriate witnesses; this is obviously an important property if the proof systems are to be used by real entities as components of other protocols. Peikert and Vaikuntanathan [PV08] gave the first *noninteractive* statistical zero-knowledge proof systems for certain lattice problems, showing that, for example, $O(\sqrt{n})$-coGapSVP has an NISZK proof. The proof systems from [PV08] also have efficient provers, although for larger $\tilde{O}(n)$ approximation factors.
+
+¹As described, the proofs from [GG98] are statistical zero knowledge against only honest verifiers, but any such proof can unconditionally be transformed to one that is statistical zero knowledge against malicious verifiers [GSV98]). We therefore ignore the distinction for the remainder of the paper.

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+[Ban95] W. Banaszczyk. Inequalites for convex bodies and polar reciprocal lattices in $\mathbb{R}^n$. *Discrete & Computational Geometry*, 13:217–231, 1995.
+
+[BDMP88] M. Blum, A. De Santis, S. Micali, and G. Persiano. Noninteractive zero-knowledge. *SIAM J. Comput.*, 20(6):1084–1118, 1991. Preliminary version in STOC 1988.
+
+[BG89] M. Bellare and S. Goldwasser. New paradigms for digital signatures and message authentication based on non-iterative zero knowledge proofs. In *CRYPTO*, pages 194–211. 1989.
+
+[BGV12] Z. Brakerski, C. Gentry, and V. Vaikuntanathan. (Leveled) fully homomorphic encryption without bootstrapping. *TOCT*, 6(3):13, 2014. Preliminary version in ITCS 2012.
+
+[BHZ87] R. B. Boppana, J. Håstad, and S. Zachos. Does co-NP have short interactive proofs? *Inf. Process. Lett.*, 25(2):127–132, 1987.
+
+[BLP<sup>+</sup>13] Z. Brakerski, A. Langlois, C. Peikert, O. Regev, and D. Stehlé. Classical hardness of learning with errors. In *STOC*, pages 575–584. 2013.
+
+[BV11] Z. Brakerski and V. Vaikuntanathan. Efficient fully homomorphic encryption from (standard) LWE. *SIAM J. Comput.*, 43(2):831–871, 2014. Preliminary version in FOCS 2011.
+
+[Can01] R. Canetti. Universally composable security: A new paradigm for cryptographic protocols. In *FOCS*, pages 136–145. 2001.
+
+[CDLP13] K. Chung, D. Dadush, F. Liu, and C. Peikert. On the lattice smoothing parameter problem. In *IEEE Conference on Computational Complexity*, pages 230–241. 2013.
+
+[DN00] C. Dwork and M. Naor. Zaps and their applications. *SIAM J. Comput.*, 36(6):1513–1543, 2007.
+
+[DPV11] D. Dadush, C. Peikert, and S. Vempala. Enumerative lattice algorithms in any norm via M-ellipsoid coverings. In *FOCS*, pages 580–589. 2011.
+
+[DR16] D. Dadush and O. Regev. Towards strong reverse Minkowski-type inequalities for lattices. In *FOCS*, pages 447–456. 2016.
+
+[GG98] O. Goldreich and S. Goldwasser. On the limits of nonapproximability of lattice problems. *J. Comput. Syst. Sci.*, 60(3):540–563, 2000. Preliminary version in STOC 1998.
+
+[GMR85] S. Goldwasser, S. Micali, and C. Rackoff. The knowledge complexity of interactive proof systems. *SIAM J. Comput.*, 18(1):186–208, 1989. Preliminary version in STOC 1985.
+
+[GMR04] V. Guruswami, D. Micciancio, and O. Regev. The complexity of the covering radius problem. *Computational Complexity*, 14(2):90–121, 2005. Preliminary version in CCC 2004.
+
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+
+[GMW91] O. Goldreich, S. Micali, and A. Wigderson. Proofs that yield nothing but their validity for all languages in NP have zero-knowledge proof systems. *J. ACM*, 38(3):691–729, 1991.
+
+[GPV08] C. Gentry, C. Peikert, and V. Vaikuntanathan. Trapdoors for hard lattices and new cryptographic constructions. In *STOC*, pages 197–206. 2008.

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+[GSV98] O. Goldreich, A. Sahai, and S. P. Vadhan. Honest-verifier statistical zero-knowledge equals general statistical zero-knowledge. In *STOC*, pages 399–408. 1998.
+
+[GSV99] O. Goldreich, A. Sahai, and S. P. Vadhan. Can statistical zero knowledge be made non-interactive? or on the relationship of SZK and NISZK. In *CRYPTO*, pages 467–484. 1999.
+
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+
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+
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+
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+
+[Kle00] P. N. Klein. Finding the closest lattice vector when it’s unusually close. In *SODA*, pages 937–941. 2000.
+
+[Len83] H. W. Lenstra. Integer programming with a fixed number of variables. *Mathematics of Operations Research*, 8(4):538–548, November 1983.
+
+[LLL82] A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. *Mathematische Annalen*, 261(4):515–534, December 1982.
+
+[MG02] D. Micciancio and S. Goldwasser. *Complexity of Lattice Problems: a cryptographic perspective*, volume 671 of *The Kluwer International Series in Engineering and Computer Science*. Kluwer Academic Publishers, Boston, Massachusetts, 2002.
+
+[MP12] D. Micciancio and C. Peikert. Trapdoors for lattices: Simpler, tighter, faster, smaller. In *EUROCRYPT*, pages 700–718. 2012.
+
+[MR04] D. Micciancio and O. Regev. Worst-case to average-case reductions based on Gaussian measures. *SIAM J. Comput.*, 37(1):267–302, 2007. Preliminary version in FOCS 2004.
+
+[MV03] D. Micciancio and S. P. Vadhan. Statistical zero-knowledge proofs with efficient provers: Lattice problems and more. In *CRYPTO*, pages 282–298. 2003.
+
+[NS01] P. Q. Nguyen and J. Stern. The two faces of lattices in cryptology. In *CaLC*, pages 146–180. 2001.
+
+[NV06] M. Nguyen and S. P. Vadhan. Zero knowledge with efficient provers. In *STOC*, pages 287–295. 2006.
+
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+
+[Odl90] A. M. Odlyzko. The rise and fall of knapsack cryptosystems. In C. Pomerance, editor, *Cryptography and Computational Number Theory*, volume 42 of *Proceedings of Symposia in Applied Mathematics*, pages 75–88. 1990.

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+[Oka96] T. Okamoto. On relationships between statistical zero-knowledge proofs. *J. Comput. Syst. Sci.*, 60(1):47–108, 2000. Preliminary version in STOC 1996.
+
+[Pei09] C. Peikert. Public-key cryptosystems from the worst-case shortest vector problem. In *STOC*, pages 333–342. 2009.
+
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+
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+
+[RS17] O. Regev and N. Stephens-Davidowitz. A reverse Minkowski theorem. In *STOC*, pages 941–953. 2017.
+
+[SV97] A. Sahai and S. P. Vadhan. A complete problem for statistical zero knowledge. *J. ACM*, 50(2):196–249, 2003. Preliminary version in FOCS 1997.
+
+[Ver12] R. Vershynin. *Introduction to the non-asymptotic analysis of random matrices*, chapter 5, pages 210–268. Cambridge University Press, 2012. Available at http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf.
+
+# A Proof of Lemma 3.3
+
+**Definition A.1.** For any $\delta > 0$, $S \subseteq \mathbb{R}^n$, we say that $A \subseteq S$ is a $\delta$-net of $S$ if for each $\mathbf{v} \in S$, there is some $\mathbf{u} \in A$ such that $\|\mathbf{u} - \mathbf{v}\| \le \delta$.
+
+**Lemma A.2.** For any $\delta > 0$, there exists a $\delta$-net of the unit sphere in $\mathbb{R}^n$ with at most $(1 + 2/\delta)^n$ points.
+
+*Proof.* Let $N$ be maximal such that $N$ points can be placed on the unit sphere in such a way that no pair of points is within distance $\delta$ of each other. Clearly, there exists a $\delta$-net of size $N$.
+
+So, it suffices to show that any collection of vectors $A$ in the unit sphere with $|A| > (1 + 2/\delta)^n$ must contain two points within distance $\delta$ of each other. Let
+
+$$ B := \bigcup_{\mathbf{u} \in A} ((\delta/2)B_2^n + \mathbf{u}) $$
+
+be the union of balls of radius $\delta/2$ centered at each point in $A$. Notice that $B \subseteq (1 + \delta/2)B_2^n$. If all of these balls were disjoint, then we would have
+
+$$ \mathrm{vol}(B_2^n) = |A| \cdot (\delta/2)^n \mathrm{vol}(B_2^n) > \mathrm{vol}((1 + \delta/2)B_2^n), $$
+
+a contradiction. Therefore, two such balls must overlap. I.e., there must be two points within distance $\delta$ of each other, as needed. $\square$
+
+We will need the following result from [Ver12, Lemma 5.4].
+
+**Lemma A.3.** For a symmetric matrix $M \in \mathbb{R}^{n \times n}$ and a $\delta$-net of the unit sphere $A$ with $\delta \in (0, 1/2)$,
+
+$$ \|M\| \le \frac{1}{1 - 2\delta} \cdot \max_{\mathbf{v} \in A} |\langle M\mathbf{v}, \mathbf{v}\rangle|. $$

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+We will also need the following result from [MP12, Lemma 2.8], which shows that the discrete Gaussian distribution is subgaussian.
+
+**Lemma A.4.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ with $\eta_{1/2}(\mathcal{L}) \le 1$, shift vector $\mathbf{t} \in \mathbb{R}^n$, unit vector $\mathbf{v} \in \mathbb{R}^n$, and any $r > 0$,
+
+$$ \Pr_{\mathbf{X} \sim D_{\mathcal{L}-\mathbf{t}}} [|\langle \mathbf{v}, \mathbf{X} \rangle| \ge r] \le 10 \exp(-\pi r^2). $$
+
+*Proof of Lemma 3.3.* Let $\{\mathbf{v}_1, \dots, \mathbf{v}_N\}$ be a (1/10)-net of the unit sphere with $N \le 25^n$, as guaranteed by Lemma A.2. By Lemma A.4, we have that for any $\mathbf{e}_i$ in the proof, any $\mathbf{v}_j$, and any $r \ge 0$, $\Pr[|\langle \mathbf{v}_j, \mathbf{e}_i \rangle| \ge r] \le 10 \exp(-\pi r^2)$. Therefore, by Lemma 2.20
+
+$$ \Pr \left[ \sum_i |\langle \mathbf{v}_j, \mathbf{e}_i \rangle|^2 \geq r \right] \leq 2^m e^{-\pi r/2}. $$
+
+Applying the union bound, we have
+
+$$ \Pr[\exists j, \sum_i |\langle \mathbf{v}_j, \mathbf{e}_i \rangle|^2 \geq r] \leq N 2^m e^{-\pi r/2}. $$
+
+Taking $r := 2m$, we see that this probability is negligible. Applying Lemma A.3 shows that
+
+$$ \left\| \sum_i \mathbf{e}_i \mathbf{e}_i^T \right\| \le 2m \cdot \frac{5}{4} < 3m, $$
+
+except with negligible probability, as needed. $\square$

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+**Gaussians and the smoothing parameter.** Gaussian measures have become an increasingly important tool in the study of lattices. For $s > 0$, the Gaussian measure of parameter (or width) $s$ on $\mathbb{R}^n$ is defined as $\rho_s(\mathbf{x}) = \exp(-\pi\|\mathbf{x}\|^2/s^2)$; for a lattice $\mathcal{L} \subset \mathbb{R}^n$, the Gaussian measure of the lattice is then
+
+$$\rho_s(\mathcal{L}) := \sum_{\mathbf{v} \in \mathcal{L}} \rho_s(\mathbf{v}).$$
+
+Gaussian measures on lattices have innumerable applications, including in worst-case to average-case reductions for lattice problems [MR04, Reg05], the construction of cryptographic primitives [GPV08], the design of algorithms for SVP and CVP [ADRS15, ADS15], and the study of the geometry of lattices [Ban93, Ban95, DR16, RS17].
+
+In all of the above applications, a key quantity is the lattice *smoothing parameter* [MR04]. Informally, for a parameter $\epsilon > 0$ and a lattice $\mathcal{L}$, the smoothing parameter $\eta_\epsilon(\mathcal{L})$ is the minimal Gaussian parameter that “smooths out” the discrete structure of $\mathcal{L}$, up to error $\epsilon$. Formally, for $\epsilon > 0$ we define
+
+$$\eta_{\epsilon}(\mathcal{L}) := \min\{s > 0 : \rho_{1/s}(\mathcal{L}^{*}) \le 1 + \epsilon\},$$
+
+where $\mathcal{L}^* := \{\mathbf{w} \in \mathbb{R}^n : \forall \mathbf{y} \in \mathcal{L}, \langle \mathbf{w}, \mathbf{y} \rangle \in \mathbb{Z}\}$ is the dual lattice of $\mathcal{L}$. All of the computational applications from the previous paragraph rely in some way on the “smoothness” of the Gaussian with parameter $s \ge \eta_\epsilon(\mathcal{L})$ where $2^{-n} \ll \epsilon < 1/2$.² For example, several of the proof systems from [PV08] start with deterministic reductions to an intermediate problem, which asks whether a lattice is “smooth” or well-separated.
+
+**The GapSPP problem.** Given the prominence of the smoothing parameter in the theory of lattices, it is natural to ask about the complexity of computing it. Chung et al. [CDLP13] formally defined the problem $\gamma$-GapSPP$_\epsilon$ of approximating the smoothing parameter $\eta_\epsilon(\mathcal{L})$ to within a factor of $\gamma \ge 1$ and gave upper bounds on its complexity in the form of proof systems for remarkably low values of $\gamma$. For example, they showed that $\gamma$-GapSPP$_\epsilon \in$ SZK for $\gamma = O(1 + \sqrt{\log(1/\epsilon)/\log n})$. This in fact subsumes the prior result that $O(\sqrt{n/\log n})$-GapSVP $\in$ SZK of [GG98], via known relationships between the minimum distance and the smoothing parameter.
+
+Chung et al. also showed a worst-case to average-case (quantum) reduction from $\tilde{O}(\sqrt{n}/\alpha)$-GapSPP to a very important average-case problem in lattice-based cryptography, Regev’s Learning With Errors (LWE), which asks us to decode from a random “q-ary” lattice under error proportional to $\alpha$ [Reg05]. Again, this subsumes the prior best reduction for GapSVP due to Regev. Most recently, Dadush and Regev [DR16] showed a similar worst-case to average-case reduction from GapSPP to the Short Integer Solution problem [Ajt96, MR04], another widely used average-case problem in lattice-based cryptography.
+
+In hindsight, the proof systems and reductions of [GG98, Reg05, MR04] can most naturally be viewed as applying to GapSPP all along. This suggests that GapSPP may be a better problem than GapSVP on which to base the security of lattice-based cryptography. However, both [CDLP13] and [DR16] left open several questions and asked for a better understanding of the complexity of GapSPP. In particular, while interactive proof systems for this problem seem to be relatively well understood, nothing nontrivial was previously known about noninteractive proof systems (whether zero knowledge or not) for this problem.
+
+²For $\epsilon = 2^{-\Omega(n)}$ the smoothing parameter is determined (up to a constant factor) by the dual minimum distance, so it is much less interesting to consider as a separate quantity. The upper bound of $1/2$ could be replaced by any constant less than one. For $\epsilon \ge 1$, $\eta_\epsilon(\mathcal{L})$ is still formally defined, but its interpretation in terms of the “smoothness” of the corresponding Gaussian measure over $\mathcal{L}$ is much less clear.

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+## 1.1 Our Results
+
+In this work we give new proof systems for lattice problems, and extend the reach of prior proof systems to new problems. Our new results, and how they compare to the previous state of the art, are as follows.
+
+Our first main result is a NISZK proof system for $\gamma$-GapSPP$_\epsilon$ with $\gamma = O(\log(n)\sqrt{\log(1/\epsilon)})$. This improves, by a $\Theta(\sqrt{n}/\log n)$ factor, upon the previous best approximation factor of $\gamma = O(\sqrt{n\log(1/\epsilon)})$, which follows from [PV08].
+
+**Theorem 1.1.** For any $\epsilon \in (0, 1/2)$, $O(\log(n)\sqrt{\log(1/\epsilon)})$-GapSPP$_\epsilon \in$ NISZK.
+
+In fact, we demonstrate two different proof systems to establish this theorem (see Section 3). The first is identical to a proof system from [PV08], but with a very different analysis that relies on a recent geometric theorem of [RS17]. However, this proof system only works for negligible $\epsilon < n^{-\omega(1)}$, so we also show an alternative that works for any $\epsilon \in (0, 1/2)$ via reduction to the NISZK-complete *Entropy Approximation problem* [GSV99].
+
+The prover in the proof system from [PV08] can be made efficient at the expense of a factor of $O(\sqrt{n \log n})$ in the approximation factor. From this we obtain the following.
+
+**Theorem 1.2.** For any negligible $0 < \epsilon < n^{-\omega(1)}$, there is a NISZK proof system with an efficient prover for $O(\sqrt{n \log^3(n) \log(1/\epsilon)})$-GapSPP$_\epsilon$.
+
+Next, we show that $O(\log n)$-GapSPP$_\epsilon \in$ coNP for any $\epsilon \in (0, 1)$. This improves, again by up to a $\Theta(\sqrt{n}/\log n)$ factor, the previous best known result of $O(1 + \sqrt{n}/\log(1/\epsilon))$-GapSPP$_\epsilon \in$ coNP, which follows from [Ban93].
+
+**Theorem 1.3.** For any $\epsilon \in (0, 1/2)$, $O(\log n)$-GapSPP$_\epsilon \in$ coNP.
+
+From this, together with the SZK protocol of [CDLP13] and the result of Nguyen and Vadhan [NV06] that any problem in SZK $\cap$ NP has an SZK proof system with an efficient prover, we obtain the following corollary. (The proof systems in [CDLP13] do not have efficient provers.)
+
+**Corollary 1.4.** For any $\epsilon \in (0, 1/2)$, there is an SZK proof system with an efficient prover for $O(\log n + \sqrt{\log(1/\epsilon)/\log n})$-coGapSPP$_\epsilon$.
+
+Finally, we observe that $O(\sqrt{n})$-GapCRP $\in$ SZK, where GapCRP is the problem of approximating the covering radius, i.e., the maximum possible distance from a given lattice. For comparison, the previous best approximation factor was from [PV08], who showed that $\gamma$-GapCRP $\in$ NISZK $\subseteq$ SZK for any $\gamma = \omega(\sqrt{n}\log n)$. We obtain this result via a straightforward reduction to $O(1)$-GapSPP$_\epsilon$ for constant $\epsilon < 1/2$, which, to recall, is in SZK [CDLP13]. Furthermore, since Guruswami, Micciancio, and Regev showed that $O(\sqrt{n})$-GapCRP $\in$ NP $\cap$ coNP [GMR04], it follows that the protocol can be made efficient.
+
+**Theorem 1.5.** We have $O(\sqrt{n})$-GapCRP $\in$ SZK. Furthermore, $O(\sqrt{n})$-GapCRP and $O(\sqrt{n})$-coGapCRP each have an SZK proof system with an efficient prover.
+
+## 1.2 Techniques
+
+**Sparse projections.** Our main technical tool will be sparse lattice projections. In particular, we use the determinant of a lattice, defined as $\det(\mathcal{L}) := |\det(\mathbf{B})|$ for any basis $\mathbf{B}$ of $\mathcal{L}$, as our measure of sparsity.³ It
+
+³This is indeed a measure of sparsity because $1/\det(\mathcal{L})$ is the average number of lattice points inside a random shift of any unit-volume body, or equivalently, the limit as $r$ goes to infinity of the number of lattice points per unit volume in a ball of radius $r$.

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+is an immediate consequence of the Poisson Summation Formula (Lemma 2.5) that $\det(\mathcal{L})^{1/n} \le 2\eta_{1/2}(\mathcal{L})$. Notice that this inequality formalizes the intuitive notion that “a lattice cannot be smooth and sparse simultaneously.”
+
+Dadush and Regev made the simple observation that the same statement is true when we consider projections of the lattice [DR16]. I.e., for any projection $\pi$ such that $\pi(\mathcal{L})$ is still a lattice, we have $\det(\pi(\mathcal{L}))^{1/\operatorname{rank}(\pi(\mathcal{L}))} \le 2\eta_{1/2}(\mathcal{L})$, where $\operatorname{rank}(\pi(\mathcal{L}))$ is the dimension of the span of $\pi(\mathcal{L})$. (Indeed, this fact is immediate from the above together with the identity $(\pi(\mathcal{L}))^* = \mathcal{L}^* \cap \operatorname{span}(\pi(\mathcal{L}))$.) Therefore, if we define
+
+$$ \eta_{\det}(\mathcal{L}) := \max_{\pi} \det(\pi(\mathcal{L}))^{1/\operatorname{rank}(\pi(\mathcal{L}))}, $$
+
+where the maximum is taken over all projections $\pi$ such that $\pi(\mathcal{L})$ is a lattice, then we have
+
+$$ \eta_{\det}(\mathcal{L}) \le 2\eta_{1/2}(\mathcal{L}). \quad (1.1) $$
+
+Dadush and Regev conjectured that Equation (1.1) is tight up to a factor of polylog(n). I.e., up to polylog factors, a lattice is not smooth if and only if it has a sparse projection. Regev and Stephens-Davidowitz proved this conjecture [RS17], and the resulting theorem, presented below, will be our main technical tool.
+
+**Theorem 1.6 ([RS17]).** For any lattice $\mathcal{L} \subset \mathbb{R}^n$,
+
+$$ \eta_{1/2}(\mathcal{L}) \le 10(\log n + 2)\eta_{\det}(\mathcal{L}). $$
+
+I.e., if $\eta_{1/2}(\mathcal{L}) \ge 10(\log n + 2)$, then there exists a lattice projection $\pi$ such that $\det(\pi(\mathcal{L})) \ge 1$.
+
+**coNP proof system.** Notice that Theorem 1.6 (together with Equation (1.1)) immediately implies that $O(\log n)$-GapSPP$_\epsilon$ is in coNP for $\epsilon = 1/2$. Indeed, a projection $\pi$ such that $\det(\pi(\mathcal{L}))^{1/\operatorname{rank}(\pi(\mathcal{L}))} \ge \eta_{1/2}(\mathcal{L})/O(\log n)$ can be used as a witness of “non-smoothness.” Theorem 1.6 shows that such a witness always exists, and Equation (1.1) shows that no such witness exists with $\det(\pi(\mathcal{L}))^{1/\operatorname{rank}(\pi(\mathcal{L}))} > 2\eta_{1/2}(\mathcal{L})$. In order to extend this result to all $\epsilon \in (0, 1)$, we use basic results about how $\eta_\epsilon(\mathcal{L})$ varies with $\epsilon$. (See Section 4.)
+
+**NISZK proof systems.** We give two different NISZK proof systems for $O(\log(n)\sqrt{\log(1/\epsilon)})$-GapSPP$_\epsilon$, both of which rely on Theorem 1.6.
+
+Our first proof system (shown in Figure 1, Section 3.1) uses many vectors $t_1, \dots, t_m$ sampled uniformly at random from a fundamental region of the lattice $\mathcal{L}$ as the common random string. The prover samples short vectors $e_i$ (for $i = 1, \dots, m$) from the discrete Gaussian distributions over the lattice cosets $e_i + \mathcal{L}$. The verifier accepts if and only if the matrix $E = \sum e_i e_i^T$ has small enough spectral norm. (I.e., the verifier accepts if the $e_i$ are “short in all directions.”) In fact, Peikert and Vaikuntanathan used the exact same proof system for the different lattice problem $O(\sqrt{n})$-coGapSVP, and their proofs of correctness and zero knowledge also apply to our setting. However, the proof of soundness is quite different: we show that, if the lattice has a sparse projection $\pi$, then $\operatorname{dist}(\pi(t_i), \pi(\mathcal{L}))$ will tend to be fairly large. It follows that $\sum ||\pi(e_i)||^2 = \operatorname{Tr}(\sum \pi(e_i)\pi(e_i)^T)$ will be fairly large with high probability, and therefore $\sum e_i e_i^T$ must have large spectral norm.
+
+Our second proof system follows from a reduction to the Entropy Approximation problem, which asks to estimate the entropy of the output distribution of a circuit on random input. Goldreich, Sahai, and Vadhan [GSV99] showed that Entropy Approximation is NISZK-complete, so that a problem is in NISZK if and only if it can be (Karp-)reduced to approximating the entropy of a circuit. If $\eta_\epsilon(\mathcal{L})$ is small, then we

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+know that a continuous Gaussian modulo the lattice will be very close to the uniform distribution, and so (a suitable discretization of) this distribution will have high entropy. On the other hand, if $\eta_\varepsilon(\mathcal{L})$ is large, then Theorem 1.6 says that most of the measure of a continuous Gaussian modulo the lattice lies in a low-volume subset of $\mathbb{R}^n/\mathcal{L}$, and so (a discretization of) this distribution must have low entropy.
+
+This second proof system works for a wider range of $\varepsilon$. In particular, the first proof system is only statistical zero knowledge when $\varepsilon$ is negligible in the input size, whereas the second proof system works for any $\varepsilon \in (0, 1/2)$.
+
+## 1.3 Organization
+
+The remainder of the paper is organized as follows.
+
+* In Section 2 we recall the necessary background on lattices, proof systems, and probability.
+
+* In Section 3 we give two different NISZK proof systems for $O(\log(n)\sqrt{\log(1/\varepsilon)})$-GapSPP$_\varepsilon$.
+
+* In Section 4 we give a coNP proof system for $O(\log n)$-GapSPP$_\varepsilon$.
+
+* In Section 5 we show that $O(\sqrt{n})$-GapCRP $\in$ SZK, via a simple reduction to $O(1)$-GapSPP$_{1/4}$.
+
+# 2 Preliminaries
+
+## 2.1 Notation
+
+For any positive integer $d$, $[\tilde{d}]$ denotes the set $\{1, \dots, d\}$. We use bold lower-case letters to denote vectors. We write matrices in capital letters. The $i$th component (column) of a vector **x** (matrix **X**) is written as $\mathbf{x}_i$ ($\mathbf{X}_i$). The function $\log$ denotes the natural logarithm unless otherwise specified. For $\mathbf{x} \in \mathbb{R}^n$, $\|\mathbf{x}\| := \sqrt{x_1^2 + x_2^2 + \cdots + x_n^2}$ is the Euclidean norm. For a matrix $\mathbf{A} \in \mathbb{R}^{n \times m}$, $\|\mathbf{A}\| := \max_{\|\mathbf{x}\|=1} \|\mathbf{A}\mathbf{x}\|$ is the operator norm.
+
+We write $rB_2^n$ for the $n$-dimensional Euclidean ball of radius $r$. A set $S \subseteq \mathbb{R}^n$ is said to be *symmetric* if $-S = S$. The distance from a point $\mathbf{x} \in \mathbb{R}^n$ to a set $S \subseteq \mathbb{R}^n$ is defined to be $\text{dist}(\mathbf{x}, S) = \inf_{s \in S} \text{dist}(\mathbf{x}, s)$. We write $S^\perp$ to denote the subspace of vectors orthogonal to $S$. For a set $S \subseteq \mathbb{R}^n$ and a point $\mathbf{x} \in \mathbb{R}^n$, $\pi_S(\mathbf{x})$ denotes the orthogonal projection of $\mathbf{x}$ onto $\text{span}(S)$. For sets $A, B \subseteq \mathbb{R}^n$, we denote their Minkowski sum by $A + B = \{\mathbf{a} + \mathbf{b} : \mathbf{a} \in A, \mathbf{b} \in B\}$. We extend a function $f$ to a countable set in the natural way by defining $f(A) := \sum_{\mathbf{a} \in A} f(a)$.
+
+Throughout the paper, we write $C$ for an arbitrary universal constant $C > 0$, whose value might change from one use to the next.
+
+## 2.2 Lattices
+
+Here we provide some backgrounds on lattices. An $n$-dimensional *lattice* $\mathcal{L} \subset \mathbb{R}^n$ of rank $d$ is the set of integer linear combinations of $d$ linearly independent vectors $\mathbf{B} := (\mathbf{b}_1, \dots, \mathbf{b}_d)$,
+
+$$ \mathcal{L} = \mathcal{L}(\mathbf{B}) = \left\{ \mathbf{B}\mathbf{z} = \sum_{i \in [\tilde{d}]} z_i \cdot \mathbf{b}_i : \mathbf{z} \in \mathbb{Z}^d \right\}. $$
+
+We usually work with full-rank lattices, where $d = n$. A *sublattice* $\mathcal{L}' \subseteq \mathcal{L}$ is an additive subgroup of $\mathcal{L}$. The *dual lattice* of $\mathcal{L}$, denoted by $\mathcal{L}^*$, is defined as the set
+
+$$ \mathcal{L}^* = \left\{ \mathbf{y} \in \mathbb{R}^n : \forall \mathbf{v} \in \mathcal{L}, \langle \mathbf{v}, \mathbf{y} \rangle \in \mathbb{Z} \right\} $$

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+of all integer vectors having integer inner products with all vectors in $\mathcal{L}$. It is easy to check that $(\mathcal{L}^*)^* = \mathcal{L}$ and that, if $\mathbf{B}$ is a basis for $\mathcal{L}$, then $\mathbf{B}^* = \mathbf{B}(\mathbf{B}^T\mathbf{B})^{-1}$ is a basis for $\mathcal{L}^*$. The fundamental parallelepiped of a lattice $\mathcal{L}$ with respect to basis $\mathbf{B}$ is the set
+
+$$ \mathcal{P}(\mathbf{B}) = \left\{ \sum_{i \in [d]} c_i \mathbf{b}_i : 0 \le c_i < 1 \right\}. $$
+
+It is easy to see that $\mathcal{P}(\mathbf{B})$ is a fundamental domain of $\mathcal{L}$. I.e., it tiles $\mathbb{R}^n$ with respect to $\mathcal{L}$. For any lattice $\mathcal{L}(\mathbf{B})$ and point $\mathbf{x} \in \mathbb{R}^n$, there exists a unique point $\mathbf{y} \in \mathcal{P}(\mathbf{B})$ such that $\mathbf{y} - \mathbf{x} \in \mathcal{L}(\mathbf{B})$. We denote this vector by $\mathbf{y} = \mathbf{x} \bmod \mathbf{B}$. Notice that $\mathbf{y}$ can be computed in polynomial time given $\mathbf{B}$ and $\mathbf{x}$. We sometimes write $\mathbf{x} \bmod \mathcal{L}$ when the specific fundamental domain is not important, and we write $\mathbb{R}^n/\mathcal{L}$ for an arbitrary fundamental domain.
+
+The determinant of a lattice $\mathcal{L}$, is defined to be $\det(\mathcal{L}) = \sqrt{\det(\mathbf{B}^T\mathbf{B})}$. It is easy to verify that the determinant does not depend on the choice of basis and that $\det(\mathcal{L})$ is the volume of any fundamental domain of $\mathcal{L}$.
+
+The minimum distance of a lattice $\mathcal{L}$, is the length of the shortest non-zero lattice vector,
+
+$$ \lambda_1(\mathcal{L}) := \min_{\mathbf{y} \in \mathcal{L} \setminus \{\mathbf{0}\}} \|\mathbf{y}\|. $$
+
+Similarly, we define
+
+$$ \lambda_n(\mathcal{L}) := \min_i \max_j \|\mathbf{y}_i\|, $$
+
+where the minimum is taken over linearly independent lattice vectors $\mathbf{y}_1, \dots, \mathbf{y}_n \in \mathcal{L}$. The covering radius of a lattice $\mathcal{L}$ is
+
+$$ \mu(\mathcal{L}) := \max_{\mathbf{t} \in \mathbb{R}^n} \mathrm{dist}(\mathbf{t}, \mathcal{L}). $$
+
+The Voronoi cell of a lattice $\mathcal{L}$ is the set
+
+$$ \nu(\mathcal{L}) := \{\mathbf{x} \in \mathbb{R}^n : \| \mathbf{t} \| \le \| \mathbf{y} - \mathbf{t} \|, \forall \mathbf{y} \in \mathcal{L} \setminus \{\mathbf{0}\}\} $$
+
+of vectors in $\mathbb{R}^n$ that are closer to **0** than any other point of $\mathcal{L}$. It is easy to check that $V(\mathcal{L})$ is a symmetric polytope and that it tiles $\mathbb{R}^n$ with respect to $\mathcal{L}$. The following claim is an immediate consequence of the fact that an $n$-dimensional unit ball has volume at most $(2\pi e/n)^{n/2}$.
+
+**Claim 2.1.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$,
+
+$$ \mu(\mathcal{L}) \geq \sqrt{n/(2\pi e)} \cdot \det(\mathcal{L})^{1/n}. $$
+
+**Lemma 2.2.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ and $r \ge 0$,
+
+$$ |\mathcal{L} \cap r B_2^n| \le (5/\sqrt{n})^n \cdot \frac{(r + \mu(\mathcal{L}))^n}{\det(\mathcal{L})}. $$
+
+*Proof.* For each vector $\mathbf{y} \in \mathcal{L} \cap rB_2^n$, notice that $\nu(\mathcal{L}) + \mathbf{y} \subseteq (r + \mu(\mathcal{L}))B_2^n$. And, for distinct vectors $\mathbf{y}, \mathbf{y}' \in \mathcal{L}, \nu(\mathcal{L}) + \mathbf{y}$ and $\nu(\mathcal{L}) + \mathbf{y}'$ are disjoint (up to a set of measure zero). Therefore,
+
+$$ \mathrm{vol}((r + \mu(\mathcal{L}))B_2^n) \geq \mathrm{vol}\left(\bigcup_{\mathbf{y} \in \mathcal{L} \cap rB_2^n} (\nu(\mathcal{L}) + \mathbf{y})\right) = |\mathcal{L} \cap rB_2^n| \mathrm{vol}(\nu(\mathcal{L})) = |\mathcal{L} \cap rB_2^n| \det(\mathcal{L}). $$
+
+The result follows by recalling that for any $r' > 0$, $\mathrm{vol}(r'B_2^n) \le (5r'/\sqrt{n})^n$. □

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+**Lemma 2.3 ([GMR04]).** For any lattice $\mathcal{L} \subset \mathbb{R}^n$,
+
+$$ \underset{\mathbf{t} \sim \mathbb{R}^n / \mathcal{L}}{\mathbb{E}} [\mathrm{dist}(\mathbf{t}, \mathcal{L})^2] \geq \mu(\mathcal{L})^2/4, $$
+
+where $\mathbf{t} \in \mathbb{R}^n/\mathcal{L}$ is sampled uniformly at random.
+
+*Proof.* Let $\mathbf{v} \in \mathbb{R}^n$ such that $\mathrm{dist}(\mathbf{v}, \mathcal{L}) = \mu(\mathcal{L})$. Notice that $\mathbf{v} - \mathbf{t} \bmod \mathcal{L}$ is uniformly distributed. And, by the triangle inequality, $\mathrm{dist}(\mathbf{v} - \mathbf{t}, \mathcal{L}) + \mathrm{dist}(\mathbf{t}, \mathcal{L}) \geq \mathrm{dist}(\mathbf{v}, \mathcal{L}) = \mu(\mathcal{L})$. So,
+
+$$ \underset{\mathbf{t} \sim \mathbb{R}^n / \mathcal{L}}{\mathbb{E}} [\mathrm{dist}(\mathbf{t}, \mathcal{L})] = \frac{1}{2} \cdot \underset{\mathbf{t} \sim \mathbb{R}^n / \mathcal{L}}{\mathbb{E}} [\mathrm{dist}(\mathbf{v} - \mathbf{t}, \mathcal{L}) + \mathrm{dist}(\mathbf{t}, \mathcal{L})] \geq \mu(\mathcal{L})/2. $$
+
+The result then follows by Markov's inequality. $\square$
+
+A *lattice projection* for a lattice $\mathcal{L} \subset \mathbb{R}^n$ is an orthogonal projection $\pi : \mathbb{R}^n \to \mathbb{R}^n$ defined by
+$\pi(\mathbf{x}) := \pi_{S^\perp}(\mathbf{x})$ for lattice vectors $S \subset \mathcal{L}$.
+
+**Claim 2.4.** For any $\mathcal{L} \subset \mathbb{R}^n$ and any lattice projection $\pi$, $\pi(\mathcal{L})$ is a lattice. Furthermore, if $\mathbf{t} \in \mathbb{R}^n/\mathcal{L}$ is sampled uniformly at random, then $\pi(\mathbf{t})$ is uniform mod $\pi(\mathcal{L})$.
+
+*Proof.* The first statement follows from the well known fact that, if $W = \operatorname{span} S$ for some set of lattice vectors $S \subset \mathcal{L}$, then there exists a basis $\mathbf{B} := (\mathbf{b}_1, \dots, \mathbf{b}_n)$ of $\mathcal{L}$ such that $\operatorname{span}(\mathbf{b}_1, \dots, \mathbf{b}_k) = W$, where $k := \dim W$. (See, e.g., [MG02].) From this, it follows immediately that $\pi(\mathbf{b}_{k+1}), \dots, \pi(\mathbf{b}_n)$ are linearly independent and $\pi(\mathcal{L})$ is the lattice spanned by these vectors, where $\pi := \pi_{S^\perp}$.
+
+The second statement follows from the following similarly well known fact. Let $\tilde{\mathbf{b}}_i := \pi_{\{\mathbf{b}_1, \dots, \mathbf{b}_{i-1}\}}^\perp (\mathbf{b}_i)$ be the Gram-Schmidt vectors of the basis $\mathbf{B}$ described above. Then, the hyperrectangle
+
+$$ \tilde{\mathbb{R}} := \left\{ \sum_i a_i \tilde{\mathbf{b}}_i : -1/2 < a_i \le 1/2 \right\} $$
+
+is a fundamental domain of the lattice. (See, e.g., [Bab85b]) I.e., for each $\mathbf{t} \in \mathbb{R}^n/\mathcal{L}$, there is a unique representative $\tilde{\mathbf{t}} \in \tilde{\mathbb{R}}$ with $\tilde{\mathbf{t}} = \mathbf{t} \bmod \mathcal{L}$. The result then follows by noting that, if $\tilde{\mathbf{t}} \in \tilde{\mathbb{R}}$ is chosen uniformly at random, then clearly $\pi(\tilde{\mathbf{t}}) \in \pi(\tilde{\mathbb{R}})$ is uniform in $\pi(\tilde{\mathbb{R}})$, which is a fundamental region of $\pi(\mathcal{L})$. $\square$
+
+## 2.3 Gaussian Measure
+
+Here we review some useful background on Gaussians over lattices. For a positive parameter $s > 0$ and vector $\mathbf{x} \in \mathbb{R}^n$, we define the Gaussian mass of $\mathbf{x}$ as $\rho_s(\mathbf{x}) = e^{-\pi\|\mathbf{x}\|^2/s^2}$. For a measurable set $A \subseteq \mathbb{R}^n$, we define $\gamma_s(A) = s^{-n} \int_A \rho_s(\mathbf{x}) d\mathbf{x}$. It is easy to see that $\gamma_s(\mathbb{R}^n) = 1$ and hence $\gamma_s$ is a probability measure. We define the discrete Gaussian distribution over a countable set $A$ as
+
+$$ D_{A,s}(\boldsymbol{x}) = \frac{\rho_s(\boldsymbol{x})}{\rho_s(A)}, \forall \boldsymbol{x} \in A. $$
+
+In all cases, the parameter $s$ is taken to be one when omitted. The following lemma is the Poisson Summation Formula for the Gaussian mass of a lattice.
+
+**Lemma 2.5.** For any (full-rank) lattice $\mathcal{L}$ and $s > 0$,
+
+$$ \rho_s(\mathcal{L}) = \frac{1}{\det(\mathcal{L})} \cdot \rho_{1/s}(\mathcal{L}^*). $$

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+We will also need Banaszczyk's celebrated lemma [Ban93, Lemma 1.5].
+
+**Lemma 2.6 ([Ban93]).** For any lattice $\mathcal{L} \subset \mathbb{R}^n$, shift vector $\mathbf{t} \in \mathbb{R}^n$, and $r \ge 1/\sqrt{2\pi}$,
+
+$$ \rho((\mathcal{L} + \mathbf{t}) \setminus \sqrt{n}B_2^n) \le (\sqrt{2\pi er^2}e^{-\pi r^2})^n \cdot \rho(\mathcal{L}). $$
+
+Micciancio and Regev introduced a lattice parameter called the smoothing parameter. For an $n$-dimensional lattice $\mathcal{L}$ and $\epsilon > 0$, the smoothing parameter $\eta_{\epsilon}(\mathcal{L})$ is defined as the smallest $s$ such that $\rho_{1/s}(\mathcal{L}^*) \le 1 + \epsilon$. The motivation for defining smoothing parameter comes from the following two facts [MR04].
+
+**Claim 2.7.** For any lattice $\mathcal{L} \subset \mathbb{R}^n$, shift vector $\mathbf{t} \in \mathbb{R}^n$, $\epsilon \in (0, 1)$, and parameter $s \ge \eta_{\epsilon}(\mathcal{L})$,
+
+$$ \frac{1 - \epsilon}{1 + \epsilon} \cdot \rho_s(\mathcal{L}) \le \rho_s(\mathcal{L} - \mathbf{t}) \le \rho_s(\mathcal{L}). $$
+
+**Lemma 2.8.** For any lattice $\mathcal{L}$, $\mathbf{c} \in \mathbb{R}^n$ and $s \ge \eta_{\epsilon}(\mathcal{L})$,
+
+$$ \Delta((D_s \bmod B), U(\mathbb{R}^n/\mathcal{L})) \le \epsilon/2, $$
+
+where $D_s$ is the continuous Gaussian distribution with parameter $s$ and $U(\mathbb{R}^n/\mathcal{L})$ denotes the uniform distribution over $\mathbb{R}^n/\mathcal{L}$.
+
+We use the following epsilon-decreasing tool which has been introduced in [CDLP13].
+
+**Lemma 2.9 ([CDLP13], Lemma 2.4).** For any lattice $\mathcal{L} \subset \mathbb{R}^n$ and any $0 < \epsilon' \le \epsilon < 1$,
+
+$$ \eta_{\epsilon'}(\mathcal{L}) \le \sqrt{\log(1/\epsilon')/\log(1/\epsilon)} \cdot \eta_{\epsilon}(\mathcal{L}). $$
+
+*Proof.* We may assume without loss of generality that $\eta_{\epsilon}(\mathcal{L}) = 1$. Notice that this implies that $\lambda_1(\mathcal{L}^*) \ge \sqrt{\log(1/\epsilon)/\pi}$. Then, for any $s \ge 1$,
+
+$$ \rho_{1/s}(\mathcal{L}^* \setminus \{\mathbf{0}\}) = \sum \exp(-\pi(s^2-1)\|\mathbf{w}\|^2) \cdot \rho(\mathbf{w}) \le \exp(-\pi(s^2-1)\lambda_1(\mathcal{L})^2)\rho(\mathcal{L}^* \setminus \{\mathbf{0}\}) \le \epsilon^{s^2}. $$
+
+Setting $s := \sqrt{\log(1/\epsilon')/\log(1/\epsilon)}$ gives the result. $\square$
+
+**Lemma 2.10.** For any lattice $\mathcal{L} \subset \mathbb{Q}^n$ with basis $\mathbf{B}$ whose bit length is $\beta$ and any $\epsilon \in (0, 1/2)$, we have $\eta_{\epsilon}(\mathcal{L}(\mathbf{B})) \le 2^{\text{poly}(\beta)}\sqrt{\log(1/\epsilon)}$, and $\lambda_n(\mathcal{L}) \le 2\mu(\mathcal{L}) \le 2^{\text{poly}(\beta)}$.
+
+## 2.4 Sampling from the Discrete Gaussian
+
+For any $\mathbf{B} = (\mathbf{b}_1, \dots, \mathbf{b}_n) \in \mathbb{R}^{n \times n}$, let
+
+$$ \| \tilde{\mathbf{B}} \| := \max_i \| \pi_{\{\mathbf{b}_1, \dots, \mathbf{b}_{i-1}\}}^\perp (\mathbf{b}_i) \|, $$
+
+i.e., $\|\tilde{\mathbf{B}}\|$ is the length of the longest Gram-Schmidt vector of $\mathbf{B}$.
+
+We recall the following result from a sequence of works due to Klein [Kle00]; Gentry, Peikert, and Vaikuntanathan [GPV08]; and Brakerski et al. [BLP$^{+}$13].

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+# Bivariate Beta Distributions and Beyond
+
+Jacobs, Rianne
+
+University of Pretoria, Department of Statistics
+cnr. Lynnwood Road and Roper Street
+Hatfield 0083, South Africa
+E-mail: rianne.jacobs@up.ac.za
+
+Bekker, Andriette
+
+University of Pretoria, Department of Statistics
+cnr. Lynnwood Road and Roper Street
+Hatfield 0083, South Africa
+E-mail: andriette.bekker@up.ac.za
+
+Human, Schalk
+
+University of Pretoria, Department of Statistics
+cnr. Lynnwood Road and Roper Street
+Hatfield 0083, South Africa
+E-mail: schalk.human@up.ac.za
+
+## Introduction
+
+Bivariate beta distributions make up a major part of statistical distribution theory. They form part of the Dirichlet family of distributions, but have become an important family of distributions in their own right. Many bivariate beta distributions have been derived out of an application or as extensions or generalizations from other well-known bivariate beta distributions. Since bivariate beta distributions are used in a wide variety of applications such as Bayesian statistics and reliability theory, there exist a huge amount of research in the development and derivation of new bivariate beta distributions. In this research, we considered the Kummer type beta distributions and more specifically, in this paper, the *bivariate Kummer-beta type IV* distribution. This distribution is an extension of the *bivariate beta type IV* distribution (or Jones' model) which has its roots in the model proposed by Libby and Novick (1982). It was, however, more explicitly derived by Jones (2001) and Olkin and Liu (2003). Furthermore, it is a special case of the model proposed by Sarabia and Castillo (2006). Although, in this paper we explicitly derive the *bivariate Kummer-beta type IV* distribution, it is also a special case of the *bimatrix variate Kummer-beta type IV distribution* defined by Bekker et al. (2010).
+
+Kummer-type distributions form an integral part of statistical distribution theory and a number of these distributions have been proposed. In the univariate case, for example, the *Kummer-gamma* and the *Kummer-beta* are introduced by Armero and Bayarri (1997) and Ng and Kotz (1995). The latter also proposed and studied the *multivariate Kummer-gamma* and *multivariate Kummer-beta* families of distributions. In the matrix variate case, there is the work by Gupta et al (2001), Nagar and Gupta (2002) and Nagar and Cardeño (2001). These authors proposed and studied matrix variate generalizations of the multivariate Kummer-beta and the multivariate Kummer-gamma families of distributions, which are called the *matrix variate Kummer-Dirichlet* (or the *matrix variate Kummer-beta*) and the *matrix variate Kummer-gamma* distributions, respectively. It should be noted that these Kummer distributions get their name from the fact that their normalizing constants are all defined in terms of one of the two so-called Kummer functions (see e.g. Rainville, 1960, p 124-126).
+
+The rest of this paper is structured as follows. First we derive the joint probability density function (pdf), $g(x_1, x_2)$, of the bivariate Kummer-beta type IV distribution. Then, the pdf's of

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+the marginal distributions, $m(x_i)$ for $i=1,2$, the pdf's of the conditional distributions, $h(x_i|x_j)$ for $i,j=1,2$ and $i \neq j$, and the product moment, $E(X_1^r X_2^s)$, of the bivariate Kummer-beta type IV are derived. We investigate the influence of the shape parameter, $\psi$, of the bivariate Kummer-beta type IV distribution. Thus, its effect on the correlation between the components of this distribution as well as its effect on the pdf of $(X_1, X_2)$ and the marginal of $X_1$. An example of a possible application will be presented.
+
+## The Bivariate Kummer-Beta type IV distribution
+
+In this section we derive the pdf of the new bivariate Kummer-beta type IV distribution, $g(x_1, x_2)$, the pdf's of the marginal distributions, $m(x_i)$ for $i = 1, 2$, the pdf's of the conditional distributions, $h(x_i|x_j)$ for $i, j = 1, 2$ and $i \neq j$, and the product moment, $E(X_1^r X_2^s)$.
+
+The pdf, $g(x_1, x_2)$, is derived by extending the kernel of the Jones' bivariate beta distribution, which has pdf
+
+$$f(x_1, x_2) = C x_1^{a_1-1} x_2^{b_1-1} (1-x_1)^{b+c-1} (1-x_2)^{a+c-1} (1-x_1 x_2)^{-(a+b+c)}$$
+
+for $0 \le x_1, x_2 \le 1$, $a, b, c > 0$ and where $C^{-1} = B(a, b, c) = \frac{\Gamma(a)\Gamma(b)\Gamma(c)}{\Gamma(a+b+c)}$ denotes the normalizing constant (Jones, 2001). Clearly, $X_1$ and $X_2$ each have a standard beta type I distribution, i.e. $X_1 \sim Beta(a, c)$ and $X_2 \sim Beta(b, c)$ over $0 \le x_1, x_2 \le 1$.
+
+**Theorem 1**
+
+The pdf of the bivariate Kummer-beta type IV distribution is given by
+
+$$ (1) \quad g(x_1, x_2) = K x_1^{a_1-1} x_2^{b_1-1} (1-x_1)^{b+c-1} (1-x_2)^{a+c-1} (1-x_1x_2)^{-(a+b+c)} e^{-\psi(x_1+x_2)} $$
+
+where $0 \le x_1, x_2 \le 1$, $a, b, c > 0$, $-\infty < \psi < \infty$, and the normalizing constant K is given by
+
+$$ (2) \quad K^{-1} = \sum_{k=0}^{\infty} \frac{(a+b+c)_k}{k!} \frac{F_1(a+k, a+b+c+k; -\psi) F_1(b+k, a+b+c+k; -\psi)}{(B(a+c,b+k)B(b+c,a+k))^{-1}} $$
+
+where $B(\cdot)$ denotes the beta function and ${}_1F_1(\cdot)$ denotes the confluent hypergeometric function (Gradshteyn, 2007, Section 9.2, p 1022). This distribution is denoted as $(X_1, X_2) \sim BKB^{IV}(a, b, c, \psi)$.
+
+**Proof**
+
+Define the bivariate Kummer-beta type IV distribution as
+
+$$ g(x_1, x_2) = K x_1^{a_1-1} x_2^{b_1-1} (1-x_1)^{b+c-1} (1-x_2)^{a+c-1} (1-x_1 x_2)^{-(a+b+c)} e^{-\psi(x_1+x_2)}. $$
+
+The normalizing constant is obtained by integrating over the full support of $g(x_1, x_2)$:
+
+$$ \begin{align*} &= \sum_{k=0}^{\infty} \frac{(a+b+c)_k}{k!} \int_{0}^{1} x_{2}^{b+k-1} (1-x_{2})^{a+c-1} e^{-\psi x_{2}} \int_{0}^{1} x_{1}^{a+k-1} (1-x_{1})^{b+c-1} e^{-\psi x_{1}} dx_{1} dx_{2} \\ &= \sum_{k=0}^{\infty} \frac{(a+b+c)_k}{k!} \int_{0}^{1} x_{2}^{b+k-1} (1-x_{2})^{a+c-1} e^{-\psi x_{2}} \frac{{}_1F_1(a+k; a+b+c+k; -\psi) {}_1F_1(b+k; a+b+c+k; -\psi)}{(B(b+c,a+k))^{-1}} dx_{2} \\ &= \sum_{k=0}^{\infty} \frac{(a+b+c)_k}{k!} \frac{{}_1F_1(a+k; a+b+c+k; -\psi) {}_1F_1(b+k; a+b+c+k; -\psi)}{(B(a+c,b+k)B(b+c,a+k))^{-1}} \end{align*} $$
+
+The above result is obtained by expanding the term $(1-x_1x_2)^{-(a+b+c)}$ as a power series and then using the integral representation of the confluent hypergeometric function, ${}_1F_1(\cdot)$ (Gradshteyn, 2007, Eq 3.383, p 347). ■

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+Note, the bivariate Kummer-beta type IV distribution may also be obtained by substituting $p = 1$ in the pdf of the bimatrix variate Kummer-beta type IV distribution defined by Bekker et al. (2010) (Section 5.3).
+
+**Theorem 2**
+
+If $(X_1, X_2) \sim BKB^{IV}(a, b, c, \psi)$, the marginal pdf of $X_1$ is given by
+
+$$ (3) \qquad m(x_1) = K x_1^{a_1-1} (1-x_1)^{b+c-1} e^{-\psi x_1} B(b, a+c) \Phi_1(b, a+b+c, a+b+c, \psi, x_1) $$
+
+$$ (4) \qquad = K x_1^{a_1-1} (1-x_1)^{b+c-1} e^{-\psi x_1} \sum_{k=0}^{\infty} \frac{(a+b+c)_k}{k!} \frac{x_1^k}{1-F_1(b+k; a+b+c+k; -\psi)} \frac{F_1(b+k; a+b+c+k; -\psi)}{(B(b+k, a+c))^{-1}} $$
+
+where $0 \le x_1, x_2 \le 1$, $a, b, c > 0$, $B(\cdot, \cdot)$ denotes the beta function, $\Phi_1(\cdot)$ denotes confluent hypergeometric series of two variables (Gradshteyn, 2007, Eq 9.261, p 1031) and $K$ is defined in (2).
+
+**Proof**
+
+Using (1), the first representation of $m(x_1)$ given in (3), is obtained by using the integral representation of the confluent hypergeometric series of two variables, $\Phi_1(\cdot)$ (Gradshteyn, 2007, Eq 3.385, p 349). The second representation of $m(x_1)$ given in (4), is obtained by expanding the term $(1-x_1x_2)^{-(a+b+c)}$ as a power series and using the integral representation of the confluent hypergeometric function, ${}_1F_1(\cdot, \cdot)$.
+
+Equation (4) is more useful (in the sense that it is easier to implement and/or program) in computer packages such as Mathematica when we want to graph the pdf $m(x_i)$ for $i = 1, 2$.
+
+The expression for the conditional pdf of $X_2|X_1$ can easily be obtained by using the definition $h(x_2|x_1) = \frac{g(x_1,x_2)}{m(x_1)}$. Note that the marginal pdf of $X_2$ and the conditional pdf of $X_1|X_2$ are obtained by substituting $x_2$ for $x_1$ in the respective expressions and interchanging the parameters $a$ and $b$.
+
+**Theorem 3**
+
+If $(X_1, X_2) \sim BKB^{IV}(a, b, c, \psi)$, the product moment, $E(X_1^r X_2^s)$, equals
+
+$$ (5) \qquad K \sum_{k=0}^{\infty} \frac{(a+b+c)_k {}_1F_1(a+k+r, a+b+c+k+r; -\psi) {}_1F_1(b+k+s, a+b+c+k+s; -\psi)}{k! (B(a+c,b+k+s)B(b+c,a+k+r))^{-1}} \\ = (A(a,b,c,0,0))^{-1} \times A(a,b,c,s,r) $$
+
+where
+
+$$ A(a,b,c,s,r) = \sum_{k=0}^{\infty} \frac{(a+b+c)_k {}_1F_1(a+k+r, a+b+c+k+r; -\psi) {}_1F_1(b+k+s, a+b+c+k+s; -\psi)}{k! (B(a+c,b+k+s)B(b+c,a+k+r))^{-1}} $$
+
+and $A(a,b,c,0,0) = K$ defined in (2).
+
+**Proof**
+
+From (1), expanding the term $(1-x_1x_2)^{-(a+b+c)}$ as a power series and using the integral representation of the confluent hypergeometric function, ${}_1F_1(\cdot, \cdot)$, we obtain (5).
+
+**Shape Analysis**
+
+Figure 1 illustrates the effect of the parameter $\psi$ for $\psi = -1.1, 0$ and $1.1$ on the bivariate Kummer-beta type IV pdf (see (1)) for $a = b = c = 2$. We also show the contour plots for easy comparison. We note that the domain of the graphs in Figure 1 are all $\mathbb{R}^2: [0, 1] \times [0, 1]$. Considering the graphs in Figure 1 we see that the parameter $\psi$ shifts the bell of the density. For negative $\psi$, the bell of the density moves away from the origin, while for positive $\psi$, the bell of the density moves towards the origin.

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+**Figure 1: Bivariate Kummer-beta type IV density function (a) and contour plots (b)**
+
+**Figure 2: Correlation between dependent components**
+
+The five curves are: thick solid line $a = 1, b = 1, c = 10$; medium solid line $a = 2, b = 4, c = 5$; thin solid line $a = 0.5, b = 0.9, c = 0.1$; dashed line $a = 2.5, b = 4, c = 0.5$; dotted line $a = 0.1, b = 0.5, c = 5$.
+
+Figure 2 illustrates the effect of the parameter $\psi \in [-10, 10]$ on the correlation between $X_1$ and $X_2$ using (5). We see that: (i) the parameter $\psi$ can both increase and decrease the correlations for different values of $a, b$ and $c$ and (ii) we can obtain a wide range of correlations between 0 and 1 - depending on the values of $a, b, c$ and $\psi$. The correlation cannot be negative because the central bivariate Kummer-beta type IV distribution is totally positive of order 2.
+
+When looking at the marginal density, we found that the parameter $\psi$ introduces skewness to the symmetric cases of the Jones' bivariate beta distribution and changes the kurtosis of the asymmetric cases.
+
+## An Application
+
+The stress-strength model in the context of reliability is a well-known application of various

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+bivariate beta distributions. This model describes the life of a component with a random strength $X_2$ subjected to a random stress $X_1$. The reliability of a component can be expressed as $P(X_1 < X_2)$ or $P(\frac{X_1}{X_2} < 1) = P(R < 1)$. One method that is frequently used in obtaining the reliability is that of obtaining the distribution of the ratio of the two components stress, $X_1$, and strength, $X_2$. In this section we derive the exact expression for the pdf of the ratio of the correlated components of the bivariate Kummer-beta type IV distribution i.e. $R = \frac{X_1}{X_2}$, in terms of Meijer's G-function (see Mathai, 1993, Definition 2.1, p 60) using the Mellin transform and the inverse Mellin transform (see Mathai, 1993, Definition 1.8, p 23).
+
+**Theorem 4**
+
+If $(X_1, X_2) \sim BKB^{IV}(a, b, c, \psi)$ and we let $R = \frac{X_1}{X_2}$, then the pdf of R is given by
+
+$$ (6) \qquad w(r) = K\Gamma(a+c)\Gamma(b+c) \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \sum_{j=0}^{\infty} (a+b+c)_k \frac{(-\psi)^{j+l}}{j!k!l!} G_{2,2}^{1,1} \begin{pmatrix} r \\ \alpha_1, \alpha_2 \\ \beta_1, \beta_2 \end{pmatrix} \text{ for } r \ge 0 $$
+
+where
+
+$$ -\alpha_1 = b+k+j \quad \alpha_2 = a+b+c+k+l-1 \quad \beta_1 = a+k+l-1 \quad -\beta_2 = a+b+c+k+j $$
+
+with *K* defined in (2).
+
+**Proof**
+
+Setting $r = h-1$ and $s = 1-h$ in (5) and using the series representation of the confluent hypergeometric function, ${}_1F_1(.)$, we obtain an expression for the Mellin transform of $g(x_1, x_2)$ as
+
+$$ 
+\begin{aligned}
+M_g(h) &= E(R^{h-1}) = E\left(\left(\frac{X_1}{X_2}\right)^{h-1}\right) \\
+&= (A(a, b, c, 0, 0))^{-1} \times A(a, b, c, h-1, 1-h) \\
+&= K \frac{\Gamma(b+c)\Gamma(a+c)}{\Gamma(a+b+c)} \sum_{k=0}^{\infty} \sum_{l=0}^{\infty} \sum_{j=0}^{\infty} \Gamma(a+b+c+k) \frac{\Gamma(1-\alpha_1-h)\Gamma(\beta_1+h)}{\Gamma(1-\beta_2-h)\Gamma(\alpha_2+h)} \frac{(-\psi)^{j+l}}{j!k!l!}
+\end{aligned}
+$$
+
+with
+
+$$ -\alpha_1 = b + k + j \quad \alpha_2 = a + b + c + k + l - 1 \quad \beta_1 = a + k + l - 1 \quad -\beta_2 = a + b + c + k + j $$
+
+Using the inverse Mellin transform, the density of R given in (6) follows. ■
+
+Using (6), we can now calculate the reliability for various combinations of parameter values. Table 1 provides the reliability of the bivariate Kummer type IV distribution for $\psi = -1.1, 0, 1.1$ with parameters $a = 1, b = c = 2$ and $a = b = c = 2$. For example, when $a = 1, b = c = 2$ and $\psi = -1.1$, we see that $P(R < 1) = 0.68471$; this implies that the probability that the component will function satisfactorily is 0.68471; or, in other words, the component will fail with probability 0.31529.
+
+**Tabel 1: Some reliability values**
+
+<table><thead><tr><th>a</th><th>b</th><th>c</th><th>ψ</th><th>P(R &lt; 1)</th></tr></thead><tbody><tr><td>1</td><td>2</td><td>2</td><td>-1.1</td><td>0.68471</td></tr><tr><td>1</td><td>2</td><td>2</td><td>0</td><td>0.74904</td></tr><tr><td>1</td><td>2</td><td>2</td><td>1.1</td><td>0.75217</td></tr><tr><td>2</td><td>2</td><td>2</td><td>-1.1</td><td>0.42325</td></tr><tr><td>2</td><td>2</td><td>2</td><td>0</td><td>0.49782</td></tr><tr><td>2</td><td>2</td><td>2</td><td>1.1</td><td>0.48213</td></tr></tbody></table>

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+Figure 3 illustrates the shape of the density of $R = \frac{X_1}{X_2}$ (see Equation 6) for the case $a = 1, b = c = 2$ and $a = b = c = 2$ for different values of $\psi$. The domain for these graphs is $\mathbb{R} : [0, \infty]$.
+
+**Figure 3: The pdf of the ratio**
+
+(i) $a=1, b=c=2$ and (ii) $a=b=c=2$. The three curves in each panel are:
+dashed line $\psi = -1.1$, solid line $\psi = 0$, dotted line $\psi = 1.1$.
+
+## Conclusion
+
+In this paper we introduced, derived and studied the new bivariate Kummer-beta type IV distribution. It was shown that the densities can take different shapes and, therefore, the bivariate Kummer-beta type IV distribution can be used to analyse skewed bivariate data sets. The expressions derived in this paper are a valuable contribution to the existing literature on *Continuous Bivariate Distributions* as the comprehensive work by Balakrishnan and Lai (2009). We also obtained an exact expression for the density function of the ratio of the components of the bivariate distribution which is useful in reliability. For future research, another possible application can be explored in the Beta-Binomial context.
+
+## REFERENCES
+
+1. Armero, C. and Bayarri, M.J. (1997). A Bayesian analysis of queueing system with unlimited service, *Journal of Statistical Planning and Inference*, **58**, 241-261.
+2. Balakrishnan, N. and Lai, C.D. (2009). *Continuous Bivariate Distributions*, 2nd Edition, Springer, New York.
+3. Bekker, A., Roux, J.J.J., Ehlers, R. and Arashi, M. (2010). Bimatrix variate beta type IV distribution relation to Wilks's statistic and bimatrix variate Kummer-beta type IV distribution, *Communications and Statistics-Theory and methods* (in press).
+4. Gradshteyn, I.S. and Ryzhik, I.M. (2007). *Table of Integrals, Series, and Products*, Academic Press, Amsterdam.
+5. Gupta, A.K., Cardeño, L. and Nagar, D.K. (2001). Matrix-variate Kummer-Dirichlet distributions, *J. Appl. Math.*, **1**(3), 117-139.
+6. Johnson, N.L., Kemp, A.W. and Kotz, S. (2005). *Univariate Discrete Distributions*, 3rd Edition, Wiley, Hoboken.
+7. Jones, M.C. (2001). Multivariate t and the beta distributions associated with the multivariate F distribution, *Metrika*, **54**, 215-231.
+8. Libby, D.L. and Novick, R.E. (1982). Multivariate generalized beta distributions with applications to utility assessment, *Journal of Educational Statistics*, **7**(4), 271-294.

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+9. Mathai, A.M. (1993). *A Handbook of Generalized Special Functions for Statistical and Physical Sciences*, Clarendon Press, Oxford.
+
+10. Nagar, D.K. and Cardeño, L. (2001). Matrix variate Kummer-Gamma distributions, *Random Oper. and Stoch. Equ.*, 9(3), 207-218.
+
+11. Nagar, D.K. and Gupta, A.K. (2002). Matrix-variate Kummer-Beta distribution, *J. Austral. Math. Soc.*, **73**, 11-25.
+
+12. Ng, K.W. and Kotz, S. (1995). Kummer-gamma and Kummer-beta univariate and multivariate distributions, *Research report*, **84**, The University of Hong Kong.
+
+13. Olkin, I. and Liu, R. (2003). A Bivariate beta distribution, *Statistics and Probability Letters*, **62**, 407-412.
+
+14. Rainville, E.D. (1960). *Special Functions*, Macmillan, New York.
+
+15. Sarabia, J.M. and Castillo, E. (2006). Bivariate Distributions Based on the Generalized Three-Parameter Beta Distribution, In *Advances in Distribution Theory, Order Statistics, and Inference*, (Ed., N. Balakrishnan, E. Castillo, J.M. Sarabia), pp. 85-110, Birkhäuser, Boston.

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+H13-65
+
+DIRECT IMPLEMENTATION OF NON-LINEAR CHEMICAL REACTION TERMS FOR OZONE CHEMISTRY
+IN CFD-BASED AIR QUALITY MODELLING
+
+Bart De Maerschalck¹, Stijn Janssen¹ and Clemens Mensink²
+
+¹VITO, Dep. Environmental Modelling, Boeretang 200, 2400 Mol, Belgium
+
+**Abstract:** In this paper we present the implementation of a chemistry model that transforms the NOx composition dynamically during transportation for a CFD-based air quality model. For that, the scalar advection equations for NO, NO2 and O3 are coupled by non-linear reaction terms and solved simultaneously. The model is implemented and tested in the Envi-met local air quality and micro climate model (Bruse 2007; De Maerschalck, Janssen *et al.* 2009).
+
+**Key words:** Key CFD-based air quality modelling, Ozone chemistry.
+
+INTRODUCTION
+
+The last decade Computational Fluid Dynamics (CFD) has gained interest as a practical tool for local air quality modelling in complex environment like street canyons, urbanized areas industrial plants. CFD-based air quality models are capable of solving complex three-dimensional flows around obstacles like buildings, trees and vehicles. After solving the wind and turbulence field, the dispersion of pollutants in the atmosphere can be simulated either in a Lagrangian approach, tracking individual particles after release, or a Eulerian approach, that is solving a 3D scalar advection equation. Until now, most CFD-based air quality models solve for an inert gas, similar to wind tunnel modelling. However, it is well understood that nitrogen oxides are reacting fast with ozone while a European air quality directive is specific for NO₂. Regarding traffic emissions about 80% of NOₓ emission is NO, but depending on the ozone background concentrations and meteorological conditions this will react and for secondary NO₂ which can have a significant effect on the local air quality.
+
+OZONE CHEMISTRY IN THE TROPOSPHERE
+
+Nitrogen oxides are ubiquitous urban air pollutants mainly emitted by traffic, power plants and industry. Nitric oxide is on mass basis the most important nitrogen compound emitted into the atmosphere. Nitric oxide is formed from atmospheric nitrogen (N) at high temperatures as in combustion processes. More than 90 percent of the emitted oxide consists of nitrogen oxide (NO), while the remaining party is emitted as nitrogen dioxide (NO₂) (Berkowicz 1998). Once emitted from the tail pipe, nitrogen oxide will react with ozone:
+
+$$
+\mathrm{NO} + \mathrm{O}_{3} \rightarrow \mathrm{NO}_{2} + \mathrm{O}_{2}, \qquad (1)
+$$
+
+Under typical tropospheric boundary layer conditions, this reaction takes place within a time span of a couple of seconds up to minutes, depending on the background concentrations NO, NO₂ and O₃ and meteorological conditions.
+NO₂ is the first reaction product of the atmospheric oxidation process of the emitted NO. However, the freshly formed nitrogen oxide will absorb solar ultraviolet radiation (200nm ≤ λ ≤ 420nm) and forms again NO and O₃:
+
+$$
+\text{NO}_2 + h\nu \rightarrow \text{NO} + \text{O}. \qquad (2)
+$$
+
+$$
+O + O_2 + M \rightarrow O_3 - M . \tag{3}
+$$
+
+Reaction (3) happens quasi immediately. Therefore, in general reactions (2) and (3) are considered as one and O2 in the
+atmosphere is accepted as being constant.
+
+The reaction of NO with O₃ and the photolysis of NO₂ form a cycle which occurs rapidly over the timescales of seconds up to minutes. Under most tropospheric conditions, NO and NO₂ will coexist as a mixture, called NOₓ. If a steady state is reached, the following equilibrium holds:
+
+$$
+\frac{[\text{NO}][\text{O}_x]}{[\text{NO}_2]} = \frac{j_{\text{NO}_2}}{k_{\text{NO}}}, \quad (4)
+$$
+
+where the parentheses indicate the number concentration of the compound in molecules/cm³. kNO is the second order or bimolecular reaction rate coefficient in (1) and is dependent on the ambient temperature (Seinfeld and Pandis 2006):
+
+$$
+k_{NO} = A_0 \exp \left( -\frac{E}{R} \frac{1}{T} \right). \qquad (5)
+$$
+
+with
+
+$$
+A_0 = 2.2 \times 10^{-12} \frac{cm^3}{molecules s},
+$$
+
+$$
+\frac{E}{R} = 1430 K . \qquad (7)
+$$
+
+Figure plots the reaction rate as a function of the temperature.

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+Figure 1: Bimolecular reaction rate coefficient as function of the temperature
+
+$j_{NO_2}$ is the photolysis coefficient of equation (2) and is dependent on the solar ultraviolet radiation. The computation is rather complicated. Theoretically one should integrate over the product of the NO₂ specific absorption cross section with the quantum yield for photolysis and the spectral actinic flux within the limits of the ultraviolet spectrum (Seinfeld and Pandis 2006). However, for a fast estimate different parameterizations are available based on solar angle, solar radiation and cloud coverage (Berkowicz and Hertel 1989; de Leeuw 1995; van Ham and Pulles 1998).
+
+For the implementation in the Envi-met model, the following empirical formulation based on the solar radiation is used:
+
+$$j_{NO_2} = 0.8 \times 10^{-3} \exp(-10/R_s) + 7.4 \times 10^{-6} R_s, \quad (8)$$
+
+with $R_s$ the solar radiation measured in [W/m²]. In ENVI-met in every cell the solar radiation is calculated based on the positions of the sun, cloud cover, local shadows and reflections. Figure 6 shows the estimated values during two different days at a location in the Netherlands based on different parameterization schemes. The red line is the one according to (8) where $R_s$ is dynamically computed by the Envi-met model.
+
+Figure 6: Computed photolysis coefficient during the day for Vaassen, The Netherlands (Left: 23/08/2006, mean cloud coverage 76%; Right: 26/09/2006, 99%)
+
+## CHEMICAL EQUILIBRIUM
+
+Assume that $[NO]_0$, $[NO_2]_0$ and $[O_3]_0$ are the initial number concentrations put in a reactor of constant volume at constant temperature and radiation. After a short time a steady state will be reached for which the photostationary state relation (4) holds. From the conservation of nitrogen and the stoichiometric reaction of O₃ with NO follows (Seinfeld and Pandis 2006):
+
+$$[NO] + NO_2 \rightleftharpoons [NO']_0 + [NO_2]_0, \qquad (9)$$
+
+$$[O_3]_0 - [O_3] = [NO']_0 - [NO]_0, \qquad (10)$$
+
+One can solve now the chemical equilibrium in the reactor and get:

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+$$
+\begin{align}
+& [NO_2] - [NO_2]_0 + \frac{1}{2} \left( O_3^{\circ} - [NO]_0 + \frac{j_{NO_2}}{k_{NO}} \right) - \frac{1}{2} \sqrt{D}, \tag{11} \\
+& [NO] = -\frac{1}{2} \left( [O_3]_0 - [NO]_0 + \frac{j_{NO_2}}{k_{NO}} \right) + \frac{1}{2} \sqrt{D}, \tag{12} \\
+& [O_3] = -\frac{1}{2} \left( [NO]_0 - [O_3]_0 + \frac{j_{NO_2}}{k_{NO}} \right) - \frac{1}{2} \sqrt{D}, \tag{13}
+\end{align}
+$$
+
+with:
+
+$$
+D = \left( [NO]_0 - [O_3]_0 - \frac{j_{NO_2}}{k_{NO}} \right)^2 + 4 \frac{j_{NO_2}}{k_{NO}} ([NO_2]_0 + [O_3]_0) \quad (14)
+$$
+
+One can assume that for a rural background concentration, NO, NO₂ and O₃ are in equilibrium. We now can verify that the parameterization in (8) together with the modelled solar radiation holds by using the computed photolysis coefficients to estimate the equilibrium state according to (11) to (14). The initial numbers are taken from nearby rural measurement stations. Theoretically, if the measured background concentration is in equilibrium and the photolysis coefficient is estimated well, the computed equilibrium should not differ from the local measurements.
+
+The measured background concentrations are compared to the modelled equilibrium state for two days at the location of Vaassen, the Netherlands (Janssen, De Maerschalck et al. 2008). The measured background concentration is the mean from three Dutch rural background concentrations. Again the red line is based on the parameterisation in (8). The red line with the bullets is the measured mean background concentration.
+
+Figure 3: Measured rural background concentrations and computed equilibrium for NO (left), NO2 (middle), and O3 (right). (Vaassen, The Netherlands, 23/08/2006, 76% cloudiness)
+
+Figure 4: Measured rural background concentrations and computed equilibrium for NO (left), NO2 (middle), and O3 (right). (Vaassen, The Netherlands, 26/09/2006, 99% cloudiness)

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+DYNAMIC CHEMICAL TRANSFORMATION PROCESSES IN CFD BASED AIR QUALITY MODELS
+
+The dispersion of a certain gas *i* can be described by a scalar dispersion equation for the concentration $C_i(x, y, z)$:
+
+$$ \frac{\partial C_i}{\partial t} + \mathbf{u} \cdot \nabla C_i + \nabla \cdot (\mathbf{K}_i \cdot \nabla C_i) = E_i - S_i + R_i, \quad (15) $$
+
+with $E_i(x, y, z)$ the local emissions of compound *i* and $S_i(x, y, z)$ the sum of all sink terms (deposition, interaction with vegetation, sedimentation, ...). $R_i$ is the chemical reaction term and is in general dependent on the concentration of all compounds involved in the reaction. The advection velocity and the turbulent reaction terms are computed by the flow solver of the CFD model.
+
+For the photochemical reactions described above the dispersion equations for NO, $NO_2$ and $O_3$ have to be solved simultaneously. The partial differential equations are coupled with the following non-linear reaction terms:
+
+$$ R_{NO} = \left( \frac{d[NO]}{dt} \right)_B = -k_{NO}[NO][O_3] - j_{NO_2}[NO_2], \quad (16) $$
+
+$$ R_{NO_2} = \left( \frac{d[NO_2]}{dt} \right)_R k_{NO} NO || O_3 | j_{NO_2} NO_2 |, \quad (17) $$
+
+$$ R_{O_3} = \left( \frac{d[O_3]}{dt} \right)_U = -k_{NO}[NO][O_3] + j_{NO_2}[NO_2]. \quad (18) $$
+
+Notice that these reaction terms are given in number concentration while equation (15) is typically describing conservation of mass. In Envi-met all concentrations are mixing ratios measured in $\mu g/kg_{air}$. Therefore, equations (16) to (18) have to be converted to mass concentrations first.
+
+Figure (Janssen, De Maerschalck et al. 2008) illustrates the local effect of oxidation of traffic emitted NO on the local air quality. The continuous lines show the modelled NO and $NO_2$ concentrations downwind of a motorway. The green lines are for a motorway with a vegetation barrier, the red lines without a vegetation barrier. The position of the driving lanes and vegetation barrier are indicated by the red and green blocks. The green and red dots with error bars are the measured concentrations.
+
+One can see that NO is decreasing faster than the $NO_2$ concentrations, both with and without a vegetation barrier. This is due to the fact the NO is reacting with ozone and forms secondary $NO_2$. One can also notice that due to the vegetation the effect is even stronger. The vegetation slows down the local wind speed, so there is more time for the chemistry. At the same time, due to increased turbulence, more fresh ozone is mixed in which enhances the oxidation process as well.
+
+Figure 5: NO and $NO_2$ concentrations downwind of a highway with and without a vegetation barrier.
+
+REFERENCES
+
+Berkowicz, R. (1998), Street Scale Models in *Urban Air Pollution - European Aspects*, J. Fenger, O. Hertel and F. Palmgren, Kluwer Academic Publishers, 223-251.
+
+Berkowicz, R. and O. Hertel (1989), Technical Report DMU LUFT - A131, National Environmental Research Institute, Roskilde, Denmark,
+
+Bruse, M. (2007), Particle filtering capacity of urban vegetation: A microscale numerical approach. *Berliner Geographische Arbeiten* 109, 61-70.

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+de Leeuw, F. A. A. M. (1995), *Parametrization of NO2 photodissociation rate*, Technical Report 722501004, Dutch National Institute for Public Health and Environment (RIVM), Bilthoven, The Netherlands
+
+De Maerschalck, B., S. Janssen, et al. (2009), *CFD Simulations Of The Impact of a Vegetation Barrier Along a Motor Way on Local Air Quality*. Air Quality, Science and Application, Istanbul, Turkey, University of Hertfordshire.
+
+Janssen, S., B. De Maerschalck, et al. (2008), *Modelanalyse van de IPL meetcampagne langs de A50 te Vaassen ter bepaling van het effect van vegetatie op luchtkwaliteit langs snelwegen*, DVS-2008-044, VITO, Mol, Belgium
+
+Seinfeld, J. H. and S. N. Pandis (2006), *Atmospheric chemistry and physics*, New Jersey, John Wiley.
+
+van Ham, J. and M. P. J. Pulles (1998), *Nieuw nationaal model*, Technical Report TNO R 98/306, TNO, KEMA, KNMI, VNONCW, RIVM, Apeldoorn, The Netherlands

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+A MULTI-PERIOD NEWSVENDOR PROBLEM WITH
+PRE-SEASON EXTENSION UNDER FUZZY DEMAND
+
+Hülya Behret¹, Cengiz Kahraman²
+
+¹,²Istanbul Technical University, Industrial Engineering Department,
+Macka, 34367, Istanbul, Turkey
+
+E-mails: ¹behreth@itu.edu.tr (corresponding author); ²kahramanc@itu.edu.tr
+
+Received 3 January 2010; accepted 1 September 2010
+
+**Abstract.** This paper proposes a fuzzy multi-period newsvendor model with pre-season extension for innovative products. The demand of the product is represented by fuzzy numbers with triangular membership function. The holding and shortage cost parameters are considered as imprecise and also represented by triangular fuzzy numbers. As the selling season draws closer, suppliers lead times shortens and thus production costs increase. In contrast, caused by the oncoming selling season, demand fuzziness decreases and more accurate demand forecasts can be maintained that lead to lower overage/underage costs. The objective of the model is to find the best order period and the best order quantity that will minimize the fuzzy expected total cost. The model is experimented with an illustrative example and supported by sensitivity analyses.
+
+**Keywords:** inventory problem, fuzzy modeling, newsvendor, innovative product, fuzzy demand, fuzzy inventory cost, pre-season.
+
+Reference to this paper should be made as follows: Behret, H.; Kahraman, C. 2010.
+A Multi-period Newsvendor Problem with Pre-season Extension under Fuzzy Demand,
+*Journal of Business Economics and Management* 11(4): 613–629.
+
+# 1. Introduction
+
+In today's competitive market conditions, development of innovative products is ex-
+tremely important for an enterprise to achieve and sustain the superiority. Innovative
+products are products that have shorter life cycles with higher innovation and fashion
+contents (Lee 2002). Fashion goods, electronic products and mass customized goods
+are examples of innovative products. Although innovative products tend to have higher
+product profit margins, the cost of obsolescence is high for them. These kinds of prod-
+ucts have much product variety and short product life cycles. Caused by the innovative
+nature of the product, usually no historical data are available for forecasting demand
+of such products. In addition to the short life cycle and high demand unpredictability
+of these products, another important feature is the long supply lead times. Because of
+having such long supply lead times and short sales seasons, procurement problem of
+these products corresponds to the single-period inventory (also known as newsvendor
+or newsboy) problem.

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+Fig. 3. Fuzzy concentration of $\tilde{X}$
+
+For example, let us order a quantity of 2000 units at the beginning of January. The unit penalty cost ($\tilde{\text{PC}}$) is a level-2 fuzzy set including imprecise demand and costs. For $x_1 = 1000$, the fuzzy penalty cost will be $\tilde{\text{PC}} = (1,2,3) \times 1000 = (1000,2000,3000)$ with possibility of 0. For $x_5 = 3000$, the fuzzy penalty cost will be $\tilde{\text{PC}} = (4,5,6) \times 1000 = (4000,5000,6000)$ with the possibility 0.80 and so on. For the ordering period January fuzzy unit penalty cost values for $Q = 2000$ is given in Table 3.
+
+The graphical representations of level-2 fuzzy sets of ($\tilde{\text{PC}}$) for January when $Q = 2000$ and the corresponding s-fuzzified set $s - \text{fuzz}(\tilde{\text{PC}})$ are shown in Figs. 4a and 4b, respectively.
+
+Table 3. Unit penalty cost values for January, $Q = 2000$
+
+<table><thead><tr><th></th><th>μ&#771;<sub>X</sub>(x<sub>i</sub>)</th><th>(OC)</th><th>(&#8494;UC)</th><th>(&#8494;PC)</th></tr></thead><tbody><tr><td>x<sub>1</sub></td><td>1000</td><td>0</td><td>(1000,2000,3000)</td><td>(0,0,0)</td><td>(1000,2000,3000)</td></tr><tr><td>x<sub>2</sub></td><td>1500</td><td>0.20</td><td>(500,1000,1500)</td><td>(0,0,0)</td><td>(500,1000,1500)</td></tr><tr><td>x<sub>3</sub></td><td>2000</td><td>0.40</td><td>(0,0,0)</td><td>(0,0,0)</td><td>(0,0,0)</td></tr><tr><td>x<sub>4</sub></td><td>2500</td><td>0.60</td><td>(0,0,0)</td><td>(2000,2500,3000)</td><td>(2000,2500,3000)</td></tr><tr><td>x<sub>5</sub></td><td>3000</td><td>0.80</td><td>(0,0,0)</td><td>(4000,5000,6000)</td><td>(4000,5000,6000)</td></tr><tr><td>x<sub>6</sub></td><td>3500</td><td>1</td><td>(0,0,0)</td><td>(6000,7500,9000)</td><td>(6000,7500,9000)</td></tr><tr><td>x<sub>7</sub></td><td>4000</td><td>0.80</td><td>(0,0,0)</td><td>(8000,10000,12000)</td><td>(8000,10000,12000)</td></tr><tr><td>x<sub>8</sub></td><td>4500</td><td>0.60</td><td>(0,0,0)</td><td>(10000,12500,15000)</td><td>(10000,12500,15000)</td></tr><tr><td>x<sub>9</sub></td><td>5000</td><td>0.40</td><td>(0,0,0)</td><td>(12000,15000,18000)</td><td>(12000,15000,18000)</td></tr><tr><td>x<sub>10</sub></td><td>5500</td><td>0.20</td><td>(0,0,0)</td><td>(14000,17500,21000)</td><td>(14000,17500,21000)</td></tr><tr><td>x<sub>11</sub></td><td>6000</td><td>1</td><td>(16,6799,24,4788)</td><td>(16,6799,24,4788)</td><td>(16,6799,24,4788)</td></tr></tbody></table>

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+**Fig. 4a.** Level-2 fuzzy set ($\tilde{PC}$)
+
+**Fig. 4b.** s-fuzz($\tilde{PC}$)
+
+According to the s-fuzzified value of the penalty cost, total cost for the given values when $Q = 2000$ is calculated as below:
+
+$$ \tilde{TC}(2000, \tilde{X}) = [3.50 \times 2000 + defuzz(s - fuzz(\tilde{PC}))]. \quad (21) $$
+
+Here, the operator “defuzz” denotes the centroid method for defuzzification. Centroid defuzzification values have been obtained by using MATLAB R2008a Fuzzy Logic Toolbox as in Fig. 5.
+
+$$ \tilde{TC}(2000, \tilde{X}) = [3.5 \times 2000 + 10,022] = \$17,022. \quad (22) $$
+
+Fuzzy total cost values for all other order quantities of the ordering period January are given in Table 4. For the ordering period January, best order quantity which gives the min fuzzy total cost is found 3000 units with the fuzzy total cost value \$16,820.
+
+The procedure continues with calculation of other periods best order quantities. Using Equation 19, the fuzzy total costs for pre-season ordering periods are calculated and given as in Table 5 and Fig. 6.
+
+As stated by Table 5, best order period ($M^*$) is found January with fuzzy total cost \$16,820 with 3000 units. Here the decrease of demand fuzziness causes a decline on the fuzzy total cost values among ordering periods. However, the change of production cost also affects the cost function. The proposed model offers best order period and corresponding order quantity for the given fuzzy variables.

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+**Table 4.** Fuzzy total costs for January, ($)
+
+<table><thead><tr><td>Order quantity</td><td>Fuzzy total cost</td></tr></thead><tbody><tr><td>1000</td><td>17,356</td></tr><tr><td>1500</td><td>17,159</td></tr><tr><td>2000</td><td>17,022</td></tr><tr><td>2500</td><td>16,930</td></tr><tr><td>3000</td><td>16,820*</td></tr><tr><td>3500</td><td>16,828</td></tr><tr><td>4000</td><td>17,258</td></tr><tr><td>4500</td><td>19,079</td></tr><tr><td>5000</td><td>21,514</td></tr><tr><td>5500</td><td>24,045</td></tr><tr><td>6000</td><td>26,681</td></tr></tbody></table>
+
+Fig. 5. Centroid defuzzification of s - fuzz($\tilde{P}C$)
+
+**Table 5.** Fuzzy total cost for pre-season ordering periods, $\tilde{c}_h = \$$[1,2,3]$, $\tilde{c}_s = \$$[4,5,6]
+
+<table><thead><tr><th>Q</th><th>T&#x0303;C(Jan)</th><th>T&#x0303;C(Feb)</th><th>T&#x0303;C(March)</th><th>T&#x0303;C(April)</th><th>T&#x0303;C(May)</th><th>T&#x0303;C(June)</th></tr></thead><tbody><tr><th>c<sub>p</sub></th><td>3,50</td><td>3,68</td><td>3,86</td><td>4,05</td><td>4,25</td><td>4,47</td></tr><tr><th>1000</th><td>17,356</td><td>17,335</td><td>17,360</td><td>17,327</td><td>17,262</td><td>17,332</td></tr><tr><th>1500</th><td>17,159</td><td>17,161</td><td>17,219</td><td>17,184*</td><td>17,098</td><td>17,202</td></tr><tr><th>2000</th><td>17,022</td><td>17,089</td><td>17,210*</td><td>17,202</td><td>17,087*</td><td>17,202*</td></tr><tr><th>2500</th><td>16,930</td><td>17,082</td><td>17,282</td><td>17,333</td><td>17,223</td><td>17,350</td></tr><tr><th>3000</th><td>16,820*</td><td>17,075*</td><td>17,375</td><td>17,515</td><td>17,448</td><td>17,617</td></tr><tr><th>3500</th><td>16,828</td><td>17,168</td><td>17,553</td><td>17,787</td><td>17,815</td><td>18,087</td></tr><tr><th>4000</th><td>17,258</td><td>17,718</td><td>18,237</td><td>18,659</td><td>19,003</td><td>19,589</td></tr><tr><th>4500</th><td>19,079</td><td>19,709</td><td>20,401</td><td>21,063</td><td>21,708</td><td>22,508</td></tr><tr><th>5000</th><td>21,514</td><td>22,252</td><td>23,059</td><td>23,861</td><td>24,678</td><td>25,636</td></tr><tr><th>5500</th><td>24,045</td><td>24,902</td><td>25,831</td><td>26,775</td><td>27,756</td><td>28,843</td></tr><tr><th>6000</th><td>26,681</td><td>27,658</td><td>28,703</td><td>29,778</td><td>30,880</td><td>32,058</td></tr></tbody></table>
+
+H. Behret, C. Kahraman. A multi-period newsvendor problem with pre-season extension under fuzzy demand

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+**Fig. 6.** Fuzzy total cost for pre-season ordering periods
+
+### 3.3. Sensitivity analysis
+
+In this section, various experiments have been performed to analyze the effect of membership function shapes to the fuzzy models. Through these experiments fuzzy unit holding cost and fuzzy unit shortage cost values have been changed (Table 6).
+
+**Table 6.** Performed experiments and values of the cost parameters
+
+<table><thead><tr><td>Experiments</td><td>c&#771;<sub>h</sub> ($)</td><td>c&#771;<sub>s</sub> ($)</td></tr></thead><tbody><tr><td>1</td><td>(1,2,3)</td><td>(4,5,6)</td></tr><tr><td>2</td><td>(1,2,3)</td><td>(4,5,8)</td></tr><tr><td>3</td><td>(1,2,3)</td><td>(4,5,10)</td></tr><tr><td>4</td><td>(1,2,3)</td><td>(4,5,12)</td></tr><tr><td>5</td><td>(1,2,5)</td><td>(4,5,6)</td></tr><tr><td>6</td><td>(1,2,5)</td><td>(4,5,8)</td></tr><tr><td>7</td><td>(1,2,5)</td><td>(4,5,10)</td></tr><tr><td>8</td><td>(1,2,5)</td><td>(4,5,12)</td></tr><tr><td>9</td><td>(1,2,7)</td><td>(4,5,6)</td></tr><tr><td>10</td><td>(1,2,7)</td><td>(4,5,8)</td></tr><tr><td>11</td><td>(1,2,7)</td><td>(4,5,10)</td></tr><tr><td>12</td><td>(1,2,7)</td><td>(4,5,12)</td></tr><tr><td>13</td><td>(1,2,9)</td><td>(4,5,6)</td></tr><tr><td>14</td><td>(1,2,9)</td><td>(4,5,8)</td></tr><tr><td>15</td><td>(1,2,9)</td><td>(4,5,10)</td></tr><tr><td>16</td><td>(1,2,9)</td><td>(4,5,12)</td></tr></tbody></table>