| { |
| "title": "Formal Power Series on Algebraic Cryptanalysis", |
| "abstract": "In the complexity estimation for an attack that reduces a cryptosystem to solving a system of polynomial equations, the degree of regularity and an upper bound of the first fall degree are often used in cryptanalysis. While the degree of regularity can be easily computed using a univariate formal power series under the semi-regularity assumption, determining an upper bound of the first fall degree requires investigating the concrete syzygies of an input system.\nIn this paper, we investigate an upper bound of the first fall degree for a polynomial system over a sufficiently large field.\nIn this case, we prove that the first fall degree of a non-semi-regular system is bounded above by the degree of regularity, and that the first fall degree of a multi-graded polynomial system is bounded above by a certain value determined from a multivariate formal power series.\nMoreover, we provide a theoretical assumption for computing the first fall degree of a polynomial system over a sufficiently large field.", |
| "sections": [ |
| { |
| "section_id": "1", |
| "parent_section_id": null, |
| "section_name": "Introduction", |
| "text": "Solving the system of polynomial equations is an important topic in computer algebra since such a system can arise from many practical applications (cryptology, coding theory, computational game theory, optimization, etc.).\nIn particular, the so-called MQ problem that solves the system of quadratic equations is NP-hard even if over the finite field of order two [35 ###reference_b35###], and the security of multivariate cryptography [19 ###reference_b19###] is based on the hardness of the problem.\nMultivariate cryptography is expected to have a particularly high potential for building post-quantum signature schemes and is investigated in NIST post-quantum cryptography (PQC) standardization project [43 ###reference_b43###].\nIn multivariate cryptography, there are several attacks reducing it to the problem solving the system of polynomial equations.\nThe direct attack [7 ###reference_b7###, 32 ###reference_b32###] generates the system of quadratic equations from the public key and a chiphertext/message and obtains its plaintext/signature by solving the system.\nMoreover, as a key recovery attack generating the system of polynomial equations, it is known such as the Rainbow-Band-Separation (RBS) attack [20 ###reference_b20###], the MinRank attack [10 ###reference_b10###, 11 ###reference_b11###, 38 ###reference_b38###], Intersection attack [10 ###reference_b10###].\nIn particular, the MinRank attack reduces a cryptosystem to the MinRank problem and solves a MinRank instance by using a method generating the system of polynomial equations, such as the Kipnis-Shamir (KS) method [38 ###reference_b38###], minors methods [6 ###reference_b6###, 33 ###reference_b33###].\nThe MinRank problem is also NP-hard [46 ###reference_b46###] and is a base of the security of the rank metric code-based cryptography, which was proposed in NIST PQC standardization project [1 ###reference_b1###, 3 ###reference_b3###].\nThe efficiency of these attacks depends on that of an algorithm, say solver, to solve the system of polynomial equations generated by an attack.\nIn cryptography, the solver with a Gröbner basis or a (matrix) kernel search is mainly used as an efficient algorithm to solve the system generated by an attack.\nThe complexity of such a solver is given by using the solving degree, which is the maximal degree of polynomials required to solve the system.\nHowever, the solving degree is an experimental value and it is practically hard to decide this value of a large-scale system arising from cryptography.\nFor the solver with a Gröbner basis, the degree of regularity [5 ###reference_b5###] and the first fall degree [27 ###reference_b27###] are used as a tool approximating to the solving degree.\nAlso, for the solver with a kernel search, an analogue of the degree of regularity is used as such a tool under some assumptions.\nFor a semi-regular polynomial system, which generalizes a regular sequence in an overdetermined case,\nthe degree of regularity introduced by M. Barded et al. [5 ###reference_b5###] is given by the minimum degree of terms with a non-positive coefficient in the power series\nwhere and are the degrees and the number of the variables of the system, respectively.\nSince the value approximates the solving degree tightly and is simply computed by the power series (1 ###reference_###), the degree of regularity is widely used for complexity estimation [9 ###reference_b9###, 14 ###reference_b14###, 15 ###reference_b15###, 25 ###reference_b25###].\nHowever, there exist non-semi-regular systems in cryptography, such as a quadratic system generated by the direct attack against HFE [44 ###reference_b44###], the RBS attack [40 ###reference_b40###] and the MinRank attack using the KS method [41 ###reference_b41###].\nA non-semi-regular system appeared in cryptography is often solved within a smaller degree than the degree of regularity in the semi-regular case.\nNamely, its solving degree is smaller than the degree of regularity.\nThe first fall degree introduced by V. Dubois and N. Gama [27 ###reference_b27###] is defined by using non-trivial syzygies of the top homogeneous component and approximates the solving degree of a non-semi-regular system generated by some attacks.\nHowever, it is hard to decide its actual value from given parameters in a cryptosystem and in previous estimations [21 ###reference_b21###, 22 ###reference_b22###, 24 ###reference_b24###, 49 ###reference_b49###] an upper bound for the first fall degree was used by investigating a concrete non-trivial syzygy. The discussion for this upper bound depends on each cryptosystem and is not given to a general system.\nMoreover, a relation with the degree of regularity is not clear mathematically.\nIn 2020, a progress has been made in the complexity estimation for an attack generating a multi-graded polynomial system. For example, [41 ###reference_b41###] shows that the first fall degree of a multi-graded polynomial system generated by the MinRank attack using the KS method is bounded above by the minimal total degree of terms with a negative coefficient in\nwhere is a multi-degree and is the size of each set of variables in the multi-graded system.\nMoreover, in NIST PQC 2nd round, the estimations [40 ###reference_b40###, 47 ###reference_b47###] in the bi-graded case influenced a parameter selection for Rainbow [26 ###reference_b26###].\nHowever, the theoretical background for such estimations it not clear." |
| }, |
| { |
| "section_id": "1.1", |
| "parent_section_id": "1", |
| "section_name": "Our contribution", |
| "text": "In this article, we mainly investigate an upper bound of the first fall degree for a polynomial system over a sufficiently large field. Upper bounds given in this article are actually that of the minimal degree at the first homology of the Koszul complex to a polynomial system (Definition 4.1 ###reference_Thm1###). The assumption for the order of a field gives that the minimal degree coincides with the first fall degree (Proposition 4.4 ###reference_Thm4###). \nFirstly, we prove that for a non-semi-regular system over a sufficiently large field, the degree of regularity is an upper bound of the first fall degree (Theorem 4.9 ###reference_Thm9###).\nIn general, we show that an upper bound of the first fall degree of a polynomial system is decided by the power series (1 ###reference_###) (Definition 4.10 ###reference_Thm10###, Theorem 4.11 ###reference_Thm11###). If , then\nthe first fall degree of a semi-regular system over a sufficient large field coincides with the degree of regularity (Corollary 4.13 ###reference_Thm13###).\nSecondly, we naturally generalize the previous discussion to the multi-graded case. Namely, we define decided by the multivariate power series (2 ###reference_###) as a generalization of (Definition 5.1 ###reference_Thm1###).\nThen we prove that the first fall degree of a multi-graded polynomial system over a sufficiently large field is bounded above by (Theorem 5.5 ###reference_Thm5###).\nIn more detail, we also see that for a multi-graded polynomial system over a sufficiently large field, the multi-graded version of is bounded above by (Definition 5.1 ###reference_Thm1###, 5.2 ###reference_Thm2###, Theorem 5.6 ###reference_Thm6###).\nThese results theoretically guarantee recently developed cryptanalyses [41 ###reference_b41###, 40 ###reference_b40###, 47 ###reference_b47###] using the first fall degree (6 ###reference_###). In particular, these cryptanalyses give examples such that is smaller than .\nFinally, we provide an application using a generalization of . We introduce a multi-graded XL algorithm with a (matrix) kernel search as a generalization of [4,46,48] and define a tool for estimating the complexity of this XL algorithm. Then we show that this tool is computed by under the assumption of a certain regularity (Proposition 5.11 ###reference_Thm11###).\nIn particular, this regularity provide computing the first fall degree of a polynomial system over a sufficiently large field (Corollary 5.13 ###reference_Thm13###)." |
| }, |
| { |
| "section_id": "1.2", |
| "parent_section_id": "1", |
| "section_name": "Organization", |
| "text": "This article is organized as follows.\nWe explain some algorithmic backgrounds for our study in Section 2, and recall some fundamental concepts in commutative ring theory and proxies of the solving degree for estimating the complexity of a solver in Section 3.\nIn Section 4, we prove that the first fall degree is smaller than the degree of regularity in the semi-regular case if the order of the coefficient field is sufficiently large.\nIn Section 5, we prove that the first fall degree of a multi-graded polynomial system is bounded by a certain value determined from its multi-degree if the order of the coefficient field is sufficiently large, and provide the theoretical assumption for applying the XL algorithm with a kernel search.\nIn Section 6, we provide actual examples that satisfy the condition for the order of the coefficient field." |
| }, |
| { |
| "section_id": "2", |
| "parent_section_id": null, |
| "section_name": "Background", |
| "text": "In this section, we explain the complexity estimation for an algorithm to solve the system of polynomial equations.\nThe Gröbner basis algorithm and the XL algorithm with a (matrix) kernel search are mainly used in cryptography such as the NIST PQC standardization project (see [9 ###reference_b9###, 14 ###reference_b14###, 15 ###reference_b15###, 25 ###reference_b25###]).\nWe explain the complexity estimation for the Gröbner basis algorithm in Subsection 2.1 ###reference_###, for the XL algorithm with a kernel search in Subsection 2.2 ###reference_###, and for a solver to a multi-graded polynomial system in Subsection 2.3 ###reference_###." |
| }, |
| { |
| "section_id": "2.1", |
| "parent_section_id": "2", |
| "section_name": "Solver with a Gröbner basis", |
| "text": "For a given polynomial system , when the codimension of the ideal is zero, the ideal includes a univariate polynomial as a generator of the Gröbner basis with respect to the lexicographic monomial order.\nBy solving this univariate polynomial, a variable in the solution to the system is fixed.\nRepeating this procedure, we obtain a solution of the system .\nThe Gröbner basis algorithm, which computes the Gröbner basis for a given ideal-generator, was discovered by Buchberger [13 ###reference_b13###], and faster algorithms are implemented such as F4 [30 ###reference_b30###], F5 [31 ###reference_b31###] and XL family [51 ###reference_b51###].\nThe F4 algorithm, for example, is the default algorithm for computing Gröbner bases in the computer algebra software Magma [12 ###reference_b12###], and the complexity of that with respect to the graded reverse lexicographic monomial order is estimated as\nwhere is the number of variables, is a linear algebra constant, and is the solving degree that is the maximal degree of polynomials required to solve the system.\nFor example, as the direct attack against GMSS [14 ###reference_b14###].\nMoreover, as a conversion algorithm to the Gröbner basis with respect to another monomial order, the FGLM algorithm returns to the Gröbner basis with respect to the lexicographic monomial order in complexity\nwhere is the number of solutions counted with multiplicity in the algebraic closure of the coefficient field [29 ###reference_b29###].\nIn order to satisfy the conditions that the codimension of an ideal is zero and that the parameter is sufficiently small, some variables are fixed before solving a polynomial system." |
| }, |
| { |
| "section_id": "2.2", |
| "parent_section_id": "2", |
| "section_name": "Solver with a kernel search", |
| "text": "For simplicity, we assume that are homogeneous of degree two.\nFor a target degree , the XL (eXtended Linearization) algorithm generates a new system of degree by multiplying the system by the monomials of degree and obtains the so-called Macaulay matrix of degree whose row components correspond to the coefficients of a polynomial in the new system.\nThe columns of the Macaulay matrix correspond to the monomials of degree .\nWe also call the solving degree of an XL algorithm the maximal degree of polynomials used in an XL algorithm.\nNote that, when arranging the columns of the Macaulay matrix according to a monomial order, the XL algorithm obtains a new pivot (corresponding to a new leading term) by computing a row echelon form of the matrix and can return Gröbner bases of degree .\nThe XL algorithm is available as an algorithm computing the Gröbner basis as explained in Subsection 2.1 ###reference_###.\nFor a solution to the system , the vector\nis a kernel vector of the Macaulay matrix of degree . The kernel space of the Macaulay matrix contains vectors corresponding to the solutions to the system. The XL algorithm with a kernel search solves the system by finding such a vector.\nWhen the algorithm randomly returns a kernel vector, its complexity is estimated as\nwhere is the complexity of the kernel search, is the number of the columns minus the rank and is the dimension of the kernel space that corresponds the solution space of the system.\nHere the term is an iteration required to find a kernel vector corresponding to a solution to the system.\nMoreover, when an algorithm that returns a basis of the kernel space is used as a kernel search, the complexity is sufficient in .\nIf holds, then and a kernel vector obtained by a kernel search always corresponds to a solution.\nFor the XL algorithm with a kernel search to solve a homogeneous system, the following value is often used as a tool to approximate the solving degree:\nIn Subsection 5.2 ###reference_###, we explain the theoretical assumption for controlling value .\nThe XL algorithm in [50 ###reference_b50###] extracts a square matrix from the obtained Macaulay matrix and uses the Wiedemann algorithm [52 ###reference_b52###] as a kernel search of the matrix.\nThe complexity of the Wiedemann algorithm is estimated as\nwhere is the number of columns, is the number of non-zero entries of the matrix and is the average number of non-zero entries in a row.\nSince the XL algorithm multiplies each quadratic polynomial in the system by a monomial, the number of non-zero entries in each row of the obtained Macaulay matrix is at most the number of non-zero coefficients in the quadratic polynomial, namely\nThe algorithm is called the Wiedemann XL algorithm and is used for the direct attack against Rainbow in [25 ###reference_b25###]." |
| }, |
| { |
| "section_id": "2.3", |
| "parent_section_id": "2", |
| "section_name": "Solver to a multi-graded polynomial system", |
| "text": "The XL algorithm with respect to a target degree generates the Macaulay matrix of a degree whose columns correspond to the monomials of degree .\nFor a multi-graded polynomial system, we can naturally consider the XL algorithm that generates the Macaulay matrix whose columns correspond to the monomials of degree .\nThen the number of columns is\nwhere , and the Macaulay matrix contains a kernel vector corresponding to a solution to the multi-graded system.\nIn the XL algorithm with a kernel search, the complexity of the Wiedemann algorithm is estimated as\nwhere is the average number of non-zero entries in a row.\nMoreover, in order to obtain the Gröbner basis for a multi-graded polynomial system, the XL algorithms performed as follows (see Subsection 3.1 ###reference_### for notation).\nLet be a multi-graded polynomial system and set .\nSet the multi-degrees to .\nMultiply an each polynomial by the monomials of degree such that . Obtain a new multi-graded system.\nGenerate the Macaulay matrix corresponding to the new system and compute its echelon form.\nNote that this approach is a natural generalization from [4 ###reference_b4###, 47 ###reference_b47###, 49 ###reference_b49###].\nWhen the Gaussian elimination is used as computing the echelon form, its complexity is estimated as\nwhere is a linear algebra constant, and .\nHere, for , the relation is defined as for each ." |
| }, |
| { |
| "section_id": "3", |
| "parent_section_id": null, |
| "section_name": "Preliminaries", |
| "text": "In this section, we recall some fundamental concepts in commutative ring theory and the complexity estimation for a Gröbner basis algorithm." |
| }, |
| { |
| "section_id": "3.1", |
| "parent_section_id": "3", |
| "section_name": "Fundamental concepts in commutative ring theory", |
| "text": "A (Noetherian) commutative ring is said to be -graded if it has a decomposition such that for any .\nIn this article, we assume always that if has a negative component, and that for every .\nThen we call a -graded commutative ring.\nAn element in is said to be -homogeneous, or simply homogeneous and then we denote by and call it the -degree of .\nIf is a field, put\nwhere and which is called a (multivariate) Hilbert series.\nFor a -graded commutative ring , its quotient ring with an ideal generated by -homogeneous elements is -graded.\nFor example, the polynomial ring is -graded by \nwhere .\nThen we have a decomposition where is the vector space over generated by the monomials of -degree .\nWhen , for a polynomial in and a well-ordering on , we consider a expression where and call its homogeneous component of degree its top homogeneous component.\nWhen , the polynomial ring is said to be standard graded and we denote as .\nIn this article, we mainly treat a homogeneous case and call an element of a system. Moreover, we call a system whose components are homogeneous a homogeneous system.\nLet be a commutative ring.\nFor , we define an -module homomorphism\nThen we denote by , or simply , the kernel of this homomorphism and its element is called a syzygy of .\nFor example,\nwhere , is such an element.\nHere, we denote by the submodule generated by the elements and its element is called a Koszul syzygy.\nLet be a -graded commutative ring.\nFor -homogeneous elements with and the free module with the standard basis , we denote by the -th exterior power and give the Koszul complex\nby \nThen we denote by the -th homology group of the complex and by its component of degree .\nNote that\nDefine as for any .\nSince is the mapping cone of\nwe have the short exact sequence (see [17 ###reference_b17###, 45 ###reference_b45###]):\nNamely, we can obtain the following long exact sequence:" |
| }, |
| { |
| "section_id": "3.2", |
| "parent_section_id": "3", |
| "section_name": "Complexity estimation for a Gröbner basis algorithm", |
| "text": "In this subsection, we treat only the standard graded case (see Subsection 3.1 ###reference_###).\nIn Subsection 2.1 ###reference_###, we explain that the complexity of the Gröbner basis algorithm depends on the solving degree (see Equation (3 ###reference_###) for example).\nHowever, the solving degree is an experimental value.\nIn order to estimate the complexity of solving a large-scale polynomial system appearing in cryptography, we need to consider a theoretical tool to approximate the solving degree.\nThe degree of regularity introduced by Bardet et al. [5 ###reference_b5###] is well-known as such a tool.\nFor polynomials in the (standard graded) polynomial ring with , we define the degree of regularity as follows:\nwhere is its top homogeneous component of .\nWhen the top homogeneous component of the system is semi-regular [5 ###reference_b5###] (also see Remark 4.6 ###reference_Thm6###), the degree of regularity coincides with the the degree defined as follows:\nFor the system with , we define as the minimum degree of terms with a non-positive coefficient in the power series\nIn [34 ###reference_b34###], Fröberg’s conjectured that the Hilbert series for the quotient ring of a “generic system” is determined by the power series (7 ###reference_###) when the characteristic of the coefficient field is zero.\nIndeed, this conjecture holds under the some conditions (see [2 ###reference_b2###, 34 ###reference_b34###, 48 ###reference_b48###]).\nMoreover, Diem in [16 ###reference_b16###] proved that, for any polynomial system that has the same number of variables and the degrees as the generic system, each coefficient of the Hilbert series for its quotient ring is larger than that for the generic system. Hence we expect the following conjecture from Fröberg’s conjecture.\nLet be a field of characteristic zero.\nFor any polynomials in such that , the following inequality holds:\nFor a semi-regular system, the degree of regularity tightly approximates the solving degree.\nHowever, for a non-semi-regular system such as a multi-graded polynomial system, it is known that its Gröbner basis is computed within a smaller degree than the degree of regularity (for example [40 ###reference_b40###, 41 ###reference_b41###]).\nBy the conjecture above, we do not expect to use the degree of regularity as a tool for the solving degree of a non-semi-regular system.\nAfter the works [32 ###reference_b32###, 36 ###reference_b36###] that observe the Gröbner basis algorithm on a non-semi-regular system generated by the direct attack against HFE [44 ###reference_b44###],\nDubois and Gama [27 ###reference_b27###] propose the first fall degree as a theoretical tool for apploximating the solving degree of a non-semi-regular system, and later Ding and Hodges [21 ###reference_b21###] reformulate the definition into an essential form (see Definition 3.4 ###reference_Thm4###). Note that the first fall degree does not always work, as it is possible to intentionally create a non-trivial syzygy [23 ###reference_b23###].\nIn order to capture a non-trivial degree fall occurs on the system, the first fall degree is defined using the non-trivial syzygies of the top homogeneous component for the polynomial system as follows: Let with the standard graded decomposition .\nFor any , we denote by the image of under the natural surjection .\nLet be a positive integer and .\nFor a positive integer , we consider\nThen, for\nwe define the following subspace of :\nFor any such that , the first fall degree is defined as\nThere are several works on the first fall degree.\nFor example, as an analysis for the direct attack against HFE [44 ###reference_b44###], Ding and Hodges [21 ###reference_b21###] give an upper bound for the first fall degree by constructing a non-trivial syzygy of the generated quadratic system and prove that it can be solved in a quasi logarithmic time.\nMoreover, as an analysis for the KS method [38 ###reference_b38###] for the MinRank problem [33 ###reference_b33###], Verbel et al. [49 ###reference_b49###] investigate a non-trivial syzygy (of the top homogeneous component) of the generated quadratic system and give a new complexity estimation with the first fall degree for this method.\nHowever, the previous evaluations for the first fall degree depend on each cryptosystem, and it is hard to apply to others.\nFor a given polynomial system, the degree of regularity and the first fall degree is defined on its top homogeneous component.\nIt suffices to show a homogeneous case for discussing them.\nThus, in this article, we always assume that a polynomial is homogeneous with respect to the standard grading." |
| }, |
| { |
| "section_id": "4", |
| "parent_section_id": null, |
| "section_name": "Estimation using the standard grading", |
| "text": "In Subsection 4.1 ###reference_###, we introduce a certain tool for apploximating the solving degree which coincides with the first fall degree of a polynomial system over a sufficiently large field. In Subsection 4.2 ###reference_###, we prove that the first fall degree of a non-semi-regular system over such a field is smaller than the degree of regularity in the semi-regular case." |
| }, |
| { |
| "section_id": "4.1", |
| "parent_section_id": "4", |
| "section_name": "Theoretical tool", |
| "text": "In this section, for a polynomial in , we only use the standard grading, i.e. .\nWe introduce the following definition for computing the first fall degree.\nFor homogeneous polynomials in the polynomial ring , we define as\nLet .\nBy the following lemmas, we see that coincides with the first fall degree for a sufficiently large .\nFor homogeneous polynomials such that , if , then we have .\nLet .\nPut .\nThere exists such that\nwhere holds as a representative.\nThen, since , we have .\nSince , it follows that .\nIf , we obtain a contradiction .\nThus . Therefore, we have .\n∎\nFor homogeneous polynomials such that , if , then we have .\nLet . Put .\nThere exists .\nIn particular, .\nAlthough an element for is contained in as generators, they do not appear since by and .\nThus, if there exists , then for some where as a representative.\nNamely, .\nBy , we have .\nThus, we obtain a contradiction .\nTherefore, .\n∎\nFor homogeneous polynomials such that , if , then we have .\nWhen , we have by Lemma 4.2 ###reference_Thm2###.\nThen , and we have by Lemma 4.3 ###reference_Thm3###.\nThus .\nSimilarly, when , we have the same result.\n∎\nIn 6 ###reference_###, we show that the assumption in this proposition, i.e. , is satisfied in attacks against actual cryptosystems.\nIn the next subsection, under the assumption, we see that the first fall degree of a non-semi-regular system is smaller than the degree of regularity in the semi-regular case." |
| }, |
| { |
| "section_id": "4.2", |
| "parent_section_id": "4", |
| "section_name": "Diem’s work", |
| "text": "In [17 ###reference_b17###], C. Diem investigates a relation between the regularity of a polynomial system and its syzygies.\nThe following definition is introduced in [17 ###reference_b17###].\nA homogeneous system is regular up to degree if the following multiplication map by is injective for each .\nIn their article [5 ###reference_b5###], Bardet et al. define a semi-regular system for a homogeneous system.\nNote that a homogeneous system is semi-regular if and only if is regular up to degree (see [17 ###reference_b17###]).\nThey also call a polynomial system semi-regular when its top homogeneous component is semi-regular.\nLet be a well-ordering on .\nFor two elements of the formal power series ring , we denote if the coefficients of these monomials of degree less than or equal to with respect to are the same.\nFor a -graded module over a -graded commutative ring , i.e. it has such that for any , we put .\nThe regularity in Definition 4.5 ###reference_Thm5### is characterized as follows:\nLet be the standard graded polynomial ring with .\nFor a homogeneous system with , the following conditions are equivalent:\nis regular up to degree\n\n\nSince , for the value in Definition 4.1 ###reference_Thm1###, note that we have\nFor a non-semi-regular system over a sufficiently large field, we see a relation between the first fall degree and the degree of regularity:\nFor a non-semi-regular system , we have\nMoreover, if and , we have\nIf for any , then the system is regular up to degree by Proposition 4.8 ###reference_Thm8###. Namely, the system is semi-regular which contradicts the assumption. Hence, there exists such that and we have .\nFurthermore, we assume that and .\nThen, since by Proposition 4.4 ###reference_Thm4###, we obtain .\n∎\nMore generally, we have a numerical upper bound of the first fall degree over a sufficiently large field.\nFor a homogeneous system , we put and define .\nFor a homogeneous system , we have\nMoreover, if and , we have\nSince it is obviously for , we may assume that . By Proposition 4.8 ###reference_Thm8###, we have . It follows that .\nMoreover, we have under the assumption and . Thus, .\n∎\nFor any homogeneous system , we have .\nSuppose .\nFor a homogeneous system , we have\nEquality holds in (8 ###reference_###) if and only if is semi-regular.\nFurthermore, if and , we have\nEquality holds in (9 ###reference_###) if and only if is semi-regular.\nIf the system is not semi-regular, then (8 ###reference_###) holds by Theorem 4.9 ###reference_Thm9###.\nWe assume that the system is semi-regular. Then, we have and . Moreover, by Theorem 4.11 ###reference_Thm11###, .\nHence, we obtain .\nConversely, if holds, then for every . By Proposition 4.8 ###reference_Thm8###, the system is regular up to degree and namely is semi-regular.\nSince holds under the assumption and , the remainder of the assertion also holds.\n∎\nBy Lemma 4.3 ###reference_Thm3###, the condition depending on the degrees and the number of variables gives that the assumption holds.\nIn this article, the assumption is only used to derive that . All our assertions about upper bounds of the first fall degree hold for any polynomial system satisfying ." |
| }, |
| { |
| "section_id": "5", |
| "parent_section_id": null, |
| "section_name": "Estimation using a multi-grading", |
| "text": "In this section, we introduce a value using a multi-degree to approximate the solving degree and, by extending Proposition 4.8 ###reference_Thm8### to a multi-grading, prove that the first fall degree of a multi-graded polynomial system over a sufficiently large field is bounded by this value.\nMoreover, we provide the theoretical assumption for applying the XL algorithm with a kernel search to a multi-graded polynomial system.\nIn this article, put and define . For and variables , denote and . For an order on , the symbol is defined as for any ." |
| }, |
| { |
| "section_id": "5.1", |
| "parent_section_id": "5", |
| "section_name": "Extend Proposition 4.8 to a multi-grading", |
| "text": "In this subsection, we show that a value in the following definition gives an upper bound for the first fall degree of a multi-graded polynomial system.\nLet be the -graded polynomial ring.\nFor -homogeneous polynomials in , we put\nand define \nMoreover, for a well-ordering on , we define .\nThe value in Definition 5.1 ###reference_Thm1### is similar to in [41 ###reference_b41###], but our value is also available for a wighted degree and is a more general concept.\nMoreover, we extend Definition 4.1 ###reference_Thm1### to the following:\nFor -homogeneous polynomial in the -graded polynomial ring with a well-ordering on , we define as\nIf , we denote this by .\nFor the standard grading, i.e. , the value coincides with in Definition 4.1 ###reference_Thm1###.\nWe can extend Definition 4.5 ###reference_Thm5### and Proposition 4.8 ###reference_Thm8### to a multi-grading as follows.\nLet be the -graded polynomial ring and be a well-ordering on .\nThen, -homogeneous system is regular up to degree d if the following multiplication map by is injective for each .\nLet be the -graded polynomial ring and be a well-ordering on compatible with on such that if .\nFor -homogeneous system with , the following conditions are equivalent:\nis regular up to degree\n\n\nThe proof proceeds in the same way as Proposition 4.8 ###reference_Thm8### ([17 ###reference_b17###]).\nFor the assertion 12, we suppose that the assertion holds for .\nThen, for any , by the injection\nwe have where .\nConversely, the map is injective if holds.\nFor the assertion 13, we suppose that the assertion holds for .\nFor any , the long exact sequence (6 ###reference_###) induces an exact sequence\nThen, by an assumption of the induction and the injection , we have\nFor the assertion 31, it suffices to prove the following statement and the case in particularly:\nIndeed, by the long exact sequence\nthe right hand condition for each gives the injection\nSuppose that there exists the minimum such that .\nThen, by and ,\nHence we have by the short exact sequence\nin the long exact sequence (6 ###reference_###).\nSince , we have .\nTherefore .\n∎\nBy this lemma, we see that the values in Definition 5.3 ###reference_Thm3### are an upper bound for the first fall degree of a multi-graded polynomial system.\nLet be a the -graded polynomial ring. Then is -graded with .\nFor -homogeneous polynomials with , we have\nMoreover, if with , and , we have\nSince the assertion is on , we may fix a well-ordering on in Lemma 5.4 ###reference_Thm4### as the graded lexicographic monomial ordering.\nWhen , the statement is obviously.\nIf , the power series (10 ###reference_###) has a negative coefficient at a certain such that .\nThen, by Lemma 5.4 ###reference_Thm4### and the positivity of the coefficients in the Hilbert series, there exists a non-Koszul syzygy of -degree equal to or less than with respect to .\nThus on -graded with .\nAssume that and it is -graded by .\nFor the standard grading, when and , we have the last assertion since by Lemma 4.4 ###reference_Thm4###.\n∎\nThe following theorem is a simple generalization of Theorem 5.5 ###reference_Thm5###, but it has an application (see 6 ###reference_###).\nLet be the -graded polynomial ring and be a well-ordering on compatible with on such that if .\nThen, for -homogeneous polynomials , we have\nSince it is obviously for , we may assume that .\nBy Lemma 5.4 ###reference_Thm4### with the same well-ordering as the statement, we have .\nIt follows that .\n∎" |
| }, |
| { |
| "section_id": "5.2", |
| "parent_section_id": "5", |
| "section_name": "Theoretical assumption for the XL algorithm with a kernel search", |
| "text": "As mentioned in Subsection 2.2 ###reference_###, the XL algorithm generates the Macaulay matrix of degree and its complexity is estimated as (4 ###reference_###).\nFor the generated Macaulay matrix, the value in (4 ###reference_###) is the number of columns minus the rank. If holds, a kernel vector obtained by a kernel search is always a solution to the system.\nIn this subsection, we consider a theoretical assumption for controlling the value .\nThe number of columns of the Macaulay matrix is that of the monomials of degree , and the row space is isomorphic to the vector space over .\nHence, the number of columns is , and the rank is .\nNamely,\nThe tool (5 ###reference_###) is rewritten as\nSince , we need to consider a case in which it is possible to combinatorially compute the Hilbert series .\nFor the dimension explained in Subsection 2.3 ###reference_###, if , then .\nHence we have to generally consider an upper bound for .\nMoreover, we discuss a more general statement applied to the multi-graded case (see Subsection 2.3 ###reference_###).\nThus we define the following tool as a generalization of (5 ###reference_###):\nFor -homogeneous polynomials , an integer and a well-ordering on , we define\nAs shown in Proposition 4.8 ###reference_Thm8### and Lemma 5.4 ###reference_Thm4###, the regularity for the polynomial system provides the partial information of the Hilbert series, and we generalize Definition 5.3 ###reference_Thm3### as follows:\nLet be a -graded system.\nFor a set , the system is regular on if, for any and , the following map is injective:\nNote that, for a -graded module , we set if has a negative component.\nThe following lemma is a generalization of the partial statement in Lemma 5.4 ###reference_Thm4###:\nLet .\nIf a -graded system is regular on , for every , the coefficient of in agrees with that in\nwhere .\nProposition 5.11 ###reference_Thm11### shows that the degree in Definition 11 ###reference_### is computed by the following value under the regularity introduced in Definition 5.8 ###reference_Thm8###.\nFor a -graded system and the well-order on , we define as follows:\nPut as\n and define\nLet be a well-ordering on .\nIf a -graded system is regular on where , then\nIn particular, we have .\nNote that if , if and is compatible with the standard degree, if and .\nFor the system arisen from an attack in multivariate cryptography, a non-positive coefficient in the multivariate power series (2 ###reference_###) is often negative, namely often holds.\nLet be homogeneous polynomials in such that .\nIf is regular on and ,\nthen we have\nSince , we have by Theorem 4.11 ###reference_Thm11###.\nMoreover, for , we have by Lemma 5.4 ###reference_Thm4###. Hence by Proposition 5.11 ###reference_Thm11###.\n∎" |
| }, |
| { |
| "section_id": "6", |
| "parent_section_id": null, |
| "section_name": "Application to multivariate cryptography", |
| "text": "In Subsection 5.1 ###reference_###, we give an upper bound for the first fall degree of a polynomial system whose top homogeneous component is -homogeneous under the assumption that and\nIn this section, we show that this assumption is satisfied in actual attacks against multivariate public-key signature schemes Rainbow [25 ###reference_b25###] and GMSS [14 ###reference_b14###] proposed in NIST PQC standardization project [43 ###reference_b43###].\nIn Subsection 6.1 ###reference_###, we recall the multivariate public-key signature scheme.\nIn Subsection 6.2 ###reference_### (resp. Subsection 6.3 ###reference_###), we explain the RBS attack (resp. the MinRank attack) as a key recovery attack against Rainbow (resp. GMSS), and show that the quadratic system generated by the attack satisfies the condition (12 ###reference_###)." |
| }, |
| { |
| "section_id": "6.1", |
| "parent_section_id": "6", |
| "section_name": "Key recovery attack", |
| "text": "For simplicity, we treat only homogeneous polynomials as polynomials and assume that the characteristic of the field is odd.\nThen, a quadratic homogeneous polynomial in corresponds to a symmetric matrix over .\nA polynomial system of gives a map by which is called a polynomial map.\nFor a set of easily invertible quadratic maps, a multivariate public-key signature scheme consists of the following three algorithms:\nKey generation: We randomly construct two invertible linear maps and , and which is called a central map, and then compute the composition The public key is given as . The tuple is a secret key.\nSignature generation: For a message , we compute .\nNext, we can compute an element of since is easily invertible. Consequently, we obtain a signature .\nVerification: We verify whether holds.\nFor a given public key , the key recovery attack generates two invertible linear maps and such that and forges a signature for any message.\nFor matrices , , and corresponding to and where and , respectively, we have\nThe RBS attack [20 ###reference_b20###] takes time to generate a part of and of UOV [39 ###reference_b39###] or its mulit-layerization, i.e. Rainbow [18 ###reference_b18###].\nFor a public key of HFE and a central matrix over , we have\nwhere , and .\nThe MinRank attack with the KS method takes time to generate a column of of HFE or its modifications, e.g. GMSS." |
| }, |
| { |
| "section_id": "6.2", |
| "parent_section_id": "6", |
| "section_name": "The RBS attack", |
| "text": "The RBS attack is an attack generating a secret key of Rainbow.\nLet and be Rainbow parameters.\nFor matrices corresponding to a public quadratic system where and , the RBS dominant system is a quadratic system in consisting of\nand the first components of\nSince and correspond to a row and a column of secret linear transformations, the RBS attack can generate a part of the secret key by solving the system.\nSince the polynomial ring is -graded by and , the top homogeneous component of the RBS dominant system is contained in and is -homogeneous.\nThen, for the -graded polynomial ring , the power series (10 ###reference_###) in Definition 5.1 ###reference_Thm1### is\nThe paper [40 ###reference_b40###] experimentally shows that the solving degree of the RBS dominant system is tightly approximated by in Definition 5.1 ###reference_Thm1### which is written as in [40 ###reference_b40###].\nAccording to [40 ###reference_b40###], for the Rainbow parameters Ia and IIIc/Vc [25 ###reference_b25###] proposed in NIST PQC 2nd round, the best complexities of the Rainbow-Based-Separation attack are given by and , respectively.\nSince and for the parameter Ia and IIIc/Vc, it follows that holds.\nIn particular, since by Theorem 5.5 ###reference_Thm5###, the assumption (12 ###reference_###) holds and the second half of Theorem 5.5 ###reference_Thm5### holds.\nNamely, the value in the paper [40 ###reference_b40###] gives an upper bound for the first fall degree .\nFurthermore, Smith-Tone and Perlner [47 ###reference_b47###] propose an XL algorithm as the bi-graded version of that explained in Subsection 2.3 ###reference_### and provide the complexity estimation for the attack by using a certain theoretical value.\nThen we can use Theorem 5.6 ###reference_Thm6### as a theoretical background of the estimation.\nHere note that their theoretical value in [47 ###reference_b47###] is defined by a non-positive coefficient appeared in (13 ###reference_###), namely, it is different from our value , and requires another theoretical background based on such as a big conjecture in Diem [16 ###reference_b16###]." |
| }, |
| { |
| "section_id": "6.3", |
| "parent_section_id": "6", |
| "section_name": "The MinRank attack using the KS method", |
| "text": "Multivariate signature scheme GMSS [14 ###reference_b14###] is a minus and vinegar modification of HFE [44 ###reference_b44###].\nThe MinRank attack with the KS method [38 ###reference_b38###] is an attack generating a secret key of a multivariate cryptosystem such as GMSS and Rainbow.\nAlthough a public quadratic system of GMSS is defined over the field of order two, the complexity of the attack is dominated by that of a Gröbner basis algorithm for solving a certain system over a very large field, say the KS system.\nLet and be GMSS parameters.\nFor matrices over corresponding to the public quadratic system , the MinRank attack finds in such that\nwhere .\nThen, since a found vector corresponds to a column vector of a certain linear transform over , the MinRank attack can generate a part of a secret key.\nFor finding , the Kinis-Shamir modeling solves the KS system in which is a quadratic system in and is the components of\nwhere , and .\nSince the polynomial ring is -graded by\nthe top homogeneous component of the KS system is contained in and is -homogeneous where .\nThen, for the -graded polynomial ring , the power series (10 ###reference_###) in Definition 5.1 ###reference_Thm1### is\nThe paper [41 ###reference_b41###] experimentally shows that the solving degree of the KS system from the Minrank problem [33 ###reference_b33###] is approximated by in Definition 5.1 ###reference_Thm1### which is written as in [41 ###reference_b41###].\nThen they also show that the value which is smaller than the order , i.e. the assumption (12 ###reference_###) holds, improves the complexity of the MinRank attack with the KS method against Rainbow.\nIn particular, the value in [41 ###reference_b41###] gives an upper bound for the first fall degree .\nMeanwhile, the GMSS parameter sets for a security of , and proposed in NIST PQC 2nd round take and [14 ###reference_b14###], respectively.\nThus, for these proposed parameter sets, the order in the definition of the first fall degree is around and , respectively, and the complexities at of the KS method are given by and , respectively.\nIt follows that holds.\nIn particular, by Theorem 5.5 ###reference_Thm5###, and the second half of Theorem 5.5 ###reference_Thm5### holds.\nTherefore, this gives an upper bound for the first fall degree ." |
| } |
| ], |
| "appendix": [], |
| "tables": {}, |
| "image_paths": {}, |
| "validation": true, |
| "references": [], |
| "url": "http://arxiv.org/html/2007.14729v3", |
| "new_table": {} |
| } |
