System Of Multivariate Polynomial Equations Research Articles

Progress in Multivariate Cryptography: Systematic Review, Challenges, and Research Directions

Multivariate Public Key Cryptosystem (MPKC) seem to be promising toward future digital security even in the presence of quantum adversaries. MPKCs derive their security from the difficulty of solving a random system of multivariate polynomial equations over a finite field, which is known to be an NP-hard problem. This article aims at presenting a comprehensive survey that covers multivariate public key encryption and signature schemes specifically targeting toward security, efficiency, and parameter choice. The survey starts by giving an overview of the existing security challenges which include structural attacks such as MinRank attack, differential attack , and finding Gröbner basis for direct attack , and so on. Additionally, it discusses the necessary algorithms for the implementation of the multivariate schemes. This study also compares the promising multivariate encryption and signature schemes. The critical open challenges that are reviewed in this survey will serve as a single comprehensive source of information on multivariate encryption and signature schemes and a ready reference for researchers working in this rising area of public key cryptography.

The complexity of solving Weil restriction systems

The solving degree of a system of multivariate polynomial equations provides an upper bound for the complexity of computing the solutions of the system via Gröbner basis methods. In this paper, we consider polynomial systems that are obtained via Weil restriction of scalars. The latter is an arithmetic construction which, given a finite Galois field extension k↪K, associates to a system F defined over K a system Weil(F) defined over k, in such a way that the solutions of F over K and those of Weil(F) over k are in natural bijection. In this paper, we find upper bounds for the complexity of solving a polynomial system Weil(F) obtained via Weil restriction in terms of algebraic invariants of the system F.

Enumeration of labelled 4-regular planar graphs II: Asymptotics

This work is a follow-up of the article (Noy et al., 2019), where the authors solved the problem of counting labelled 4-regular planar graphs. In this paper, we obtain a precise asymptotic estimate for the number gn of labelled 4-regular planar graphs on n vertices. Our estimate is of the form gn∼g⋅n−7/2ρ−nn!, where g>0 is a constant and ρ≈0.24377 is the radius of convergence of the generating function ∑n≥0gnxn/n!, and conforms to the universal pattern obtained previously in the enumeration of several classes of planar graphs. In addition to analytic methods, our solution needs intensive use of computer algebra in order to deal with large systems of multivariate polynomial equations. We also obtain asymptotic estimates for the number of 2- and 3-connected 4-regular planar graphs, and for the number of 4-regular simple maps, both connected and 2-connected.

On generalizing Descartes' rule of signs to hypersurfaces

We give partial generalizations of the classical Descartes' rule of signs to multivariate polynomials (with real exponents), in the sense that we provide upper bounds on the number of connected components of the complement of a hypersurface in the positive orthant. In particular, we give conditions based on the geometrical configuration of the exponents and the sign of the coefficients that guarantee that the number of connected components where the polynomial attains a negative value is at most one or two. Our results fully cover the cases where such an upper bound provided by the univariate Descartes' rule of signs is one. This approach opens a new route to generalize Descartes' rule of signs to the multivariate case, differing from previous works that aim at counting the number of positive solutions of a system of multivariate polynomial equations.

A Column Space Based Approach to Solve Systems of Multivariate Polynomial Equations

We propose a novel approach to solve systems of multivariate polynomial equations, using the column space of the Macaulay matrix that is constructed from the coefficients of these polynomials. A multidimensional realization problem in the null space of the Macaulay matrix results in an eigenvalue problem, the eigenvalues and eigenvectors of which yield the common roots of the system. Since this null space based algorithm uses well-established numerical linear algebra tools, like the singular value and eigenvalue decomposition, it finds the solutions within machine precision. In this paper, on the other hand, we determine a complementary approach to solve systems of multivariate polynomial equations, which considers the column space of the Macaulay matrix instead of its null space. This approach works directly on the data in the Macaulay matrix, which is sparse and structured. We provide a numerical example to illustrate our new approach and to compare it with the existing null space based root-finding algorithm.

Globally Optimal H2-Norm Model Reduction: A Numerical Linear Algebra Approach

We show that the H2-norm model reduction problem for single-input/single-output (SISO) linear time-invariant (LTI) systems is essentially an eigenvalue problem (EP), from which the globally optimal solution(s) can be retrieved. The first-order optimality conditions of this model reduction problem constitute a system of multivariate polynomial equations that can be converted to an affine (or inhomogeneous) multiparameter eigenvalue problem (AMEP). We solve this AMEP by using the so-called augmented block Macaulay matrix, which is introduced in this paper and has a special (block) multi-shift invariant null space. The set of all stationary points of the optimization problem, i.e., the (2r)-tuples (r is the order of the reduced model) of affine eigenvalues and eigenvectors of the AMEP, follows from a standard EP related to the structure of that null space. At least one of these (2r)-tuples corresponds to the globally optimal solution of the H2-norm model reduction problem. We present a simple numerical example to illustrate our approach.

Extended Galbraith’s test on the anonymity of IBE schemes from higher residuosity

At PKC 2019, Clear and McGoldrick presented the first identity-based encryption (IBE) scheme that is group homomorphic for addition modulo a poly-sized prime e. Assuming that deciding solvability of a special system of multivariate polynomial equations is hard, they proved that their scheme for $$e>2$$ is anonymous. In this paper, we review the classical Galbraith’s test on the anonymity of the first pairing-free IBE scheme due to Cocks. With the eye of the reciprocity law for $$\mathbb {F}_\mathtt {q}[x]$$ , we can have a profound understanding of the test and naturally extend it to give a practical attack on the anonymity of the Clear–McGoldrick IBE scheme. Furthermore, we believe that our technique plays a crucial role in anonymizing IBE schemes from higher residuosity.

On the termination of the general XL algorithm and ordinary multinomials

The XL algorithm is an algorithm for solving overdetermined systems of multivariate polynomial equations, which was initially introduced for quadratic equations. However, the algorithm works for polynomials of any degree, and in this paper we will focus on the performance of XL for polynomials of degree ≥3, where the optimal termination value of the parameter D is still unknown. We prove that the XL algorithm terminates at a certain value of D in the case that the number of equations exceeds the number of variables by 1 or 2. We also give strong evidence that this value is best possible, and we show that this value is smaller than the degree of regularity. Part of our analysis requires proving that ordinary multinomials are strongly unimodal, and this result may be of independent interest.

Globally Optimal Least-Squares ARMA Model Identification is an Eigenvalue Problem

We show that globally optimal least-squares identification of autoregressive moving-average (ARMA) models is an eigenvalue problem (EP). The first order optimality conditions of this identification problem constitute a system of multivariate polynomial equations, in which most variables appear linearly. This system is basically a multiparameter eigenvalue problem (MEP), which we solve by iteratively building a so-called block Macaulay matrix, the null space of which is block multi-shift-invariant. The set of all stationary points of the optimization problem, i.e., the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -tuples of eigenvalues and eigenvectors of the MEP, follows from a standard EP related to the multidimensional realization problem in that null space. At least one of these <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> -tuples corresponds to the global minimum of the original least-squares objective function. Contrary to existing heuristic techniques, this approach yields the globally optimal parameters of the ARMA model. We provide a numerical example to illustrate the new identification method.

Algebraic Techniques for Covering Arrays and related Structures

In this paper, we extend an existing algebraic modelling technique for covering arrays by considering additional properties which are required when these structures are applied in practice in a branch of software testing called combinatorial testing. Corresponding to these properties, we give semantically equivalent systems of multivariate polynomial equations.

A Green's function based analytical method for forward and inverse modeling of quasi-periodic nanostructured surfaces

We present an efficient Green's function based analytical method for forward but particularly also for the inverse modeling of light scattering by quasi-periodic and aperiodic surface nanostructures. In the forward modeling, good agreement over an important texture amplitude range is achieved between the developed formalism and exact rigorous calculations on the one hand and angle resolved light scattering measurements of complex quasi-periodic SiO2-Au nanopatterned interfaces on the other hand. Exploiting our formalism, we demonstrate for the first time how the inverse problem of quasi-periodic surface textures for a desired multiresonant absorption response can be expressed in terms of coupled systems of multivariate polynomial equations of the height profile's Fourier amplitudes. A good estimate of the required surface profile can thus be obtained in a computationally cheap manner via solving the multivariate polynomial equations. In principle, the inverse modeling formalism introduced here can be implemented in conjunction with any scattering model that provides expressions of the coupling coefficients between different modes in terms of the surface texture height profile.

Algebraic Cryptanalysis Scheme of AES-256 Using Gröbner Basis

The zero-dimensional Gröbner basis construction is a crucial step in Gröbner basis cryptanalysis on AES-256. In this paper, after performing an in-depth study on the linear transformation and the system of multivariate polynomial equations of AES-256, the zero-dimensional Gröbner basis construction method is proposed by choosing suitable term order and variable order. After giving a detailed construction process of the zero-dimensional Gröbner basis, the necessary theoretical proof is presented. Based on this, an algebraic cryptanalysis scheme of AES-256 using Gröbner basis is proposed. Analysis shows that the complexity of our scheme is lower than that of the exhaustive attack.

Comparative Study of Algebraic Attacks

Cryptographic schemes have an algebraic structure and can be described as multivariate polynomial equations. Even though algebra is the default tool in the cryptanalysis of asymmetric cryptosystems, there has been recently an increase in interest in the use of algebraic cryptanalysis techniques in the analysis of symmetric cryptosystems. The basic idea behind the algebraic attack is to express the whole cryptosystem as a large system of multivariate polynomial equations, then considers methods for solving the system to recover the key. Solving multivariate polynomial systems is a typical problem studied in Algebraic Geometry and Computational Algebra. Computing Grobner basis is the best well known method to solve this problem. Finding grobner bases is a difficult task which requires lots of computational resources. This paper discusses and explains in depth different algorithms to compute grobner bases using examples. This paper also, compares these algorithms from the point of views of accuracy and efficiency (the required resources: time and effort) to get the accurate results. Finally, the worthiness of these algorithms to be applied to cryptanalysis has been discussed.

On polynomial semantics for propositional logics

Some properties and an algorithm for solving systems of multivariate polynomial equations over finite fields are presented. It is then shown how formulas of propositional logics (particularly of finite-valued logics and paraconsistent logics) can be translated into polynomials over finite fields in such a way that several logic problems are expressed in terms of algebraic problems. Consequently, algebraic properties and algorithms can be used to solve the algebraically-represented logic problems. The methods described herein combine and generalise those of various previous works.

On the Complexity of Computing Critical Points with Gröbner Bases

Computing the critical points of a polynomial function $q\in\mathbb{Q}[X_1,\ldots,X_n]$ restricted to the vanishing locus $V\subset\mathbb{R}^n$ of polynomials $f_1,\ldots, f_p\in\mathbb{Q}[X_1,\ldots, X_n]$ is of first importance in several applications in optimization and in real algebraic geometry. These points are solutions of a highly structured system of multivariate polynomial equations involving maximal minors of a Jacobian matrix. We investigate the complexity of solving this problem by using Grobner basis algorithms under genericity assumptions on the coefficients of the input polynomials. The main results refine known complexity bounds (which depend on the maximum $D=\max(\deg(f_1),\ldots,\deg(f_p),\deg(q))$) to bounds which depend on the list $(\deg(f_1),\ldots,\deg(f_p),\deg(q))$: we prove that the Grobner basis computation can be performed in $\delta^{O(\log(A)/\log(G))}$ arithmetic operations in $\mathbb{Q}$, where $\delta$ is the algebraic degree of the ideal vanishing on the critical poin...

Parallel Gaussian elimination for XL family over GF(2)

ABSTRACTThe security of many cryptographic systems is based on the difficulty of solving systems of quadratic multivariate polynomial equations. XL family (which stands for eXtended Linearization) is a traditional type of algorithm for solving systems of multivariate polynomial equations over finite fields. The main idea of these algorithms is to extend the initial system by monomials multiplications and use linear algebra techniques to solve the system. The overall complexity of XL family is essentially determined by the workload of Gaussian elimination step. In this paper, we use the structured Gaussian elimination and parallel fast Gaussian elimination to reduce the complexity of XL family over GF(2). Because of the sparsity of the extended system, structured Gaussian elimination is applied to reduce the dimensions of the extended system. Thus, a great reduction of the storage can be obtained. By slightly modifying the logic of ordinarily Gaussian elimination, we parallelly compute the reduced system. Theory analysis indicates that the complexity of our improved XL, XL_SP, is (λT)2, which is far lower than (7/64)T2.8, the complexity of XL, where T is the number of monomials and λ is the ratio of the dimensions of the reduced system to the extended system. Further experiments to Hidden Field Equation cryptosystems show that about 90% reduction of the extended system is achieved by using structured Gaussian elimination, which means λ is about 1/10 in our experiments. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Spherical Radon transform and the average of the condition number on certain Schubert subvarieties of a Grassmannian

We study the average complexity of certain numerical algorithms when adapted to solving systems of multivariate polynomial equations whose coefficients belong to some fixed proper real subspace of the space of systems with complex coefficients. A particular motivation is the study of the case of systems of polynomial equations with real coefficients. Along these pages, we accept methods that compute either real or complex solutions of these input systems. This study leads to interesting problems in Integral Geometry: the question of giving estimates on the average of the normalized condition number along great circles that belong to a Schubert subvariety of the Grassmannian of great circles on a sphere. We prove that this average equals a closed formula in terms of the spherical Radon transform of the condition number along a totally geodesic submanifold of the sphere.

Computing the torsion points of a variety defined by lacunary polynomials

We present an algorithm for computing the set of torsion points satisfying a given system of multivariate polynomial equations. Its complexity is quasilinear in the logarithm of the degree of the input equations and exponential in their number of non zero terms and variables.

Efficient polynomial system-solving by numerical methods

These pages contain a short overview on the state of the art of efficient Numerical Analysis methods that solve systems of multi-variate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Steve Smale and Mike Shub, which set the foundations of this approach to polynomial system-solving.

Rational Univariate Reduction via toric resultants

We describe algorithms for solving a given system of multivariate polynomial equations via the Rational Univariate Reduction (RUR). We compute the RUR from the toric resultant of the input system. Our algorithms derandomize several of the choices made in similar prior algorithms. We also propose a new derandomized algorithm for solving an overdetermined system. Finally, we analyze the computational complexity of the algorithm, and discuss its implementation and performance.