System Of Quadratic Equations Research Articles

Exact formula for the solution of degenerate system of quadratic equations

Abstract. The paper devoted to the solution of nonlinear system of equations F(x) = 0, where the mapping F is quadratic, acting from ℝn→ℝn. We consider the case, when the derivative F' is degenerate at the solution point. Based on the constructions of the p-regularity theory was proposed a 2-factor method for solving singular system of equations, which converges at a quadratic rate. Moreover, an exact formula is obtained for solving this quadratic system of equations in the 2-regular case of the mapping F(x).

Pole dynamics and an integral of motion for multiple SLE(0)

We describe the Loewner chains of the real locus of a class of real rational functions whose critical points are on the real line. Our main result is that the poles of the rational function lead to explicit formulas for the dynamical system that governs the driving functions. Our formulas give a simple method for mapping the class of rational functions into solutions to a non-trivial system of quadratic equations, and for directly showing that the curves in the real locus satisfy geometric commutation. These results are entirely self-contained and have no reliance on probabilistic objects, but make use of an integral of motion for the Loewner chain that is motivated by ideas from conformal field theory. We also show that the dynamics of the driving functions are a special case of the Calogero–Moser integrable system, restricted to a particular submanifold of phase space carved out by the Lax matrix. Our approach complements a recent result of Peltola and Wang, who showed that the real locus is the deterministic κ→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\kappa \\rightarrow 0$$\\end{document} limit of the multiple SLE(κ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\kappa )$$\\end{document} curves.

Novel Weakness Multivariate Quadratic Structures Detected within Macaulay Matrix

The security of a Multivariate Public-Key Cryptosystem (MPKC) is based on the hard mathematical problem of solving Multivariate Quadratic (MQ) equations over finite fields, also known as the MQ problem. An MPKC has the potential to be a post-quantum cryptosystem. In this paper, we identify new weaknesses in the Macaulay matrix identified via Wang's technique, which was initially designed for solving multivariate quadratic equation systems. This new weakness occurs in the case of random coefficients in any column vector for different variables of monomials and random coefficients are assigned to other monomials. The weakness is exposed through the use of Gaussian elimination to obtain a univariate equation. We illustrate our findings using a random example.

Exact Formula for Solving a Degenerate System of Quadratic Equations

Exact Formula for Solving a Degenerate System of Quadratic Equations

On security of GIS systems with N-tier architecture and family of graph based ciphers

Discovery of q-regular tree description in terms of an infinite system of quadratic equations over finite field Fq had an impact on Computer Science, theory of graph based cryptographic algorithms in particular. It stimulates the development of new graph based stream ciphers. It turns out that such encryption instruments can be efficiently used in GIS protection systems which use N-Tier Architecture. We observe known encryption algorithms based on the approximations of regular tree, their modifications defined over arithmetical rings and implementations of these ciphers. Additionally new more secure graph based ciphers suitable for GIS protection will be presented.Algorithms are constructed using vertices of bipartite regular graphs D(n,K) defined by a finite commutative ring K with a unit and a non-trivial multiplicative group K*. The partition of such graphs are n-dimensional affine spaces over the ring K. A walk of even length determines the transformation of the transition from the initial to the last vertex from one of the partitions of the graph. Therefore, the affine space Kn can be considered as a space of plaintexts, and walking on the graph is a password that defines the encryption transformation.With certain restrictions on the password the effect when different passwords with K*)2s, s <[(n+5)/2]/2 correspond to different ciphertexts of the selected plaintext with Kn can be achieved.In 2005, such an algorithm in the case of the finite field F127 was implemented for the GIS protection. Since then, the properties of encryption algorithms using D(n, K) graphs (execution speed, mixing properties, degree and density of the polynomial encryption transform) have been thoroughly investigated.The complexity of linearization attacks was evaluated and modifications of these algorithms with the resistance to linearization attacks were found. It turned out that together with D(n, K) graphs, other algebraic graphs with similar properties, such as A(n, K) graphs, can be effectively used.The article considers several solutions to the problem of protecting the geological information system from possible cyberattacks using stream ciphers based on graphs. They have significant advantages compared to the implemented earlier systems.

On the Finite Complexity of Solutions in a Degenerate System of Quadratic Equations: Exact Formula.

The paper describes an application of the p-regularity theory to Quadratic Programming (QP) and nonlinear equations with quadratic mappings. In the first part of the paper, a special structure of the nonlinear equation and a construction of the 2-factor operator are used to obtain an exact formula for a solution to the nonlinear equation. In the second part of the paper, the QP problem is reduced to a system of linear equations using the 2-factor operator. The solution to this system represents a local minimizer of the QP problem along with its corresponding Lagrange multiplier. An explicit formula for the solution of the linear system is provided. Additionally, the paper outlines a procedure for identifying active constraints, which plays a crucial role in constructing the linear system.

Robust feasibility of systems of quadratic equations using topological degree theory

Robust feasibility of systems of quadratic equations using topological degree theory

Algebraic signature algorithms with a hidden group group, based on hardness of solving systems of quadratic equations

A new-type algebraic digital signature schemes on non-commutative associative algebras are developed using technique of performing exponentiation operations in a hidden group. The signature contains two elements: a randomization integer e and a vector S. The used verification equations are characterized in multiple entries of the signature element S. The post-quantum security of the introduced signature algorithms is provided by the computational difficulty of solving a system of many quadratic equations in many variables, like in the public-key multivariate cryptosystems. However in the former case the quadratic equations are set over the finite fields having the order of significantly larger size.

Post-quantum signature algorithms on noncommutative algebras, using difficulty of solving systems of quadratic equations

A recently proposed new concept for constructing algebraic signature schemes with a hidden group is used to develop two new post-quantum signature algorithms on four-dimensional and six-dimensional finite noncommutative associative algebras. As in the case of the post-quantum signature schemes of multivariate cryptography, the security of the introduced algorithms is based on the computational difficulty of solving systems of many quadratic equations (44 and 42) with many unknowns (40 and 36). The signature represents a pair of a natural number e and a vector S. The latter enters three times in the verification equation used, providing resistance to the forging signature attacks by using the value S as a fitting parameter. A public key is generated in the form of a set of vectors, each of which is calculated as the product of triples of secret vectors. With a special choice of these triples, a signer has the possibility of calculating a signature that satisfies the verification equation. The developed post-quantum signature algorithms are practical, having sufficiently small signatures (97 and 109 bytes), public keys (387 and 291 bytes), and secret keys (315 and 451 bytes). A significant difference from the public-key algorithms of multivariate cryptography is that in the developed signature schemes, the system of quadratic equations is derived from the formulas for generating the public-key elements in the form of a set of vectors of m-dimensional finite noncommutative algebra with an associative vector multiplication operation. The formulas define the system of n quadratic vector equations, which reduces to the system of m quadratic equations over a finite field.

The Legendre pseudorandom function as a multivariate quadratic cryptosystem: security and applications

AbstractSequences of consecutive Legendre and Jacobi symbols as pseudorandom bit generators were proposed for cryptographic use in 1988. Major interest has been shown towards pseudorandom functions (PRF) recently, based on the Legendre and power residue symbols, due to their efficiency in the multi-party setting. The security of these PRFs is not known to be reducible to standard cryptographic assumptions. In this work, we show that key-recovery attacks against the Legendre PRF are equivalent to solving a specific family of multivariate quadratic (MQ) equation system over a finite prime field. This new perspective sheds some light on the complexity of key-recovery attacks against the Legendre PRF. We conduct algebraic cryptanalysis on the resulting MQ instance. We show that the currently known techniques and attacks fall short in solving these sparse quadratic equation systems. Furthermore, we build novel cryptographic applications of the Legendre PRF, e.g., verifiable random function and (verifiable) oblivious (programmable) PRFs.

A New Analysis of the Kipnis-Shamir Method Solving the MinRank Problem

The MinRank problem is investigated as a problem related to rank attacks in multivariate cryptography and the decoding of rank codes in coding theory. The Kipnis-Shamir method is one of the methods to solve the problem, and recently, significant progress has been made in its complexity estimation by Verbel et al. As this method reduces the problem to an MQ problem, which asks for a solution to a system of quadratic equations, its complexity depends on the solving degree of a quadratic system deduced from the method. A theoretical value introduced by Verbel et al. approximates the minimal solving degree of the quadratic systems in the method although their value is defined under a certain limit for the system considered. A quadratic system outside their limitation often has a larger solving degree, but the solving complexity is not always higher because it has a smaller number of variables and equations. Thus, in order to discuss the best complexity of the Kipnis-Shamir method, a theoretical value is needed to approximate the solving degree of each quadratic system deduced from the method. A quadratic system deduced from the Kipnis-Shamir method always has a multi-degree, and the solving complexity is influenced by this property. In this study, we introduce a theoretical value defined by such a multi-degree and show that it approximates the solving degree of each quadratic system. Thus, the systems deduced from the method are compared, and the best complexity is discussed. As an application, for the MinRank attack using the Kipnis-Shamir method against the multivariate signature scheme Rainbow, we show a case in which a deduced quadratic system outside Verbel et al.'s limitation is the best. In particular, the complexity estimation of the MinRank attack using the KS method against the Rainbow parameter sets I, III and V is reduced by about 172, 140 and 212 bits, respectively, from Verbel et al.'s estimation.

On a random entanglement problem

Abstract We study a model for the entanglement of a two-dimensional reflecting Brownian motion in a bounded region divided into two halves by a wall with three or more small windows. We map the Brownian motion into a Markov Chain on the fundamental groupoid of the region. We quantify entanglement of the path with the length of the appropriate element in this groupoid. Our main results are a law of large numbers and a central limit theorem for this quantity. The constants appearing in the limit theorems are expressed in terms of a coupled system of quadratic equations.

Recent progress in the security evaluation of multivariate public‐key cryptography

AbstractMultivariate public‐key cryptography (MPKC) is considered a leading candidate for post‐quantum cryptography (PQC). It is based on the hardness of the multivariate quadratic polynomial (MQ) problem, which is a problem of finding a solution to a system of quadratic equations over a finite field. In this paper, we survey some recent progress in the security analysis of MPKC. Among various existing multivariate schemes, the most important one is the Rainbow signature scheme proposed by Ding et al. in 2005, which was later selected as a finalist in the third round of the PQC standardization project by the National Institute of Standards and Technology. Under the circumstances, some recent research studies in MPKC have focussed on the security analysis of the Rainbow scheme. In this paper, the authors first explain efficient algorithms for solving the MQ problem and the research methodology for estimating their complexity in MPKC. Then, the authors survey some recent results related to the security analysis of the Rainbow scheme. In particular, the authors provide a detailed description of the complexity analysis for solving the bi‐graded polynomial systems studied independently by Nakamura et al. and Smith‐Tone et al., and then expound the rectangular MinRank attack against Rainbow proposed by Beullens.

Алгебраические алгоритмы ЭЦП, основанные на трудности решения систем уравнений

Purpose of work is the development of post-quantum digital signature algorithms with comparatively small sizes of the public and secret keys and the signature. Research method is the use of a new concept for constructing signature algorithms on finite non-commutative associative algebras, which is distinguished by the multiple occurrences of the signature S in the power verification equation. A public key is generated in the form of a set of vectors every of which is calculated as the product of triples of secret vectors. With a special choice of these triples, it is possible to calculate a signature that satisfies the verification equation. Results of the study are two developed algebraic post-quantum digital signature algorithms of a new type, security of which is based on the computational difficulty of solving systems of many quadratic equations with many unknowns. The difference from the public-key algorithms of multivariate cryptography is that the system of quadratic equations is derived from the formulas for generating the public-key elements in the form of a set of vectors of m-dimensional finite non-commutative algebra with an associative vector multiplication operation. The said formulas define the system of n quadratic vector equations, which reduces to the system of mn quadratic equations over a finite field. Thanks to the “natural” mechanism for the occurrence of the specified system, it is set above the field, the order of which has a large size (97 and 129 bits). The used procedures for generating the public key and signature include the exponentiation operations to the degree of a large size (96 and 128 bits), which are performed over the elements of the secret (hidden) commutative group contained in the algebra. The signature is formed in the form of two elements: a randomizing natural number e and a “fitting” vector S. The signature authentication equation includes a multiple occurrence of the S element and every entry of the vector S is associated with the formation of a product that is exponentiated to a degree dependent on the value of the e element. A significant reduction in the size of public and secret keys and signatures has been achieved, as well as an increase in performance compared to foreign analogues, considered currently as basic algorithms for the adoption of post-quantum digital signature standards.

Post-Quantum Signature Algorithms on Non-Commutative Algebras, Using Difficulty of Solving Systems of Quadratic Equations

Post-Quantum Signature Algorithms on Non-Commutative Algebras, Using Difficulty of Solving Systems of Quadratic Equations

Killing Tensor Fields of Third Rank on a Two-Dimensional Riemannian Torus

A rank m symmetric tensor field on a Riemannian manifold is called a Killing field if the symmetric part of its covariant derivative is equal to zero.
 
 Such a field determines the first integral of the geodesic flow which is a degree m homogeneous polynomial in velocities. There exist global isothermal coordinates on a two-dimensional Riemannian torus such that the metric is of the form ds^2= λ(z)|dz|^2 in the coordinates. The torus admits a third rank Killing tensor field if and only if the function λ satisfies the equation R(∂/∂z（λ(c∆^-1λ_zz+a))= 0 with some complex constants a and c≠0. The latter equation is equivalent to some system of quadratic equations relating Fourier coefficients of the function λ. If the functions λ and λ + λ_0 satisfy the equation for a real constant λ0, 0, then there exists a non-zero Killing vector field on the torus.

New complexity estimation on the Rainbow-Band-Separation attack

Multivariate public key cryptography is a candidate for post-quantum cryptography, and it allows generating particularly short signatures and fast verification. The Rainbow signature scheme proposed by Ding and Schmidt is such a multivariate cryptosystem, and it is considered secure against all known attacks. The Rainbow-Band-Separation attack recovers a secret key of Rainbow by solving certain systems of quadratic equations, and its complexity is estimated by the well-known theoretical value called the degree of regularity. However, the degree of regularity is generally larger than the solving degree in experiments, and an accurate estimation cannot be obtained. In this article, we propose a new theoretical value for the complexity of the Rainbow-Band-Separation attack using a Gröbner basis algorithm, which provides a more precise estimation compared to that using the degree of regularity. This theoretical value is deduced by the two-variable power series∏i=1m(1−t1di1t2di2)(1−t1)n1(1−t2)n2. Since the two-variable power series coincides with the one-variable power series at t1=t2 deriving the degree of regularity, the theoretical value is less than or equal to the degree of regularity under a certain condition. Moreover, we show a relation between the Rainbow-Band-Separation attack using the hybrid approach and the HighRank attack. By considering this relation and our theoretical value, we obtain a new complexity estimation for the Rainbow-Band-Separation attack. Furthermore, applying our theoretical value to the complexity formula used in the NIST PQC 2nd round, we show that a slight modification of the proposed Rainbow parameter sets is required. Consequently, we provide a new theoretical value for generally estimating the solving degree of a bi-graded polynomial system, which can influence the parameter selection of Rainbow in the NIST PQC standardization project.

Analysis of security of post-quantum algorithm of Rainbow electronic signature against potential attacks

Multidimensional public key cryptography is a candidate for post-quantum cryptography, and it makes it possible to generate particularly short signatures and quick verification. The Rainbow signature scheme proposed by J. Dean and D. Schmidt is such a multidimensional cryptosystem and it is considered to be protected against all known attacks. The need for research on Rainbow ES is justified by the fact that there is a need to develop and adopt a post-quantum national securities standard, and that in the process of the US NIST competition on the mathematical basis of cryptographic transformation method Rainbow, promising results. Therefore, it is considered important to take them into account and use them in Ukraine. The Rainbow signature scheme can be implemented simply and efficiently using linear algebra methods over a small finite field and, in particular, creates shorter signatures than those used in RSA and other post-quantum signatures [1]. In the 2nd round of NIST PQC, protected sets of Rainbow parameters are offered and several attacks on them are analyzed [1]. When comparing ES, preference is given to ES algorithms that have been selected according to unconditional criteria, as well as those that have better indicators for integral conditional criteria, because such a technique is more rational. In particular, the Rainbow-Band-Separation (RBS) attack [2] is the best known Rainbow attack with a certain set of parameters and is important. The Rainbow-Band-Separation attack restores the Rainbow secret key by solving certain systems of quadratic equations, and its complexity is measured by a well-known measure called the degree of regularity. However, as a rule, the degree of regularity is greater than the degree of solution in experiments, and it is impossible to obtain an accurate estimate. The paper proposes a new indicator of the complexity of the Rainbow-Band-Separation attack using F4 algorithm, which gives a more accurate estimate compared to the indicator that uses the degree of regularity.
 The aim of the work is a comparative analysis of ES based on MQ-transformations on the criterion of stability-complexity and an attempt to understand the security of Rainbow against RBS attack using F4.

Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval

We aim to find a solution \({\varvec{x}}\in {\mathbb {C}}^n\) to a system of quadratic equations of the form \(b_i=|{\varvec{a}}_i^*{\varvec{x}}|^2\), \(i=1,2,\ldots ,m\), e.g., the well-known NP-hard phase retrieval problem. As opposed to recently proposed state-of-the-art nonconvex methods, we revert to the semidefinite relaxation (SDR) PhaseLift convex formulation and propose a successive and incremental nonconvex optimization algorithm, termed as IncrePR, to indirectly minimize the resulting convex problem on the cone of positive semidefinite matrices. Our proposed method overcomes the excessive computational cost of typical SDP solvers as well as the need of a good initialization for typical nonconvex methods. For Gaussian measurements, which is usually needed for provable convergence of nonconvex methods, restart-IncrePR solving three consecutive PhaseLift problems outperforms state-of-the-art nonconvex gradient flow based solvers with a sharper phase transition of perfect recovery and typical convex solvers in terms of computational cost and storage. For more challenging structured (non-Gaussian) measurements often occurred in real applications, such as transmission matrix and oversampling Fourier transform, IncrePR with several consecutive repeats can be used to find a good initial guess. With further refinement by local nonconvex solvers, one can achieve a better solution than that obtained by applying nonconvex gradient flow based solvers directly when the number of measurements is relatively small. Extensive numerical tests are performed to demonstrate the effectiveness of the proposed method.

A structural attack to the DME-(3,2,q) cryptosystem

We present a structural attack on the DME cryptosystem with parameters (3,2,q). The attack recovers 10 of the 12 coefficients of the first linear map. We also show that, if those 12 coefficients were known, the rest of the private key can be efficiently obtained by solving systems of quadratic equations with just two variables.