Computing Grobner Bases Research Articles

A Gröbner Basis Database

In response to the increased interest in machine learning to solve mathematical problems combined with the complexity inherent in computing Grobner bases, we create a Grobner basis database. Through this database we aim to provide a platform where researchers and practitioners can: obtain accurate and verified data on Grobner bases, submit interesting and important examples of Grobner bases, and obtain curated datasets for benchmarking, learning, and other common problems in the research community. Given the nascent stage of the database, new features are likely to be added on as it grows.

Cryptanalysis of the extension field cancellation cryptosystem

In this article, we present algebraic attacks against the Extension Field Cancellation ( $$\texttt {EFC}$$ ) scheme, a multivariate public-key encryption scheme which was published at PQCRYPTO’2016. First, we present a successful Grobner basis message-recovery attack on the first and second proposed parameters of the scheme. For the first challenge parameter, a Grobner-based hybrid attack has a $$2^{65}$$ bit complexity which beats the claimed 80 bit security level. We further show that the algebraic system arising from an $$\texttt {EFC}$$ public-key is much easier to solve than a random system of the same size. Briefly, this is due to the apparition of many lower degree equations during the Grobner basis computation. We present a polynomial-time method to recover such lower-degree relations and also show their usefulness in improving the Grobner basis attack complexity on $$\texttt {EFC}$$ . Thus, we show that there is an algebraic structural weakness in the system of equations coming from $$\texttt {EFC}$$ and hence makes the scheme not suitable for encryption.

Solving Multivariate Polynomial Matrix Diophantine Equations with Gröbner Basis Method

Different from previous viewpoints, multivariate polynomial matrix Diophantine equations are studied from the perspective of modules in this paper, that is, regarding the columns of matrices as elements in modules. A necessary and sufficient condition of the existence for the solution of equations is derived. Using powerful features and theoretical foundation of Grobner bases for modules, the problem for determining and computing the solution of matrix Diophantine equations can be solved. Meanwhile, the authors make use of the extension on modules for the GVW algorithm that is a signature-based Grobner basis algorithm as a powerful tool for the computation of Grobner basis for module and the representation coefficients problem directly related to the particular solution of equations. As a consequence, a complete algorithm for solving multivariate polynomial matrix Diophantine equations by the Grobner basis method is presented and has been implemented on the computer algebra system Maple.

A Strongly-Consistent Difference Scheme for 3D Nonlinear Navier-Stokes Equations

This paper constructs a strongly-consistent explicit finite difference scheme for 3D constant viscosity incompressible Navier-Stokes equations by using of symbolic algebraic computation. The difference scheme is space second order accurate and temporal first order accurate. It is proved that difference Grobner basis algorithm is correct. By using of difference Grobner basis computation method, an element in Grobner basis of difference scheme for momentum equations is a difference scheme for pressure Poisson equation. The authors find that the truncation errors expressions of difference scheme is consistent with continuous errors functions about modified version of above difference equation. The authors prove that, for strongly consistent difference scheme, each element in the difference Grobner basis of such difference scheme always approximates a differential equation which vanishes on the analytic solutions of Navier-Stokes equations. To prove the strongly-consistency of this difference scheme, the differential Thomas decomposition theorem for nonlinear differential equations and difference Grobner basis theorems for difference equations are applied. Numerical test certifies that strongly-consistent difference scheme is effective.

Bifurcations of Periodic Solutions of a Hamiltonian System with a Discrete Symmetry Group

An autonomous Hamiltonian system with two degrees of freedom that is invariant under the Klein four-group of linear canonical automorphisms of the extended phase space of the system is studied. A sequence of symplectic transformations of the monodromy matrix of the symmetric periodic solution of this system is constructed. Using these transformations, the structure and bifurcation of the phase flow in the vicinity of this solution is investigated. It is shown that the bifurcations corresponding to the multiple period increase are different for the solutions with double symmetry and the solutions with a single symmetry. An example of critical periodic solutions of the family of doubly symmetric orbits of the plane circular Hill problem is discussed. The majority of tedious analytical calculations are performed using packages for the computation of Grobner bases and for the work with polynomial ideals in the computer algebra system Maple.

Modular Techniques for Noncommutative Gröbner Bases

We extend modular techniques for computing Grobner bases from the commutative setting to the vast class of noncommutative G-algebras. As in the commutative case, an effective verification test is only known to us in the graded case. In the general case, our algorithm is probabilistic in the sense that the resulting Grobner basis can only be expected to generate the given ideal, with high probability. We have implemented our algorithm in the computer algebra system Singular and give timings to compare its performance with that of other instances of Buchberger’s algorithm, testing examples from D-module theory as well as classical benchmark examples. A particular feature of the modular algorithm is that it allows parallel runs.

On the Consistency Analysis of Finite Difference Approximations

Finite difference schemes are widely used in applied mathematics to numerically solve partial differential equations. However, for a given solution scheme, it is usually difficult to evaluate the quality of the underlying finite difference approximation with respect to the inheritance of algebraic properties of the differential problem under consideration. In this paper, we present an appropriate quality criterion of strong consistency for finite difference approximations to systems of nonlinear partial differential equations. This property strengthens the standard requirement of consistency of difference equations with differential ones. We use a verification algorithm for strong consistency, which is based on the computation of difference Grobner bases. This allows for the evaluation and construction of solution schemes that preserve some fundamental algebraic properties of the system at the discrete level. We demonstrate the suggested approach by simulating a Karman vortex street for the two-dimensional incompressible viscous flow described by the Navier–Stokes equations.

Speeding Up the GVW Algorithm via a Substituting Method

The GVW algorithm is an efficient signature-based algorithm for computing Grobner bases. In this paper, the authors consider the implementation of the GVW algorithm by using linear algebra, and speed up GVW via a substituting method. As it is well known that, most of the computing time of a Grobner basis is spent on reductions of polynomials. Thus, linear algebraic techniques, such as matrix operations, have been used extensively to speed up the implementations. Particularly, one-direction (also called signature-safe) reduction is used in signature-based algorithms, because polynomials (or rows in matrices) with larger signatures can only be reduced by polynomials (rows) with smaller signatures. The authors propose a new method to construct sparser matrices for signature-based algorithms via a substituting method. Specifically, instead of only storing the original polynomials in GVW, the authors also record many equivalent but sparser polynomials at the same time. In matrix construction, denser polynomials are substituted by sparser equivalent ones. As the matrices get sparser, they can be eliminated more efficiently. Two specifical algorithms, Block-GVW and LMGVW, are presented, and their combination is the Sub-GVW algorithm. The correctness of the new proposed method is proved, and the experimental results demonstrate the efficiency of this new method.

Constructing orthogonal designs in powers of two via symbolic computation and rewriting techniques

In the past few decades, design theory has grown to encompass a wide variety of research directions. It comes as no surprise that applications in coding theory and communications continue to arise, and also that designs have found applications in new areas. Computer science has provided a new source of applications of designs, and simultaneously a field of new and challenging problems in design theory. In this paper, we revisit a construction for orthogonal designs using the multiplication tables of Cayley-Dixon algebras of dimension 2n. The desired orthogonal designs can be described by a system of equations with the aid of a Grobner basis computation. For orders greater than 16 the combinatorial explosion of the problem gives rise to equations that are unfeasible to be handled by traditional search algorithms. However, the structural properties of the designs make this problem possible to be tackled in terms of rewriting techniques, by equational unification. We establish connections between central concepts of design theory and equational unification where equivalence operations of designs point to the computation of a minimal complete set of unifiers. These connections make viable the computation of some types of orthogonal designs that have not been found before with the aforementioned algebraic modeling.

Usage of Modular Techniques for Efficient Computation of Ideal Operations

Modular techniques are widely applied to various algebraic computations. In this paper, we discuss how modular techniques can be efficiently applied to computation of Grobner basis of the ideal generated by a given set, and extend the techniques for further ideal operations such as ideal quotient, saturation and radical computation. In order to make modular techniques very efficient, we focus on the most important issue, how to efficiently guarantee the correctness of computed results. Unifying notions of luckiness of primes proposed by several authors, we give precise and comprehensive explanation on the techniques which shall be a certain basis for further development.

Application of computer algebra methods for investigation of stationary motions of a gyrostat satellite

With the help of computer algebra methods, properties of a non-linear algebraic system that determines the equilibrium orientations of a gyrostat satellite moving along a circular orbit were investigated. The main attention is paid to the study of equilibrium orientations of the satellite with given principal central moments of inertia and given gyrostatic torque in special cases, when the projection of the vector of gyrostatic torque is located in one of the principal central planes of inertia of the satellite. Computer algebra method based on the algorithm for the construction of a Grobner basis were applied to reduce the satellite stationary motion system of nine algebraic equations with nine variables to a single algebraic equation in one variable that determines all the equilibrium orientations of the satellite. Bifurcation curves in the space of system parameters that determine boundaries of domains with a fixed number of equilibria of the satellite were obtained symbolically. A comparative analysis of the effectiveness of different algorithms of Grobner bases computation in solving the problem was carried out.

Computing Gröbner bases associated with lattices

We specialize Moller's algorithm to the computation of Grobner bases related to lattices. We give the complexity analysis of our algorithm. Then we provide experiments showing that our algorithm is more efficient than Buchberger's algorithm for computing the associated Grobner bases. Furthermore we show that the binomial ideal associated to the lattice can be constructed from a set of binomials associated with a set of generators of the corresponding label code. This result is presented in a general way by means of three ideal constructions associated with group codes that constitute the same ideal. This generalizes earlier results for specific cases of group codes such as linear codes, codes over ${\mathbb Z}_m$ and label codes of lattices.

Efficient Symbolic Computation for Word-Level Abstraction From Combinational Circuits for Verification Over Finite Fields

This paper introduces a technique to derive a word-level abstraction of the function implemented by a combinational logic circuit. The abstraction provides a canonical representation of the function as a polynomial ${Z} {= {\mathcal {F}}(A)}$ over the finite field $ {{\mathbb {F}}_{2^{k}}}$ , where ${Z}$ and $ {A}$ represent the ${k}$ -bit output and input bit-vectors (words) of the circuit, respectively. This canonical abstraction can be utilized for formal verification and equivalence checking of combinational circuits. Our approach to abstraction is based upon concepts from computational commutative algebra and algebraic geometry. We show that the abstraction ${Z} {= {\mathcal {F}}(A)}$ can be derived by computing a Grobner basis of the polynomials corresponding to the circuit, using a specific elimination term order derived from the circuit’s topology. Computing Grobner bases using elimination term orders is infeasible for large circuits. To overcome this limitation, we describe an efficient symbolic computation to derive the word-level polynomial. Our algorithms exploit: 1) the structure of the circuit; 2) the properties of Grobner bases; 3) characteristics of finite fields $ {{\mathbb {F}}_{2^{k}}}$ ; and 4) modern algorithms from symbolic algebra, to derive the canonical polynomial representation. This approach is employed to verify (and detect bugs in) large combinational finite field arithmetic circuits, where contemporary verification techniques are known to be infeasible.

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 implementing the symbolic preprocessing function over Boolean polynomial rings in Gröbner basis algorithms using linear algebra

Some techniques using linear algebra was introduced by Faugere in F4 to speed up the reduction process during Grobner basis computations. These techniques can also be used in fast implementations of F5 and some other signature-based Grobner basis algorithms. When these techniques are applied, a very important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function (given in F4). Since multiplications of monomials and polynomials are involved in the Symbolic Preprocessing function, this step can be very costly when the number of involved polynomials/monomials is huge. In this paper, multiplications of monomials and polynomials for a Boolean polynomial ring are investigated and a specific method of implementing the Symbolic Preprocessing function over Boolean polynomial rings is reported. Many examples have been tested by using this method, and the experimental data shows that the new method is very efficient.

Sparse Gröbner bases

Toric (or sparse) elimination theory is a framework developed during the last decades to exploit monomial structures in systems of Laurent polynomials. We describe sparse variants of the F5 and FGLM algorithms that allow us to compute efficiently sparse Grobner bases of sparse systems. In the case where the generating subset of monomials corresponds to the points with integer coordinates in a normal lattice polytope and under some regularity assumptions, we prove complexity bounds which depend only on the combinatorial properties of the polytope. These bounds yield new estimates on the complexity of solving zerodimensional systems where all polynomials share the same Newton polytope. For instance, we generalize the bound min(n1, n2)+2 on the maximal degree in a Grobner basis of a zero-dim. bilinear system with blocks of variables of sizes (n1, n2) to the multi-homogeneous case: for polynomials of multi-degree (d1,d2, . . .) w.r.t. a partition of the variables in blocks of sizes (n1, n2, . . .) the maximal degree occurring in a Grobner basis computation is bounded by 2+ ∑ ni−max( ⌈ ni+1 di ⌉ ). From a practical point of view, a proof-of-concept implementation shows large speed-ups compared to optimized Grobner bases implementation.

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

Algebraic algorithms for LWE problems

No abstract available.

A Hybrid Gröbner bases approach to computing power integral bases

Bettale, Faugere, and Perret [3] present and analyze a hybrid method for solving multivariate polynomial systems over finite fields that mixes Grobner bases computations with an exhaustive search. Inspired by their method, we use a hybrid approach to characterize all power integral bases in the pth cyclotomic field \({\mathbb{Q}(\zeta_p)}\) for the regular primes p = 29, 31, 41. For each prime p this involves solving a system of (p−1)/2 multivariate polynomial equations of degree (p−1)/2 in (p−1)/2 variables over the finite field \({\mathbb{Z}/p\mathbb{Z}}\).

Compact representation of polynomials for algorithms for computing Gröbner and involutive bases

In the computation of involutive and Grobner bases with rational coefficients, the major part of the memory is occupied by precision numbers; however, in the case of modular operations (especially, in the computation of Grobner bases), of most importance is the problem of compact representation of monomials composing polynomials of the system. For this purpose, for example, ZDD diagrams and other structures are used, which make execution of typical operations--multiplication by a monomial and reduction of polynomials--more complicated. In this paper, an attempt is made to develop convenient (in the sense of computation of bases) and compact representation of polynomials that is based on hash-tables. Results of test runs are presented.