This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo computations. 1

Abstract. In this paper, we review and explain the existing algebraic cryptanalysis of multivariate cryptosystems from the hidden field equation (HFE) family. These cryptanalysis break cryptosystems in the HFE family by solving multivariate systems of equations. In this paper we present a new and efficient attack of this cryptosystem based on fast algorithms for computing Gröbner basis. In particular it was was possible to break the first HFE challenge (80 bits) in only two days of CPU time by using the new algorithm F5 implemented in C. From a theoretical point of view we study the algebraic properties of the equations produced by instance of the HFE cryptosystems and show why they yield systems of equations easier to solve than random systems of quadratic equations of the same sizes. Moreover we are able to bound the maximal degree occuring in the Gröbner basis computation. As a consequence, we gain a deeper understanding of the algebraic cryptanalysis against these cryptosystems. We use this understanding to devise a specific algorithm based on sparse linear algebra. In general, we conclude that the cryptanalysis of HFE can be performed in polynomial time. We also revisit the security estimates for existing schemes in the HFE family. 1

This paper investigates polynomial interpolation with respect to a finite set of appropriate linear functionals and the close relations to the Gröbner basis of the associated finite dimensional ideal.

this paper is to generalize FGLM algorithm to noncommutative polynomial rings

We present a solution for optimal triangulation in three views. The solution is guaranteed to find the optimal solution because it computes all the stationary points of the (maximum likelihood) objective function. Internally,

under the SFB grants F1302 and F1308. A new approach for symbolically solving linear boundary value problems is presented. Rather than using general–purpose tools for obtaining parametrized solutions of the underlying ODE and fitting them against the specified boundary conditions (which may be quite expensive), the problem is interpreted as an operator inversion problem in a suitable Banach space setting. Using the concept of the oblique Moore–Penrose inverse, it is possible to transform the inversion problem into a system of operator equations that can be attacked by virtue of non– commutative Gröbner bases. The resulting operator solution can be represented as an integral operator having the classical Green's function as its kernel. Although, at this stage of research, we cannot yet give an algorithmic formulation of the method and its domain of admissible inputs, we do believe that it has promising perspectives of automation and generalization; some of these perspectives are discussed. KEYWORDS: Symbolic Methods for ODE, Linear BVP, Moore–Penrose Equations 1.

The aim of this paper is to demonstrate a specific application of Computer Algebra to bifurcation theory with symmetry. The classification of different bifurcation phenomena in case of several parameters is automated, based on a classification of Grobner bases of possible tangent spaces. The computations are performed in new coordinates of fundamental invariants and fundamental equivariants, with the induced weighted ordering. In order to justify the approach the theory of intrinsic modules is applied. Results for the groups D 3 ; Z 2 ; and Z 2 \Theta Z 2 demonstrate that the algorithm works independent of the group and that new results are obtained. Keywords. bifurcation theory, singularity theory, symmetry, systems with several parameters, Computer Algebra, Grobner bases. AMS subject classification. 13 P10, 34 A47, 34 C20, 58 E09 1 Introduction We present a systematic and algorithmic approach for the theory of bifurcations with symmetry. In applications the solutions often depen...

In this paper, we propose an efficient method to solve polynomial systems whose equations are left invariant by the action of a finite group G. The idea is to simultaneously compute a truncated SAGBI-Gröbner bases (a generalisation of Gröbner bases to ideals of subalgebras of polynomial ring) and a Gröbner basis in the invariant ring K[σ1,..., σn] where σi is the i-th elementary symmetric polynomial. To this end, we provide two algorithms: first, from the F5 algorithm we can derive an efficient and easy to implement algorithm for computing truncated SAGBI–Gröbner bases of the ideals in invariant rings. A first implementation of this algorithm in C enable us to estimate the practical efficiency: for instance, it takes only 92s to compute a SAGBI basis of Cyclic 9 modulo a small prime. The second algorithm is inspired by the FGLM algorithm: from a truncated SAGBI–Gröbner basis of a zero-dimensional ideal we can compute efficiently a Gröbner basis in some invariant rings K[h1,..., hn]. Finally, we will show how this two algorithms can be combined to find the complex roots of such invariant polynomial systems.

Polynomial system solving is one of the important area of Computer Algebra with many applications in Robotics, Cryptology, Computational Geometry, etc. To this end computing a Gröbner basis is often a crucial step. The most efficient algorithms [6, 7] for computing Gröbner bases [2] rely heavily on linear algebra techniques. In this paper, we present a new linear algebra package for computing Gaussian elimination of Gröbner bases matrices. The library is written in C and contains specific algorithms [11] to compute Gaussian elimination as well as specific internal representation of matrices (sparse triangular blocks, sparse rectangular blocks and hybrid rectangular blocks). The efficiency of the new software is demonstrated by showing computational results fr well known benchmarks as well as some crypto-challenges. For instance, for a medium size problem such as Katsura 15, it takes 849.7 sec on a PC with 8 cores to compute a DRL Gröbner basis modulo p < 216; this is 88 faster than Magma (V2-16-1). Categories and Subject Descriptors I.1.2 [Computing Methodologies]: Symbolic and Algebraic Manipulation—Algorithms:

Abstract. We describe a design for a system for mathematical theory exploration that can be extended by implementing new reasoners using the logical input language of the system. Such new reasoners can be applied like the built-in reasoners, and it is possible to reason about them, e.g. proving their soundness, within the system. This is achieved in a practical and attractive way by adding reflection, i.e. a representation mechanism for terms and formulae, to the system’s logical language, and some knowledge about these entities to the system’s basic reasoners. The approach has been evaluated using a prototypical implementation called Mini-Tma. It will be incorporated into the Theorema system. 1