A survey on signature-based Gr\"obner basis computations

This paper is a survey on the area of signature-based Gr\"obner basis algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table

Gröbner Bases of Ideals Invariant under a Commutative Group: the Non-Modular Case

International audienceWe propose efficient algorithms to compute the Gröbner basis of an ideal $I\subset k[x_1,\dots,x_n]$ globally invariant under the action of a commutative matrix group $G$, in the non-modular case (where $char(k)$ doesn't divide $|G|$). The idea is to simultaneously diagonalize the matrices in $G$, and apply a linear change of variables on $I$ corresponding to the base-change matrix of this diagonalization. We can now suppose that the matrices acting on $I$ are diagonal. This action induces a grading on the ring $R=k[x_1,\dots,x_n]$, compatible with the degree, indexed by a group related to $G$, that we call $G$-degree. The next step is the observation that this grading is maintained during a Gröbner basis computation or even a change of ordering, which allows us to split the Macaulay matrices into $|G|$ submatrices of roughly the same size. In the same way, we are able to split the canonical basis of $R/I$ (the staircase) if $I$ is a zero-dimensional ideal. Therefore, we derive \emph{abelian} versions of the classical algorithms $F_4$, $F_5$ or FGLM. Moreover, this new variant of $F_4/F_5$ allows complete parallelization of the linear algebra steps, which has been successfully implemented. On instances coming from applications (NTRU crypto-system or the Cyclic-n problem), a speed-up of more than 400 can be obtained. For example, a Gröbner basis of the Cyclic-11 problem can be solved in less than 8 hours with this variant of $F_4$. Moreover, using this method, we can identify new classes of polynomial systems that can be solved in polynomial time

A survey on signature-based algorithms for computing Gröbner basis computations

International audienceThis paper is a survey on the area of signature-based Gröbner basis algorithms that was initiated by Faugère's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research

Analysis of a key distribution scheme in secure multicasting

This article presents an analysis of the secure key broadcasting scheme proposed by Wu, Ruan, Lai and Tseng [Proceedings of the 25th Annual IEEE Conference on Local Computer Networks (2000), 208-212]. The study of the parameters of the system is based on a connection with a special type of symmetric equations over finite fields. We present two different attacks against the system, whose efficiency depends on the choice of the parameters. In particular, a time-memory tradeoff attack is described, effective when a parameter of the scheme is chosen without care. In such a situation, more than one third of the cases can be broken with a time and space complexity in the range of the square root of the complexity of the best attack suggested by Wu et al. against their system. This leads to a feasible attack in a realistic scenari

Analysis of a key distribution scheme in secure multicasting

This article presents an analysis of the secure key broadcasting scheme proposed by Wu, Ruan, Lai and Tseng [Proceedings of the 25th Annual IEEE Conference on Local Computer Networks (2000), 208-212]. The study of the parameters of the system is based on a connection with a special type of symmetric equations over finite fields. We present two different attacks against the system, whose efficiency depends on the choice of the parameters. In particular, a time-memory tradeoff attack is described, effective when a parameter of the scheme is chosen without care. In such a situation, more than one third of the cases can be broken with a time and space complexity in the range of the square root of the complexity of the best attack suggested by Wu et al. against their system. This leads to a feasible attack in a realistic scenari

Faster real root decision algorithm for symmetric polynomials

In this paper, we consider the problem of deciding the existence of real solutions to a system of polynomial equations having real coefficients, and which are invariant under the action of the symmetric group. We construct and analyze a Monte Carlo probabilistic algorithm which solves this problem, under some regularity assumptions on the input, by taking advantage of the symmetry invariance property. The complexity of our algorithm is polynomial in $d^s, {{n+d} \choose d}$, and ${{n} \choose {s+1}}$, where $n$ is the number of variables and $d$ is the maximal degree of $s$ input polynomials defining the real algebraic set under study. In particular, this complexity is polynomial in $n$ when $d$ and $s$ are fixed and is equal to $n^{O(1)}2^n$ when $d=n$

Symmetry Preserving Interpolation

International audienceThe article addresses multivariate interpolation in the presence ofsymmetry. Interpolation is a prime tool in algebraic computationwhile symmetry is a qualitative feature that can be more relevantto a mathematical model than the numerical accuracy of the pa-rameters. The article shows how to exactly preserve symmetryin multivariate interpolation while exploiting it to alleviate thecomputational cost. We revisit minimal degree and least interpo-lation with symmetry adapted bases, rather than monomial bases.This allows to construct bases of invariant interpolation spaces inblocks, capturing the inherent redundancy in the computations.We show that the so constructed symmetry adapted interpolationbases alleviate the computational cost of any interpolation problemand automatically preserve any aquivariance of thir interpolation problem might have