Magali Bardet

Magali Bardet
  • Université de Rouen Normandie

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15
Publications
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948
Citations
Current institution
Université de Rouen Normandie

Publications

Publications (15)
Article
In this paper, we consider a family of closed planar algebraic curves $\mathcal{C}$ which are given in parametrization form via a trigonometric polynomial $p$. When $\mathcal{C}$ is the boundary of a compact convex set, the polynomial $p$ represents the support function of this set. Our aim is to examine properties of the degree of the defining pol...
Article
We study the complexity of Gr\"obner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system. We give a bound on the number of polynomials of degree d in a Gr\"obner basis computed by Faug\`ere's F5 algorithm (Fau02) in this generic case for the grevlex ordering (...
Article
A fundamental problem in computer science is to find all the common zeroes of $m$ quadratic polynomials in $n$ unknowns over $\mathbb{F}_2$. The cryptanalysis of several modern ciphers reduces to this problem. Up to now, the best complexity bound was reached by an exhaustive search in $4\log_2 n\,2^n$ operations. We give an algorithm that reduces t...
Article
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The C-Algorithm introduced in [5] is designed to determine isochronous centers for Lienard-type differential systems, in the general real analytic case. However, it has a large complexity that prevents computations, even in the quartic polynomial case.The main result of this paper is an efficient algorithmic implementation of C-Algorithm, called Re...
Article
Full-text available
We study the isochronicity of centers at O∈R2 for systems where A,B∈R[x,y], which can be reduced to the Liénard type equation. When deg(A)⩽4 and deg(B)⩽4, using the so-called C-algorithm we found 36 new multiparameter families of isochronous centers. For a large class of isochronous centers we provide an explicit general formula for linearization....
Article
We revisit in this paper the concept of decoding binary cyclic codes with Grobner bases. These ideas were rst introduced by Cooper, then Chen, Reed, Helleseth and Truong, and eventually by Orsini and Sala. We discuss here another way of putting the decoding problem into equations: the Newton's identities. Although these identities have been extensi...
Article
Full-text available
We adress the problem of the algebraic decoding of any cyclic code up to the true minimum distance. For this, we use the classical formulation of the problem, which is to find the error locator polynomial in terms of the syndroms of the received word. This is usually done with the Berlekamp-Massey algorithm in the case of BCH codes and related code...
Article
Les bases de Gröbner constituent un outil important pour la résolution de systèmes d'équations algébriques, et leur calcul est souvent la partie difficile de la résolution. Cette thèse est consacrée à des analyses de complexité de calculs de bases de Gröbner pour des systèmes surdéterminés (le nombre m d'équations est supérieur au nombre n d'inconn...
Article
We extend the notion of regular sequence ((Mac16)) to overdetermined system of al- gebraic equations. We study generic properties of Gröbner bases and analyse precisely the behavior of the F5 (Fau02) algorithm. Sharp asymptotic estimates of the degree of regularity are given.
Article
Full-text available
The problem of decoding cyclic codes can be rewritten into an algebraic system of equations, whose solutions are closely related to the error that occured. Extensive work has been done previously, where it has been shown that the computation of a Gröbner basis of this algebraic system enables to decode up to the true minimum distance. The Gröbner b...
Article
We present complexity results for solving "typical"overdetermined algebraic systems over GF(2) with solutions in GF(2) using Gröbner bases. They are useful for instance to predictthe complexity of an algebraic cryptanalysis over a cryptosystem,they give a priori upper bounds. We define semi-regularsequences and their associated notion of degree of...
Article
While the computation of Grobner bases is known to be an expspace-complete problem, the generic behaviour of algorithms for their computation is much better. We study generic properties of Grobner bases and analyse precisely the best algorithm currently known, F5.
Article
Abstract We compute the asymptotic expansion of the index of regularity for overdetermined quadratic semi-regular sequences of algebraic equations. This implies bounds for the generic complexity of Gr obner bases algorithms, in particular the F5 [Fau02] algorithm. Bounds can also be derived for the XL [SPCK00] family of algorithms used by the cryp...

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