Magali Bardet

On the complexity of the F5 Gröbner basis algorithm

We study the complexity of Grobner 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 Grobner basis computed by Faugeres F5F5 algorithm (2002) in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Grobner basis, independently of the algorithm used). Next, we analyse more precisely the structure of the polynomials in the Grobner bases with signatures that F5F5 computes and use it to bound the complexity of the algorithm. Our estimates show that the version of F5F5 we analyse, which uses only standard Gaussian elimination techniques, outperforms row reduction of the Macaulay matrix with the best known algorithms for moderate degrees, and even for degrees up to the thousands if Strassens multiplication is used. The degree being fixed, the factor of improvement grows exponentially with the number of variables.

Efficient decoding of (binary) cyclic codes above the correction capacity of the code using grobner bases

This paper revisits the topic of decoding cyclic codes with Grobner bases. We introduce. new algebraic systems, for which the Grobner basis com- putation is easier. We show that formal decoding for- mulas are too huge to be useful, and that the most ef- ficient technique seems to be to recompute a Grobner basis for each word (online decoding). We use new Grobner basis algorithms and “trace preprocessing” to gain in efficiency.

Isochronicity conditions for some planar polynomial systems II

Abstract We study the isochronicity of centers at O ∈ R 2 for systems x ˙ = − y + A ( x , y ) , y ˙ = x + B ( x , y ) , where A , B ∈ R [ x , y ] , which can be reduced to the Lienard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers.

On the decoding of binary cyclic codes with the Newton identities

We revisit in this paper the concept of decoding binary cyclic codes with Grobner bases. These ideas were first 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 identities. Although these identities have been extensively used for decoding, the work was done manually, to provide formulas for the coefficients of the locator polynomial. This was achieved by Reed, Chen, Truong and others in a long series of papers, for decoding quadratic residue codes, on a case-by-case basis. It is tempting to automate these computations, using elimination theory and Grobner bases. Thus, we study in this paper the properties of the system defined by the Newton identities, for decoding binary cyclic codes. This is done in two steps, first we prove some facts about the variety associated with this system, then we prove that the ideal itself contains relevant equations for decoding, which lead to formulas. Then we consider the so-called online Grobner basis decoding, where the work of computing a Grobner basis is done for each received word. It is much more efficient for practical purposes than preprocessing and substituting into the formulas. Finally, we conclude with some computational results, for codes of interesting length (about one hundred).

Algebraic properties of polar codes from a new polynomial formalism

Polar codes form a very powerful family of codes with a low complexity decoding algorithm that attains many information theoretic limits in error correction and source coding. These codes are closely related to Reed-Muller codes because both can be described with the same algebraic formalism, namely they are generated by evaluations of monomials. However, finding the right set of generating monomials for a polar code which optimises the decoding performances is a nontrivial task and is channel dependent. The purpose of this paper is to reveal some universal properties of these monomials. We will namely prove that there is a way to define a nontrivial (partial) order on monomials so that the monomials generating a polar code devised for a binary-input symmetric channel always form a decreasing set. We call such codes decreasing monomial codes. The fact that polar codes are decreasing monomial codes turns out to have rather deep consequences on their structure. Indeed, we show that decreasing monomial codes have a very large permutation group by proving that it contains a group called lower triangular affine group. Furthermore, the codewords of minimum weight correspond exactly to the orbits of the minimum weight codewords that are obtained from evaluations of monomials of the generating set. In particular, it gives an efficient way of counting the number of minimum weight codewords of a decreasing monomial code and henceforth of a polar code.

Cryptanalysis of the McEliece Public Key Cryptosystem Based on Polar Codes

Polar codes discovered by Arikan form a very powerful family of codes attaining many information theoretic limits in the fields of error correction and source coding. They have in particular much better decoding capabilities than Goppa codes which places them as a serious alternative in the design of both a public-key encryption scheme i la McEliece and a very efficient signature scheme. Shrestha and Kim proposed in 2014 to use them in order to come up with a new code-based public key cryptosystem. We present a key-recovery attack that makes it possible to recover a description of the permuted polar code providing all the information required for decrypting any message.

On formulas for decoding binary cyclic codes

We address 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 syndromes of the received word. This is usually done with the Berlekamp-Massey algorithm in the case of BCH codes and related codes, but for the general case, there is no generic algorithm to decode cyclic codes. Even in the case of the quadratic residue codes, which are good codes with a very strong algebraic structure, there is no available general decoding algorithm. For this particular case of quadratic residue codes, several authors have worked out, by hand, formulas for the coefficients of the locator polynomial in terms of the syndromes, using the Newton identities. This work has to be done for each particular quadratic residue code, and is more and more difficult as the length is growing. Furthermore, it is error-prone. We propose to automate these computations, using elimination theory and Grobner bases. We prove that, by computing appropriate Grobner bases, one automatically recovers formulas for the coefficients of the locator polynomial, in terms of the syndromes.

Complexity reduction of C-Algorithm

Abstract 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 ReCA (Reduced C-Algorithm). Moreover, an adapted version of it is proposed in the rational case. It is called RCA (Rational C-Algorithm) and is widely used in [1] , [2] to find many new examples of isochronous centers for the Lienard type equation.

On the complexity of the F 5 Gröbner basis algorithm

We study the complexity of Grobner 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 Grobner basis computed by Faugeres F5F5 algorithm (2002) in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Grobner basis, independently of the algorithm used). Next, we analyse more precisely the structure of the polynomials in the Grobner bases with signatures that F5F5 computes and use it to bound the complexity of the algorithm. Our estimates show that the version of F5F5 we analyse, which uses only standard Gaussian elimination techniques, outperforms row reduction of the Macaulay matrix with the best known algorithms for moderate degrees, and even for degrees up to the thousands if Strassens multiplication is used. The degree being fixed, the factor of improvement grows exponentially with the number of variables.

On the complexity of the F5F5 Gröbner basis algorithm

We study the complexity of Grobner 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 Grobner basis computed by Faugeres F5F5 algorithm (2002) in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Grobner basis, independently of the algorithm used). Next, we analyse more precisely the structure of the polynomials in the Grobner bases with signatures that F5F5 computes and use it to bound the complexity of the algorithm. Our estimates show that the version of F5F5 we analyse, which uses only standard Gaussian elimination techniques, outperforms row reduction of the Macaulay matrix with the best known algorithms for moderate degrees, and even for degrees up to the thousands if Strassens multiplication is used. The degree being fixed, the factor of improvement grows exponentially with the number of variables.