Décodage des codes algébriques et cryptographie

Abstract : I discuss the decoding problem of two important families of algebraic
codes: binary cyclic codes and $q$-ary Reed-Solomon codes (and also
algebraic geometry codes). Concerning cyclic codes, they do not have a
generic decoding algorithm, except for the case of the BCH codes and
related codes (Hartmann-Tzeng, Roos bound). Among these codes are the
quadratic residue codes, for which there is no generic decoding
algorithm, but which have good parameters. I present and study a
system of equations related the syndrom decoding of cyclic codes.
These equations can be solved by Gröbner tools. We thus obtain
decoding algorithms with good complexity for these codes. This work
was a part of the PHD thesis of Magali Bardet.

Regarding Reed-Solomon codes, they can be seen as {\em evaluation
codes}, and the associated decoding problem amounts to finding
approximations of a function by low degree polynomials. A major
progress has been achieved by Guruswami and Sudan, who have produced
an algorithm which can decode much more errors than classical ones, by
relaxing the hypothesis that the solution be unique. I have found
improvements of this algorithm, which speed it, and make it
deterministic (notably by removing a fectorization over finite
fields), either in the Reed-Solomon case, or in the algebraic geometry
case. This achievements were made when I directed Lancelot Pecquet PHD

From the theoretical point of view, I have studied mutlivariate
generalizations, which correspond either to products of Reed-Solomon
codes, or to Reed-Muller codes. I then obtain a good decoding radius,
for codes with small rate. In the case of Reed-Muller codes over the
binary field, Cédric Tavernier, in his PHD thesis under my direction,
has found and implemented an efficient algorithm, more than the
algorithms based on the Guruswami-Sudan method.

I have also studied the negative aspects of syndrom decoding of
general linear codes, and of the the decoding of Reed-Solomon codes,
when the number of errors is high, aiming at applications in
cryptography. In the first case, I have built a cryptographic hash
function with a security reduction, that is to say, to find an attack
on the hash function is equivalent to solve a difficult coding
problem. I have also built a new primitive for public key encryption,
relying on the difficulty of decoding Reed-Solomon codes.

In a more applied domain, I have proposed, with Raghav Bhaskar, a new
protocol for multiusers group key agreement, founded on the discrete
logarithm problem. Raghav Bhaskar has given a security proof of this
protocol, in his PHD thesis under my direction. We have also discussed
how to adapt the protocol to messages losses, since our protocol is
among the few which can resist such losses.
Document type :
Habilitation à diriger des recherches
Software Engineering. Université Pierre et Marie Curie - Paris VI, 2007

Contributor : Daniel Augot <>
Submitted on : Monday, July 2, 2007 - 2:20:32 PM
Last modification on : Monday, July 2, 2007 - 2:47:35 PM


  • HAL Id : tel-00159149, version 1



Daniel Augot. Décodage des codes algébriques et cryptographie. Software Engineering. Université Pierre et Marie Curie - Paris VI, 2007. <tel-00159149>




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