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Titre : Algebraic techniques for decoding Reed-Solomon codes and cryptanalyzing McEliece-like cryptosystems
Date : 7 avril 2023
Heure: 14:00
Résumé : Algebraic error-correcting codes respond to numerous needs that emerge from the world of digital communications. The mathematical structure of these families of codes allows to design efficient encoding and decoding algorithms, thus enabling a variety of applications in modern technologies. In this manuscript, we delve into two of these aspects. First, we tackle the decoding problem for Reed-Solomon codes, by solving a polynomial system, related to the key-equations from power decoding, with Gröbner bases techniques. Then we study the security of Goppa codes in code-based cryptography. We investigate an algebraic method that allows to distinguish alternant and Goppa codes from random ones, for high enough rates. This study brings us to develop a polynomial-time attack on high-rate alternant codes. Again, we exploit Gröbner bases to solve a polynomial system modeling the key-recovery problem. Finally, we give a procedure to enhance in some cases the range of distinguishable parameters.
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