Étude des systèmes algébriques surdéterminés. Applications aux codes correcteurs et à la cryptographie

Gröbner bases constitute an important tool for solving algebraic systems of equations, and their computation is often the hard part of the resolution. This thesis is devoted to the complexity analysis of Gröbner basis computations for overdetermined algebraic systems (the number m of equations is greater than the number n of variables). In the generic (”random”) case, tools exist to analyze the complexity of Gröbner basis computations for a non overdetermined system (regular sequences, Macaulay bound). We extend these results to the overdetermined case, by defining the semiregular sequences and the degree of regularity for which we give a precise asymptotic analysis. For example as soon as m > n we gain a factor 2 on the Macaulay bound, and a factor 11,65 whenm = 2n (these factors are reflected on the exponent of global complexity). We determine the complexity of the F5 Algorithm (J-C. Faugère) for computing Gröbner bases. These results are applied in information theory, where the systems are then considered modulo 2 : analysis of the complexity of the algebraic attacks on cryptosystems, algorithms for the decoding of cyclic codes. In this last case, a new equation set-up of this problem leads to use systems of positive dimension for which the resolution is in a surprising way faster. We thus obtain an effective algorithm for decoding codes previously undecodable, allowing list decoding and applicable to any cyclic code.