Sieve algorithms for the discrete logarithm in medium characteristic finite fields. (Algorithmes de crible pour le logarithme discret dans les corps finis de moyenne caractéristique)

‘e security of public-key cryptography relies mainly on the diculty to solvesome mathematical problems, among which the discrete logarithm problem on€nite €eldsFpn . In this thesis, we study the variants of the number €eld sieve(NFS) algorithm, which solve the most eciently this problem, in the case wherethe characteristic of the €eld is medium.‘e NFS algorithm can be divided into four main steps: the polynomial selec-tion, the relation collection, the linear algebra and the computation of an individ-ual logarithm. We describe these steps and focus on the relation collection, oneof the most costly steps. A way to perform it eciently is to make use of sievealgorithms.Contrary to the classical case for which the relation collection takes place ina two-dimensional space, the €nite €elds we target require the enumeration ofelements in a higher-dimensional space to reach the best theoretical complexity.‘ere exist ecient sieve algorithms in two dimensions, but only a few in higherdimensions. We propose and study two new sieve algorithms allowing us to treatany dimensions, with an emphasis on the three-dimensional case.We have provided a complete implementation of the relation collection forsome variants of the NFS in three dimensions. ‘is implementation relies on ournew sieve algorithms and is distributed in the CADO-NFS so‰ware. We validatedits performances by comparing with examples from the literature. We also estab-lish two new discrete logarithm record computations, one in a 324-bitFp5 andone in a 422-bitFp6 .