Théorie algorithmique des nombres et applications à la cryptanalyse de primitives cryptographiques. (Algorithmic Number Theory and Applications to the Cryptanalysis of Cryptographical Primitives)

Le probleme de la factorisation et celui du logarithme discret sont deux fondements essentiels de nombreux algorithmes de la cryptographie a cle publique. Dans le champ des algorithmes pour attaquer ces problemes eminemment ardus, le crible algebrique et ses algorithmes cousins occupent une place de premiere importance. La premiere partie de ce memoire est consacree a la presentation de la " famille " du crible algebrique, et a plusieurs de mes contributions dans ce domaine. D'autres travaux sont abordes dans la partie suivante, notamment en lien avec le probleme du logarithme discret sur les jacobiennes de courbes, et a ma contribution a de nouveaux algorithmes pour ce probleme dans certains cas particuliers. La partie 3 du memoire aborde mes travaux sur le theme de l'algebre lineaire creuse sur les corps finis, motives par le contexte d'application des algorithmes precedemment cites. La partie 4, enfin, traite de mes travaux dans le domaine de l'arithmetique, notamment concernant l'arithmetique des polynomes sur GF(2). La proximite des travaux apparaissant dans ces parties 3 et 4 avec des problematiques d'implantation indique le souci permanent, dans mes travaux, de ne pas laisser de cote cet aspect.

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