Algorithmes pour la factorisation d'entiers et le calcul de logarithme discret

Cyril Bouvier 1
1 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : In this thesis, we study the problems of integer factorization and discrete logarithm computation in finite fields. First, we study the ECM algorithm for integer factorization and present a method to analyze the elliptic curves used in this algorithm by studying the Galois properties of division polynomials. Then, we present in detail the NFS algorithm for integer factorization and we study in particular the polynomial selection step for which we propose improvements of existing algorithms. Next, we present two algorithms for computing discrete logarithms in finite fields: NFS-DL and FFS. We also give some details of two computations of discrete logarithms carried out during this thesis, one with NFS-DL and the other with FFS. Finally, we study a common step of the NFS algorithm for integer factorization and the NFS-DL and FFS algorithms for discrete logarithm computations: the filtering step. We study this step thoroughly and present an improvement for which we study the impact using data from several computations of discrete logarithms and factorizations.
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Cyril Bouvier. Algorithmes pour la factorisation d'entiers et le calcul de logarithme discret. Cryptographie et sécurité [cs.CR]. Université de Lorraine, 2015. Français. ⟨NNT : 2015LORR0053⟩. ⟨tel-01751648v2⟩

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