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Theses

Le logarithme discret dans les corps finis

Cécile Pierrot 1
1 ALMASTY - ALgorithms for coMmunicAtion SecuriTY
LIP6 - Laboratoire d'Informatique de Paris 6
Abstract : Cryptography is the study of techniques for secure communication in the presence of third parties, also called adversaries. Such techniques are detailed in cryptosystems, explaining how to securely encode and decode messages. They are designed around computational hardness assumptions related to mathematical properties, making such algorithms hard to break in practice by any adversary. These protocols are based on the computational difficulty of various problems which often come from number theory, such as integer factorization or discrete logarithms computations. This manuscript focuses on the discrete logarithm problem in finite fields and revolves around three axes.First we detail classical results about the problem without any consideration to the target group. We deal with complexity classes and some general methods that do not need any information on the group.The study of the discrete logarithm problem in finite fields starts with small characteristic ones. The aim is to present a Frobenius representation algorithm that leads to the current discrete logarithm record in characteristic 3.For medium or large characteristics finite fields, another approach is required. The multiple number field sieve reaches the best asymptotic heuristic complexities for this double range of characteristics. We also introduce the notion of nearly sparse matrices. Designing a new algorithm dedicated to explicitly give the kernel of such a matrix eases in practice the linear algebra step of any variant of the number field sieve.
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Cécile Pierrot. Le logarithme discret dans les corps finis. Cryptographie et sécurité [cs.CR]. Université Pierre et Marie Curie - Paris VI, 2016. Français. ⟨NNT : 2016PA066495⟩. ⟨tel-01633583⟩

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