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Applications des fonctions thêta à la cryptographie sur courbes hyperelliptiques

Abstract : Since the mid 1980's, abelian varieties have been widely used in cryptography: the discrete logarithm problem and the protocols that rely on it allow asymmetric encryption, signatures, authentification... For cryptographic applications, one of the most interesting examples of principally polarized abelian varieties is given by the Jacobians of hyperelliptic curves. The theory of theta functions provides efficient algorithms to compute with abelian varieties. In particular, using decomposable curves of genus 2, we present a generalization of the ECM algorithm. In this thesis, we also study the correspondences between Mumford coordinates and theta functions. This led to the construction of complete addition laws in genus 2. Finally we present an algorithm to compute isogenies between abelian varieties. Most of the results of this thesis are valid for hyperelliptic curves of arbitrary genus. More specifically we emphasize on genus 2 hyperelliptic curves, which is the most relevant case in cryptography. These results have been implemented in a Magma package called AVIsogenies
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Romain Cosset. Applications des fonctions thêta à la cryptographie sur courbes hyperelliptiques. Autre [cs.OH]. Université Henri Poincaré - Nancy 1, 2011. Français. ⟨NNT : 2011NAN10145⟩. ⟨tel-01746303⟩



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