# Fonction thêta et applications à la cryptographie

Abstract : The discrete logarithm on elliptic curves give the standard protocols in public key cryptography: asymmetric encryption, signatures, ero-knowledge authentification. To extends the discrete logarithm to hyperelliptic curves of higher genus we need efficient methods to generate secure curves. The aim of this thesis is to give new algorithms to compute with abelian varieties. For this we use the theory of algebraic theta functions in the framework of Mumford. In particular, we give a full generalization of Vélu's formulas for the computation of isogenies on abelian varieties. We also give a new algorithm for the computation of pairings using theta coordinates. Finally we present a point compression method to manipulate These applications follow from the analysis of Riemann relations on theta functions for the addition law. If the results of this thesis are valid for any abelian variety, for the applications a special emphasis is given to Jacobians of hyperelliptic genus~$2$ curves, since they are the most significantly relevant case in cryptography.
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Cited literature [122 references]

https://hal.univ-lorraine.fr/tel-01748199
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SCD_T_2010_0037_ROBERT.pdf
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• HAL Id : tel-01748199, version 1

### Citation

Damien Robert. Fonction thêta et applications à la cryptographie. Autre [cs.OH]. Université Henri Poincaré - Nancy 1, 2010. Français. ⟨NNT : 2010NAN10037⟩. ⟨tel-01748199⟩

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