Fonctions thêta et applications à la cryptographie

Damien Robert 1, 2
1 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
2 CACAO - Curves, Algebra, Computer Arithmetic, and so On
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : The discrete logarithm on elliptic curves gives the standard protocols in public key cryptography: asymmetric encryption, signatures, zero-knowledge authentification. To extend 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 \name{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 more efficiently theta coordinates of high level. These applications follow from the analysis of Riemann relations on theta functions for the addition law. Whereas 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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Damien Robert. Fonctions thêta et applications à la cryptographie. Informatique [cs]. Université Henri Poincaré - Nancy I, 2010. Français. ⟨tel-00528942⟩

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