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Anneaux d'endomorphismes en cryptographie

Abstract : Modern communications heavily rely on cryptography to ensure data integrity and privacy. Over the past two decades, very efficient, secure, and featureful cryptographic schemes have been built on top of abelian varieties defined overfinite fields. This thesis contributes to several computational aspects of ordinary abelian varieties related to their endomorphism ring structure. This structure plays a crucial role in the construction of abelian varieties with desirable properties, such as pairings, and we show that more such varieties can be constructed than expected. We also address the inverse problem, that of computing the endomorphism ring of a prescribed abelian variety. Prior state-of-the-art methods could only solve this problem in exponential time, and we design several algorithms of subexponential complexityfor solving it in the ordinary case. For elliptic curves, we rigorously bound the complexity of our algorithms assuming solely the extended Riemann hypothesis, and demonstrate that they are very effective in practice. As a subroutine, we design in particular a memory-less algorithm to solve a generalization of the subset sum problem. We also generalize our method to higher-dimensional abelian varieties. Practically speaking, we develop a library enabling the computation of isogenies between abelian varieties; this building block enables us to apply a generalization of our algorithm to cases that were previously not computable.
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Gaetan Bisson. Anneaux d'endomorphismes en cryptographie. Ordinateur et société [cs.CY]. Institut National Polytechnique de Lorraine, 2011. Français. ⟨NNT : 2011INPL047N⟩. ⟨tel-01749554v1⟩

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