Calcul de polynômes modulaires en dimension 2

Enea Milio 1, 2
2 LFANT - Lithe and fast algorithmic number theory
IMB - Institut de Mathématiques de Bordeaux, Inria Bordeaux - Sud-Ouest
Abstract : Modular polynomials on elliptic curves are a fundamental tool used for the computation of graph of isogenies, class polynomials or for point counting. Thus, they are fundamental for the elliptic curve cryptography. A generalization of these polynomials for principally polarized abelian surfaces has been introduced by Régis Dupont in 2006, who has also described an algorithm to compute them, while theoretical results can been found in an article of Bröker– Lauter of 2009. But these polynomials being really big, they have been computed only in the minimal case p = 2. In this thesis, we continue the work of Dupont and Bröker–Lauter by defining and giving theoretical results on modular polynomials with new invariants, based on theta constants. Using these invariants, we have been able to compute the polynomials until p = 7 but bigger examples look intractable. Thus we define a new kind of modular polynomials where we restrict on the surfaces having real multiplication by the maximal order of a real quadratic field. We present many examples and theoretical results.
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Theses
Cryptographie et sécurité [cs.CR]. Université de Bordeaux, 2015. Français. 〈NNT : 2015BORD0285〉
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Enea Milio. Calcul de polynômes modulaires en dimension 2. Cryptographie et sécurité [cs.CR]. Université de Bordeaux, 2015. Français. 〈NNT : 2015BORD0285〉. 〈tel-01240690v2〉

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