Description
Isogenies are an essential tool in Elliptic Curves cryptography, where they are used in a wide variety of area: fast point counting, complex multiplication methods... Velu's formulas give an efficient method for computing such isogenies, but there are no formula known for curves of higher genera or general abelian varieties (except some special case for isogenies of degree 2). In this talk we will present the framework of the theta structure on an abelian variety, developped by Mumford in 1967, which allows us to compute isogenies. For this we lift a theta null point of level $l$, corresponding to an abelian variety B, to the modular space of theta null points of level $lk$. We use a specialized Groebner algorithm that considerably speed-up this phase, and we show how to detect degenerate solutions. To each lifted point corresponds an isogeny of degree $\pi: A \to B$ of degree $k$. We then explain how to compute their dual efficiently.<br/> This is a joint work with Jean-Charles Faugere and David Lubicz.
Next sessions
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Séminaire C2 à INRIA Paris
Emmanuel Thomé et Pierrick Gaudry Rachelle Heim Boissier Épiphane Nouetowa Dung Bui Plus d'infos sur https://seminaire-c2.inria.fr/ -
Attacking the Supersingular Isogeny Problem: From the Delfs–Galbraith algorithm to oriented graphs
Speaker : Arthur Herlédan Le Merdy - COSIC, KU Leuven
The threat of quantum computers motivates the introduction of new hard problems for cryptography.One promising candidate is the Isogeny problem: given two elliptic curves, compute a “nice’’ map between them, called an isogeny.In this talk, we study classical attacks on this problem, specialised to supersingular elliptic curves, on which the security of current isogeny-based cryptography relies. In[…]-
Cryptography
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