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Article Dans Une Revue LMS Journal of Computation and Mathematics Année : 2014

On the quaternion $\ell$-isogeny path problem

Résumé

Let $\mathcal{O}$ be a maximal order in a definite quaternion algebra over $\mathbb{Q}$ of prime discriminant $p$, and $\ell$ a small prime. We describe a probabilistic algorithm, which for a given left $O$-ideal, computes a representative in its left ideal class of $\ell$-power norm. In practice the algorithm is efficient, and subject to heuristics on expected distributions of primes, runs in expected polynomial time. This breaks the underlying problem for a quaternion analog of the Charles-Goren-Lauter hash function, and has security implications for the original CGL construction in terms of supersingular elliptic curves.

Dates et versions

hal-01257092 , version 1 (15-01-2016)

Identifiants

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David Kohel, Kristin Lauter, Christophe Petit, Jean-Pierre Tignol. On the quaternion $\ell$-isogeny path problem. LMS Journal of Computation and Mathematics, 2014, 17 (A), pp.418-432. ⟨10.1112/S1461157014000151⟩. ⟨hal-01257092⟩
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