# On the quaternion $\ell$-isogeny path problem

2 Cryptography group
Microsoft Research [Redmond]
Abstract : 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.
Type de document :
Article dans une revue
LMS Journal of Computation and Mathematics, London Mathematical Society, 2014, 17 (A), pp.418-432. 〈10.1112/S1461157014000151〉

https://hal.archives-ouvertes.fr/hal-01257092
Contributeur : David Kohel <>
Soumis le : vendredi 15 janvier 2016 - 17:14:31
Dernière modification le : jeudi 18 janvier 2018 - 01:32:41

### Citation

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

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