Cryptology ePrint Archive: Report 2014/505

On the quaternion $\ell$-isogeny path problem

David Kohel, Kristin Lauter, Christophe Petit, Jean-Pierre Tignol

Abstract: Let $\cO$ 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 $\cO$-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.

Category / Keywords: number theory

Original Publication (in the same form): To appear in the LMS Journal of Computation and Mathematics, as a special issue for ANTS (Algorithmic Number Theory Symposium) conference.