On the quaternion `-isogeny path problem

Abstract. Let O be a maximal order in a definite quaternion algebra over Q of prime discriminant p, and ` a small prime. We describe a probabilistic algorithm, which for a given left O-ideal, computes a repre-sentative in its left ideal class of `-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.