On the quaternion ℓ-isogeny path problem

Let O be a maximal order in a definite quaternion algebra over Q of prime discriminant p, and ℓ a small prime. We describe a probabilistic algorithm which, for a given left O-ideal, computes a representative in its left ideal class of ℓ-power norm. In practice the algorithm is efficient and, subject to heuristics on expected distributions of primes, runs in expected polynomial time. This solves 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.