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Title:
On the quaternion \ell-isogeny path problem
Authors: 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.
ePrint: https://eprint.iacr.org/2014/505
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