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arXiv · 1406.0981

On the quaternion $\ell$-isogeny path problem

David Kohel·Kristin Lauter·Christophe Petit·Jean-Pierre Tignol
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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 $O$-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.

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David Kohel, Kristin Lauter, Christophe Petit, Jean-Pierre Tignol. 2014-06-04. On the quaternion $\ell$-isogeny path problem. https://arxiv.org/abs/1406.0981

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