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On the quaternion $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\ell $-isogeny path problem

Published online by Cambridge University Press:  01 August 2014

David Kohel ,
Kristin Lauter ,
Christophe Petit  and
Jean-Pierre Tignol
David Kohel
Affiliation:
Institut de Mathématiques de Marseille, Université d’Aix-Marseille, 163, avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France email David.Kohel@univ-amu.fr
Kristin Lauter
Affiliation:
Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA email klauter@microsoft.com
Christophe Petit
Affiliation:
UCL Crypto Group, Université catholique de Louvain, Place du Levant 3, B1348 Louvain-la-Neuve, Belgium email christophe.petit@uclouvain.be
Jean-Pierre Tignol
Affiliation:
UCL – ICTEAM/INMA, Université catholique de Louvain, Avenue G. Lemaitre 4, box L4.05.01, B1348 Louvain-la-Neuve, Belgium email Jean-Pierre.Tignol@uclouvain.be

Abstract

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Let $\mathcal{O}$ 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 $\mathcal{O}$-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 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.

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