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Mathematics of Computation

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Pairing the volcano

Authors: Sorina Ionica and Antoine Joux
Journal: Math. Comp. 82 (2013), 581-603
MSC (2010): Primary 14H52; Secondary 14K02
Published electronically: July 24, 2012
MathSciNet review: 2983037
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Abstract: Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are $\ell$-isogenies. Algorithms allowing to travel on these graphs were developed by Kohel in his thesis (1996) and later on, by Fouquet and Morain (2001). However, up to now, no method was known, to predict, before taking a step on the volcano, the direction of this step. Hence, in Kohel’s and Fouquet-Morain’s algorithms, many steps are taken before choosing the right direction. In particular, ascending or horizontal isogenies are usually found using a trial-and-error approach. In this paper, we propose an alternative method that efficiently finds all points $P$ of order $\ell$ such that the subgroup generated by $P$ is the kernel of a horizontal or an ascending isogeny. In many cases, our method is faster than previous methods. This is an extended version of a paper published in the proceedings of ANTS 2010. In addition, we treat the case of 2-isogeny volcanoes and we derive from the group structure of the curve and the pairing a new invariant of the endomorphism class of an elliptic curve. Our benchmarks show that the resulting algorithm for endomorphism ring computation is faster than Kohel’s method for computing the $\ell$-adic valuation of the conductor of the endomorphism ring for small $\ell$.

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Additional Information

Sorina Ionica
Affiliation: Laboratoire d’Informatique de l’Ecole Polytechnique (LIX) 91128 Palaiseau CEDEX, France
Address at time of publication: LORIA (UMR 7503), Equipe-projet CARAMEL, Bâtiment A, Campus Scientifique – BP 239, 54506 Vandœuvre-lès-Nancy Cedex, France

Antoine Joux
Affiliation: DGA and Université de Versailles Saint-Quentin-en-Yvelines, 45 avenue des États-Unis, 78035 Versailles CEDEX, France
MR Author ID: 316495

Received by editor(s): November 16, 2010
Received by editor(s) in revised form: August 30, 2011
Published electronically: July 24, 2012
Additional Notes: This work has been carried out at Prism Laboratory, University of Versailles and is part of the author’s PhD thesis.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.