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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Pairing the volcano
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by Sorina Ionica and Antoine Joux PDF
Math. Comp. 82 (2013), 581-603 Request permission


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
  • Email:
  • 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
  • Email:
  • 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.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 82 (2013), 581-603
  • MSC (2010): Primary 14H52; Secondary 14K02
  • DOI:
  • MathSciNet review: 2983037