Pairing the Volcano

  • Sorina Ionica
  • Antoine Joux
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)


Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are ℓ-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 algorithms, we take many steps 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 ℓ such that the subgroup generated by P is the kernel of an horizontal or an ascending isogeny. In many cases, our method is faster than previous methods.


Elliptic Curve Elliptic Curf Stability Level Discrete Logarithm Endomorphism Ring 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Sorina Ionica
    • 1
  • Antoine Joux
    • 1
    • 2
  1. 1.Université de Versailles Saint-Quentin-en-YvelinesVersailles CEDEXFrance
  2. 2.DGA 

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