Algorithmic Number Theory

Volume 6197 of the series Lecture Notes in Computer Science pp 201-218

Pairing the Volcano

  • Sorina IonicaAffiliated withUniversité de Versailles Saint-Quentin-en-Yvelines
  • , Antoine JouxAffiliated withUniversité de Versailles Saint-Quentin-en-YvelinesDGA

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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.