Pairing the Volcano
Sorina Ionica (INRIA Saclay - Ile de France), Antoine Joux (PRISM)

TL;DR
This paper introduces an efficient method to identify the direction of steps on isogeny volcanoes, improving the speed of navigating these graphs and computing endomorphism rings of elliptic curves.
Contribution
It presents a novel approach to predict isogeny directions on volcanoes and derives a new invariant for endomorphism class determination, enhancing existing algorithms.
Findings
Faster identification of horizontal and ascending isogenies.
Improved algorithm for endomorphism ring computation.
Effective for small prime degrees .
Abstract
Isogeny volcanoes are graphs whose vertices are elliptic curves and whose edges are -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, 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 of order such that the subgroup generated by is the kernel of an 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…
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Taxonomy
TopicsCryptography and Residue Arithmetic · Polynomial and algebraic computation · Algebraic Geometry and Number Theory
