Isogeny volcanoes of elliptic curves and sylow subgroups

Abstract

International Conference on Cryptology and Information Security in Latin America LATINCRYPT 2014: Progress in Cryptology - LATINCRYPT 2014 pp 162-175

Given an ordinary elliptic curve over a finite field located in the floor of its volcano of ℓ-isogenies, we present an efficient procedure to take an ascending path from the floor to the level of stability and back to the floor. As an application for regular volcanoes, we give an algorithm to compute all the vertices of their craters. In order to do this, we make use of the structure and generators of the ℓ-Sylow subgroups of the elliptic curves in the volcanoes.

The authors thank the reviewers for their valuable comments and specially Sorina Ionica for her suggestions which have improved this article. Research of the second and third authors was supported in part by grants MTM2013-46949-P (Spanish MINECO) and 2014 SGR1666 (Generalitat de Catalunya).