Paper 2025/1243

Improved algorithms for ascending isogeny volcanoes, and applications

Abstract

Given two elliptic curves over F_q, computing an isogeny mapping one to the other is conjectured to be classically and quantumly hard. This problem plays an important role in the security of elliptic curve cryptography. In 2024, Galbraith applied recently developed techniques for isogenies to improve the state-of-the-art for this problem. In this work, we focus on computing ascending isogenies with respect to an orientation. Our results apply to both ordinary and supersingular curves. We give a simplified framework for computing self-pairings, and show how they can be used to improve upon the approach from Galbraith to recover these ascending isogenies and eliminate a heuristic assumption from his work. We show that this new approach gives an improvement to the overall isogeny recovery when the curves have a small crater (super-polynomial in size). We also study how these self-pairings affect the security of the (PEARL)SCALLOP group action, gaining an improvement over the state-of-the-art for some very particular parameter choices. The current SCALLOP parameters remain unaffected.