A Subexponential Algorithm for Evaluating Large Degree Isogenies

TL;DR

This paper introduces a subexponential algorithm for efficiently evaluating large degree isogenies between elliptic curves, significantly improving over previous exponential-time methods under certain heuristics.

Contribution

It presents a novel subexponential time algorithm for computing large degree isogenies, based on factoring ideals and building on prior work to produce explicit equations.

Findings

Algorithm runs in subexponential time under heuristics

Enables efficient computation of large degree isogenies

Produces explicit isogeny equations from minimal data

Abstract

An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same endomorphism ring, the previous best known algorithm has a worst case running time which is exponential in the length of the input. In this paper we show this problem can be solved in subexponential time under reasonable heuristics. Our approach is based on factoring the ideal corresponding to the kernel of the isogeny, modulo principal ideals, into a product of smaller prime ideals for which the isogenies can be computed directly. Combined with previous work of Bostan et al., our algorithm yields equations for large degree isogenies in quasi-optimal time given only the starting curve and the kernel.