Fast algorithms for computing isogenies between elliptic curves

TL;DR

This paper surveys existing algorithms for elliptic curve isogenies and introduces a new quasi-linear time algorithm for computing isogenies of degree , leveraging fast power series expansions of elliptic functions.

Contribution

The paper presents a novel quasi-linear time algorithm for computing elliptic curve isogenies of degree , improving efficiency over previous methods.

Findings

New quasi-linear time algorithm for isogeny computation

Application of fast power series expansion techniques

Enhanced efficiency in elliptic curve isogeny calculations

Abstract

We survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime. We introduce a new algorithm that computes an isogeny of degree $\ell$ ($\ell$ different from the characteristic) in time quasi-linear with respect to $\ell$. This is based in particular on fast algorithms for power series expansion of the Weierstrass $\wp$-function and related functions.