On Elkies subgroups of l-torsion points in elliptic curves defined over a finite field

TL;DR

This paper introduces a new algorithm for computing l-torsion points on elliptic curves over finite fields that is efficient for all primes l and characteristics p, improving upon previous limitations.

Contribution

It combines existing isogeny computation methods with p-adic techniques to create a universally applicable, efficient algorithm for elliptic curve point counting.

Findings

Algorithm works without restrictions on l and p

Complexity is comparable to previous specialized algorithms

First such general algorithm with similar efficiency

Abstract

As a subproduct of the Schoof-Elkies-Atkin algorithm to count points on elliptic curves defined over finite fields of characteristic p, there exists an algorithm that computes, for l an Elkies prime, l-torsion points in an extension of degree l-1 at cost O(l max(l, \log q)^2) bit operations in the favorable case where l < p/2. We combine in this work a fast algorithm for computing isogenies due to Bostan, Morain, Salvy and Schost with the p-adic approach followed by Joux and Lercier to get for the first time an algorithm valid without any limitation on l and p but of similar complexity.