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

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 ` an Elkies prime, `-torsion points in an extension of degree `−1 at cost Õ(` max(`, log q)) bit operations in the favorable case where ` 6 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 an algorithm valid without any limitation on ` and p but of similar complexity. For the sake of simplicity, we precisely state here the algorithm in the case of finite fields with characteristic p > 5. We give experiment results too.

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