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

International audienceAs 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 $\ell$ an Elkies prime, $\ell$-torsion points in an extension of degree $\ell-1$ at cost $\tilde{Ot}(\ell \, \max(\ell, \log q)^2)$ bit operations in the favorable case where $\ell\leqslant p/2$.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 $\ell$ 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 $p\geqslant 5$. We give experiment results too