Paper 2006/176

Counting points on elliptic curves in medium characteristic

Antoine Joux and Reynald Lercier

Abstract

In this paper, we revisit the problem of computing the kernel of a separable isogeny of degree $\ell$ between two elliptic curves defined over a finite field $\GF{q}$ of characteristic $p$. We describe an algorithm the asymptotic time complexity of which is equal to $\SoftO(\ell^2(1+\ell/p)\log q)$ bit operations. This algorithm is particularly useful when $\ell > p$ and as a consequence, we obtain an improvement of the complexity of the SEA point counting algorithm for small values of $p$. More precisely, we obtain a heuristic time complexity $\SoftO(\log^{4} q)$ and a space complexity $O(\log^{2} q)$, in the previously unfavorable case where $p \simeq \log q$. Compared to the best previous algorithms, the memory requirements of our SEA variation are smaller by a $\log^2 q$ factor.