Computing l-isogenies using the p-torsion

Abstract

Computing l-isogenies between elliptic curves defined over a finite field Fq of small characteristic is of some importance for the computation of the cardinalities of elliptic curves using Schoof-Atkin-Elkies method. In a previous publication [3] we showed that this could be achieved in O(l3+e) multiplications in \(\mathbb{F}_q \)using formal groups. This method has been implemented by Lercier and Morain who obtained spectacular results in this direction [7, 8]. Nevertheless, the use of formal groups seems to be a serious deterent both for man and machine to perform such a work. More recently Lercier proposed an algorithm specific for characteristic 2 that has the same assymptotic complexity but is faster by some significative constant factor. In this paper we propose a general algorithm which does not use formal groups. Instead we take advantage of the elementary Galois properties of the p-torsion. This algorithm has the same complexity as the previous ones if we don't use fast multiplication techniques. But, contrary to the previous methods, it allows the use of fast multiplication for polynomials and then turns out to run in O(l2+e) multiplications in the field \(\mathbb{F}_q \). Our algorithm has also the advantage that it is made exclusively of very classical routines in polynomial and elliptic curve arithmetic. Also one may expect that the implementation of this method should require less work than the previous ones thus bringing new people to this kind of calculation.