Reynald Lercier

R. Lercier and F. Morain. Counting points on elliptic curves over GF(pⁿ) using Couveignes's algorithm. Research report LIX/RR/95/09, Laboratoire d'Informatique de l'École polytechnique (LIX), 1995.

The heart of the improvements of Elkies to Schoof's algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves. Elkies' approach is well suited for the case where the characteristic of the field is large. Couveignes showed how to compute isogenies in small characteristic. The aim of this paper is to describe the first successful implementation of Couveignes's algorithm and to give numerous computational examples. In particular, we describe the use of fast algorithms for performing incremental operations on series. We will also insist on the particular case of the characteristic 2.

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