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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