Computing the cardinality of CM elliptic curves using torsion points
Abstract
Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Omega of the order. If the prime p splits completely in Omega, then we can reduce E modulo one the factors of p and get a curve Ep defined over GF(p). The trace of the Frobenius of Ep is known up to sign and we need a fast way to find this sign. For this, we propose to use the action of the Frobenius on torsion points of small order built with class invariants a la Weber, in a manner reminiscent of the Schoof-Elkies-Atkin algorithm for computing the cardinality of a given elliptic curve modulo p. We apply our results to the Elliptic Curve Primality Proving algorithm (ECPP).
Cite
@article{arxiv.math/0210173,
title = {Computing the cardinality of CM elliptic curves using torsion points},
author = {F. Morain},
journal= {arXiv preprint arXiv:math/0210173},
year = {2007}
}
Comments
Revised and shortened version, including more material using discriminants of curves and division polynomials