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).