Counting Points on Elliptic Curves over Nite Elds

We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large ; it is based on Shanks's baby-step-giant-step strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is based on calculations with the torsion points of the elliptic curve [18]. This deterministic polynomial time algorithm was impractical in its original form. We discuss several practical improvements by Atkin and Elkies.