Quasi-quadratic elliptic curve point counting using rigid cohomology

Abstract

Let E be a nonsupersingular elliptic curve over the finite field with p n elements. We present a deterministic algorithm that computes the zeta function and hence the number of points of such a curve E in time quasi-quadratic in n . An older algorithm having the same time complexity uses the canonical lift of E , whereas our algorithm uses rigid cohomology combined with a deformation approach. An implementation in small odd characteristic turns out to give very good results.