Counting rational points on curves over finite fields

Abstract

We consider the problem of counting the number of points on a plane curve, given by a homogeneous polynomial F/spl isin/F/sub p/[x, y, z], which is rational over the ground field F/sub p/. More precisely, we show that if we are given a projective plane curve C of degree n, and if C has only ordinary multiple points, then one can compute the number of F/sub p/-rational points on C in randomized time (log p)/sup /spl Delta// where /spl Delta/=(degF)/sup O(1/). The complexity of this construction improves previously known bounds for this problem by at least an order of magnitude. >