Counting rational points on curves over finite fields

We consider the problem of counting the number of points on a plane curve, given by a homogeneous polynomial F∈F_p [x, y, z], which is rational over the ground field F_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_p -rational points on C in randomized time (log p)^Δ where Δ=(degF)^O(1). The complexity of this construction improves previously known bounds for this problem by at least an order of magnitude