Modular Counting of Rational Points over Finite Fields

Let Fq be the finite field of q elements, where q = ph. Let f(x) be a polynomial over Fq in n variables with m nonzero terms. Let N(f) denote the number of solutions of f(x) = 0 with coordinates in Fq. In this paper we give a deterministic algorithm which computes the reduction of N(f) modulo pb in O(n(8m)p(h+b)p) bit operations. This is singly exponential in each of the parameters {h, b, p}, answering affirmatively an open problem proposed by Gopalan, Guruswami, and Lipton.