Counting points on hyperelliptic curves in average polynomial time

Abstract

Let g 1, and let Q2 Z[x] be a monic, squarefree polynomial of degree 2g + 1. For an odd prime p not dividing the discriminant of Q, let Zp(T ) denote the zeta function of the hyperelliptic curve of genus g over the nite eld Fp obtained by reducing the coecients of the equation y 2 = Q(x) modulo p. We present an explicit deterministic algorithm that given as input Q and a positive integer N, computes Zp(T ) simultaneously for all such primes p < N, whose average complexity per prime is polynomial in g, logN, and the number of bits required to represent Q.