Counting points on hyperelliptic curves in average polynomial time

TL;DR

This paper introduces an explicit deterministic algorithm that efficiently computes the zeta functions of hyperelliptic curves over finite fields for many primes simultaneously, with average polynomial complexity.

Contribution

It presents a novel algorithm that computes zeta functions for hyperelliptic curves over multiple primes with average polynomial time complexity.

Findings

Algorithm computes zeta functions for all primes p < N efficiently

Average complexity per prime is polynomial in key parameters

Applicable to hyperelliptic curves of genus g over finite fields

Abstract

Let g >= 1 and let Q be a monic, squarefree polynomial of degree 2g + 1 in Z[x]. For an odd prime p not dividing the discriminant of Q, let Z_p(T) denote the zeta function of the hyperelliptic curve of genus g over the finite field F_p obtained by reducing the coefficients 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 Z_p(T) simultaneously for all such primes p < N, whose average complexity per prime is polynomial in g, log N, and the number of bits required to represent Q.