Counting points on hyperelliptic curves in average polynomial time
Number Theory
2013-09-27 v3
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.
Cite
@article{arxiv.1210.8239,
title = {Counting points on hyperelliptic curves in average polynomial time},
author = {David Harvey},
journal= {arXiv preprint arXiv:1210.8239},
year = {2013}
}
Comments
17 pages, some simplifications, main theorem strengthened slightly, to appear in the Annals of Mathematics