Point counting in families of hyperelliptic curves in characteristic 2

Abstract:

Let E_Γ be a family of hyperelliptic curves over F₂^{alg cl} with general Weierstrass equation given over a very small field F. The author of this paper describes an algorithm for computing the zeta function of E_γ, with γ in a degree n extension field of F, which has time complexity O(n^{3 + ε}) bit operations and memory requirements O(n²) bits. Using a slightly different algorithm, one can get time O(n^2.667) and memory O(n^2.5), and the computation for n curves of the family can be done in time O(n^3.376). All of these algorithms are polynomial-time in the genus.