Point counting in families of hyperelliptic curves in characteristic 2
Hendrik Hubrechts
Abstract
Let EG be a family of hyperelliptic curves over F2(alg cl) with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm to compute the zeta function of Eg for g in a degree n extension field of F, which has as time complexity O(n3) and memory requirements O(n2). With a slightly different algorithm we can get time O(n2,667) and memory O(n2,5), and the computation of O(n) curves of the family can be done in time and space O(n3). All these algorithms are polynomial in the genus.
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