Abstract
In this note, we present the first results in our overall program aiming to improve the state-of-the-art for general-purpose hyperelliptic curvearithmetic, where the curve may have any genus, be defined over any field, and is given in standard Weierstrass form as opposed to a special non-generic form. Our first contribution is an adaptation of Shanks' NUCOMP algorithm for divisor class group arithmetic on split model hyperelliptic curves of arbitrary genus. Our algorithm works for any split model curve given in general Weierstrass form defined over any field. Our Magma implementations for both ramified and split models over prime finite fields are the method of choice for genus greater than 7. In addition, our split model implementation performs better than Cantor's approach and closes the previously-observed performance gap with the ramified model. Our second contribution is a series of practical improvements to speed-up addition and doubling operations for generic hyperelliptic curves of genus 2 in Weierstrass form defined over any field. Our algorithms, as is typically the case for small genus, are presented as explicit formulas.