Explicit addition formulae on hyperelliptic curves of genus 2 for isogeny-based cryptography

Abstract

Some isogeny-based cryptosystems use addition and doubling on the Jacobian over genus-2 sextic and non-monic hyperelliptic curves. In this study, we generalized some formulae for quintic and monic curves to sextic curves using projective coordinates and then compared them. For sextic curves and projective coordinates, the formulae based on Lange’s were faster than those based on Costello–Lauter’s, in contrast to quintic curves. The formulae based on Lange’s take 64M + 6S for addition and 59M + 9S for doubling, where M and S denote the computational costs of multiplication and squaring, respectively.