Jacobian Coordinates on Genus 2 Curves

Abstract

This paper presents a new projective coordinate system and new explicit algorithms which together boost the speed of arithmetic in the divisor class group of genus 2 curves. The proposed formulas generalise the use of Jacobian coordinates on elliptic curves and their application improves the speed of performing cryptographic scalar multiplications in Jacobians of genus 2 curves over prime fields by an approximate factor of 1.25x. For example on a single core of an Intel Core i7-3770M (Ivy Bridge) we show that replacing the previous best formulas with our new set improves the cost of generic scalar multiplications from 243000 to 195000 cycles and drops the cost of specialised GLV-style scalar multiplications from 166000 to 129000 cycles.