Montgomery Addition for Genus Two Curves

Abstract

Hyperelliptic curves of low genus obtained a lot of attention in the recent past for cryptographic applications. They were shown to be competitive with elliptic curves in speed and security. In practice, one also needs to prevent from side channel analysis, a method using information leaked during the process of computing to attack the system. For elliptic curves the curve arithmetic proposed by Montgomery requires a comparably small number of field operations to perform a scalar multiplication but at the same time achieves security against non-differential side channel attacks.

This paper studies the generalization of Montgomery arithmetic for genus 2 curves. We do not give the explicit formulae here, but together with the explicit formulae for affine or projective group operations the results show how to implement it. The divisor classes can be represented using only their first polynomials, a feature that is important for actual implementations. Our method applies to arbitrary genus two curves over arbitrary fields of odd characteristic which have at least one rational Weierstraß point.