Optimization of the arithmetic of the ideal class group for genus 4 hyperelliptic curves over projective coordinates

The aim of this paper is to reduce the number of operations in Cantor’s algorithm for the Jacobian group of hyperelliptic curves for genus 4 in projective coordinates. Specifically, we developed explicit doubling and addition formulas for genus 4 hyperelliptic curves over binary fields with h(x) = 1. For these curves, we can perform a divisor doubling in 63M + 19S, while the explicit adding formula requires 203M + 18S, and the mixed coordinates addition (in which one point is given in affine coordinates) is performed in 165M + 15S. These formulas can be useful for public key encryption in some environments where computing the inverse of a field element has a high computational cost (either in time, power consumption or hardware price), in particular with embedded microprocessors.