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.