Classification of genus 2 curves over $\mathbb{F}_{2^n}$ and optimization of their arithmetic

Abstract

To obtain efficient cryptosystems based on hyperelliptic curves, we studied genus 2 isomorphism classes of hyperelliptic curves in characteristic 2. We found general and optimal form for these curves, just as the short Weierstrass form for elliptic curves. We studied the security and the arithmetic on their jacobian. We also rewrote and optimized the formulas of Lange in characteristic 2, and we introduced a new system of coordinate. Therefore, we deduced the best form of hyperelliptic curves of genus 2 in characteristic 2 to use in cryptography