Computation on Jacobians of Hyperelliptic Curves of Genus 3

Abstract

In this article, we give an easy method to distinguish different cases of additions on Jacobians of hyperelliptic curves of genus 3. In addition, we give an advanced algorithm for group laws on Jacobian of hyperelliptic curves of genus 3. By this method, our algorithm can handle all kinds of inputs without recalling a generic algorithm. Our method is mainly based on Harley's algorithm. However, we use linear algebra over finite fields, instead of Chinese Reminder Theorem over function fields. Moreover, We did $2\times 10^{8}$ experiments in the finite field $\mathbb{F}_{2^{61}-1}$, our algorithm runs 0.033% faster than previous works in general addition.