A new cryptosystem based on a twisted Hessian curve $H^{4}_{a,d}$

Abstract

In this paper, we are going to study the twisted Hessian curves on the local ring $\mathbb {F}_{q}[\epsilon ]$, $\epsilon ^{4}=0$, with $\mathbb {F}_{q}$ is a finite field of order $q=p^{b}$, where p is a prime number $\ge 5$ and $b\in \mathbb {N}^{*}$. In a first time, we study the arithmetic of the ring $\mathbb {F}_{q}[\epsilon ]$, $\epsilon ^{4}=0$, which will be used in the remainder of this work. After, we define the twisted Hessian curves $H^{4}_{a,d}$ over this ring and we give essential properties and the classification of these elements. In addition, we define the group extension $H^{4}_{a,d}$ of $H_{a_{0},d_{0}}$ by $Ker \ \tilde{\pi }$. We finish this work by introducing a new public key cryptosystem which is a variant of Cramer-Shoup public key cryptosystem on a twisted Hessian curves and study its security and efficiency. Our future work will focus on the generalist these studies for all integers $n>4$, $\epsilon ^{n}=0$, which is beneficial and interesting in cryptography.