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

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.