Twisted Hessian curve over a local ring

Abstract

In this paper, we study the twisted Hessian curve denoted $H^{n}_{a,d}$ over the ring $\mathbb{F}_{q} [X] / (X^{n})$, where $\mathbb{F}_{q}$ is a finite field of q elements, with q is a power of a prime number $p \geq 5$ and $n \geq 5$. In a first time, we describe these curves over this ring. In addition, we prove that when $p$ doesn't divide $\# ( H_{\pi(a), \ \pi(d)})$, then $H^{n}_{a,d}$ is a direct sum of $ H_{\pi(a), \ \pi(d)}$ and $\mathbb{F}_{q}^{n-1}$, where $ H_{\pi(a), \ \pi(d)}$ is the twisted Hessian curve over $\mathbb{F}_{q}$. Other results are deduced from, we cite the equivalence of the discrete logarithm problem on the twisted Hessian curves $H^{n}_{a,d}$ and $ H_{\pi (a), \ \pi (d)}$, which is beneficial for cryptography and cryptanalysis as well.