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.

Let Fq be a finite field of q elements, where q is a power of a prime number p. In this paper, we study the Huff curves over the ring Fq[epsilon] where epsilon^{2}=epsilon, denoted by Ha,b (Fq[epsilon]), (a,b) in (Fq[epsilon])^{2}.Using the Huff equation, we define the Huff curves Ha,b (Fq[epsilon]) and we will show that H pi_{0}(a),pi_{0}(b)(Fq) and H pi_{1}(a),pi_{1}(b)(Fq) are two Huff curves over the field Fq, where pi_{0} and pi_{1} are respectively the canonical projection and the sum projection of coordinates from Fq[epsilon] to Fq. Precisely, we give a bijection between the sets Ha,b (Fq[epsilon]) and H pi_{0}(a),pi_{0}(b)(Fq) X H pi_{1}(a),pi_{1}(b)(Fq).