Twisted Hessian curve over a local ring

In this paper, we study the twisted Hessian curve denoted Hⁿa,d over the ring Fq [X] / (Xⁿ), where Fq is a finite field of q elements, with q is a power of a prime number p ≥ 5 and n ≥ 5. In a first time, we describe these curves over this ring. In addition, we prove that when p doesn't divide # ( Hπ(a), \ π(d)), then Hⁿa,d is a direct sum of Hπ(a), \ π(d) and Fqⁿ⁻¹, where Hπ(a), \ π(d) is the twisted Hessian curve over Fq. Other results are deduced from, we cite the equivalence of the discrete logarithm problem on the twisted Hessian curves Hⁿa,d and Hπ (a), \ π (d), which is beneficial for cryptography and cryptanalysis as well.