Twisted Hessian curve over a local ring

In this paper, we study the twisted Hessian curve denoted H^n₀, ₃ over the ring Fₐ X / (X^n), where Fₐ 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 \# (H (₀), \ (₃) ), then H^n₀, ₃ is a direct sum of H (₀), \ (₃) and Fₐ^n-1, where H (₀), \ (₃) is the twisted Hessian curve over Fₐ. Other results are deduced from, we cite the equivalence of the discrete logarithm problem on the twisted Hessian curves H^n₀, ₃ and H (₀), \ (₃), which is beneficial for cryptography and cryptanalysis as well.