Binary Twisted Hessian curve over a local ring

Binary Twisted Hessian curve

Abstract

Let F_2ⁿ be a finite field, where n is a positive integer. In this article, we will study the twisted Hessian curve over the ring A₂ = F_2ⁿ [e], where the relation e²= 0. More precisely, we will give many various explicit formulas, which describe the binary operations calculus in HB²_a,d where HB²_a,d is the binary twisted Hessian curve over A₂, and we will reduce the cost of the complexity of the calculus in HB²_a,d.  In a first time, we describe these curves over this ring. In addition, we prove that when $2$ doesn't divide # ( HB_{\pi(a), \ \pi(d)})$, then HB²_a,d is a direct sum of $ HB_{\pi(a), \ \pi(d)}$ and F_2ⁿ, where $ HB_{\pi(a), \ \pi(d)}$ is the twisted Hessian curve over F_2ⁿ. Other results are deduced from, we cite the equivalence of the discrete logarithm problem on the binary twisted Hessian curves HB²_a,d and $ HB_{\pi (a), \ \pi (d)}$, which is beneficial for cryptography and cryptanalysis as well.