Huff curve over the ring $\mathbb{F}_{q}[\epsilon], \epsilon^{2}=\epsilon$

Abstract

Let F_q  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 F_q[epsilon] where epsilon^{2}=epsilon, denoted by H_a,b (F_q[epsilon]), (a,b) in (F_q[epsilon])^{2}.
Using the Huff equation, we define the Huff curves H_a,b (F_q[epsilon]) and we will show that H _{pi_{0}(a),pi_{0}(b)}(F_q)  and H _{pi_{1}(a),pi_{1}(b)}(F_q) are two Huff curves over the field F_q, where pi_{0} and pi_{1} are respectively the canonical projection and the sum projection of coordinates from F_q[epsilon] to F_q. Precisely, we give a bijection between the sets H_a,b (F_q[epsilon]) and H _{pi_{0}(a),pi_{0}(b)}(F_q)  X H _{pi_{1}(a),pi_{1}(b)}(F_q).