Study of elliptic curve over a finite ring F_{3^d}[ε], ε^4 = ε^3

Abstract

Let F_3^d be a finite field of order 3^d with d∈ N^*. In this paper, we study the elliptic curve over the finite ring F_3^d[ε] :=F_3^d[X] / (X⁴ -X³), where ε⁴ = ε³ of characteristic 3 given by the homogeneous Weierstrass equation of the form Y²Z = X³ + aX²Z + bZ³, where a, b ∈F_3^d[ε], such that we study the arithmetic operations of this ring and define the elliptic curve over it. Next, we show that E_{Π0(a), Π0(b)}(F_3^d) and E_{Π1(a), Π1(b)}(F_3^d) are two elliptic curves over the finite field F_3^d, such that Π₀ is a canonical projection and Π₁ is a sum projection of coordinate of element in F_3^d[ε] and we conclude by given a classification of elements in elliptic curve over the finite ring F_3^d[ε].