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

Abstract

Let F3d be a finite field of order 3d with d∈ N*. In this paper, we study the elliptic curve over the finite ring F3d[ε] :=F3d[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 ∈F3d[ε], such that we study the arithmetic operations of this ring and define the elliptic curve over it. Next, we show that EΠ0(a), Π0(b)(F3d) and EΠ1(a), Π1(b)(F3d) are two elliptic curves over the finite field F3d, such that Π0 is a canonical projection and Π1 is a sum projection of coordinate of element in F3d[ε] and we conclude by given a classification of elements in elliptic curve over the finite ring F3d[ε].