Elliptic curves over the ring R

Abstract

Let Fq be a ï¬nite ï¬eld of q elements, where q is a power of a prime number p greater than or equal to 5. In this paper, we study the elliptic curve denoted Ea,b(Fq[e]) over the ring Fq[e], where e2 = e and (a,b) ∈ (Fq[e])2. In a ï¬rst time, we study the arithmetic of this ring. In addition, using the Weierstrass equation, we deï¬ne the elliptic curve Ea,b(Fq[e]) and we will show that EÏ€0(a),Ï€0(b)(Fq) and EÏ€1(a),Ï€1(b)(Fq) are two elliptic curves over the ï¬eld Fq, where Ï€0 and Ï€1 are respectively the canonical projection and the sum projection of coordinates of X ∈Fq[e]. Precisely, we give a bijection between the sets Ea,b(Fq[e]) and EÏ€0(a),Ï€0(b)(Fq)×EÏ€1(a),Ï€1(b)(Fq).