Twisted Edwards Curve Over the Ring

Abstract

Abstract Let 𝔽q be a finite field of q elements, where q is a power of an odd prime number. In this paper, we study the twisted Edwards curves denoted EEa,d over the local ring 𝔽q[e], where e2 = 0. In the first time, we study the arithmetic of the ring 𝔽q[e], e2 = 0. After that we define the twisted Edwards curves EEa,d over this ring and we give essential properties and we define the group EEa,d , these properties. Precisely, we give a bijection between the groups EEa,d and EEa,d0 × Fq,where EEa,d0 is the twisted Edwards curves over the finite field 𝔽q.