Twisted Edwards Curve Over the Ring

Let 𝔽_q 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 E_Ea,d over the local ring 𝔽_q[e], where e² = 0. In the first time, we study the arithmetic of the ring 𝔽_q[e], e² = 0. After that we define the twisted Edwards curves E_Ea,d over this ring and we give essential properties and we define the group E_Ea,d , these properties. Precisely, we give a bijection between the groups E_Ea,d and E_Ea,d0 × F_q,where E_Ea,d0 is the twisted Edwards curves over the finite field 𝔽_q.