Twisted Edwards Curve Over the Ring

Abstract

Abstract 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 E a,d over the local ring 𝔽 q [e], where e 2 = 0. In the first time, we study the arithmetic of the ring 𝔽 q [e], e 2 = 0. After that we define the twisted Edwards curves E E a,d over this ring and we give essential properties and we define the group E E a,d , these properties. Precisely, we give a bijection between the groups E E a,d and E E a,d 0 × F q ,where E E a,d 0 is the twisted Edwards curves over the finite field 𝔽 q .