TWISTED EDWARDS CURVE OVER THE RING

Abstract

Let Fq be a finite field ofqelements, whereqis 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], wheree2= 0. In the first time, we study the
arithmetic of the ringFq[e],e2= 0. After that we define the twisted Edwards
curvesEEa,dover this ring and we give essential properties and we define the
groupEEa,d, these properties. Precisely, we give a bijection between the groups
EEa,dandEEa0,d0×Fq, whereEEa0,d0is the twisted Edwards curves over the
finite fieldFq.