Huff’s form for elliptic curves over a local ring

Let F q be a finite field of order q = p c , where p is a prime number ≥ 3 , c ∈ N ∗ . In this paper, we introduce Huff curves denoted H a , b 2 over the local ring R 2 = F q [ X ] / ( X 2 ) . At first, we recall the arithmetic of the ring R 2 . After that, we define Huff curves H a , b 2 over this ring and we study the group H a , b 2 , its properties and the classification of its elements. Precisely, we give a bijection between the groups H a , b 2 and H a 0 , b 0 × F q , where H a 0 , b 0 is the Huff curves over the finite field F q .