Explicit Formulas of the Addition Law on the Elliptic Curve Over a Special Nonlocal Ring

In this paper, we come to study some properties of elliptic curves over a nonlocal ring \(A=\mathbb {F}_{q}[\varepsilon ];\varepsilon ^{3}=\varepsilon ^2,\) where \(\mathbb {F}_{q}\) is the finite field of q elements, and q is a power of a prime integer \(p\geq 5.\) We discuss, and we define here a new addition law on such elliptic curve over this special ring. More precisely we show that such elliptic curve over A is isomorphic to the product of an elliptic curve over the field \(F_q\) and one over the local ring \(\frac {\mathbb {F}_{2^d}[X]}{(X^2)}.\) This allows us to compute the addition points over these elliptic curves, and to propose such encryption protocol.