New elliptic group over a nonlocal ring $\mathbb{F}_{2^d}[\varepsilon],\,\varepsilon^{3}=\varepsilon^2$

Abstract

In this paper, we consider the set of elliptic curves over an extended nonlocal ring of characteristic two $A=\frac{\mathbb{F}_{2^d}[X]}{(X^3- X^2)}$.
Then by studying the arithmetic operation of this ring, and define such elliptic curves, we come to classify their elements.
More precisely, we define a new group law structure on this elliptic curve by using one of the explicit bijection
$E_{\pi_{1}(a),\pi_{1}(b)}(\mathbb{F}_{2^d})\times E_{\pi_{2}(a),\pi_{2}(b)}(A_2)\simeq E_{a,b}(A),$
where $A_2=\frac{\mathbb{F}_{2^d}[X]}{(X^2)}$ is a local ring, $\pi_{1}$ is a sum projection of the coordinates elements in A,
and $\pi_{2}$ is the surjective morphism defined by: $$ \pi_2: A\longrightarrow A_2=\frac{\mathbb{F}_{2^d}[X]}{(X^2)}$$ $$ x_0+x_1\varepsilon+x_2\varepsilon^2\longmapsto x_0+x_1\sigma\,\,\mbox{ where}\,\,\sigma^2=0.$$