Elliptic curves over a nonlocal ring 𝔽2d[𝜀],𝜀2 = 𝜀

Abstract

In this work, we came to present the elliptic curves over a nonlocal ring [Formula: see text] where [Formula: see text] is a finite field of characteristic 2, and [Formula: see text] is a prime integer. We consider an elliptic curve given by the Weierstrass equation of the form [Formula: see text] such that [Formula: see text] are in the ring [Formula: see text] and b is invertible in [Formula: see text] see Tadmori et al. [Normal form of the elliptic curves over the finite ring, J. Math. Syst. Sci. 4 (2014) 194–196]. More precisely, we give a classification of elements of the elliptic curve over this ring, and we construct the group law over it, which allows us to prove same properties of the elliptic group [Formula: see text] by showing the isomorphism [Formula: see text]