Geodesic
Classification of Elements in Elliptic Curve Over the Ring \(\mathbb{F}_{q}[\varepsilon]\)
Discussiones Mathematicae. General Algebra and Applications, Tome 41 (2021) no. 2, pp. 283-298

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Let 𝔽_q[ε] := 𝔽_q[X]/(X^4 − X^3) be a finite quotient ring where ε^4 = ε^3, with 𝔽_q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽_q[ε], ε^4 = ε^3 of characteristic p ≠ 2, 3 given by homogeneous Weierstrass equation of the form Y^2Z = X^3 + aXZ^2 + bZ^3 where a and b are parameters taken in 𝔽_q[ε]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve E_a,b(𝔽_q[ε]) and we will show that E_π_0(a),π_0(b)(𝔽_q) and E_π_1(a),π_1(b)(𝔽_q) are two elliptic curves over the finite field 𝔽_q, such that π_0 is a canonical projection and π_1 is a sum projection of coordinate of element in 𝔽_q[ε]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽_q[ε].
Keywords: elliptic curves, finite ring, finite field, projective space 
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Selikh, Bilel; Mihoubi, Douadi; Ghadbane, Nacer. Classification of Elements in Elliptic Curve Over the Ring \(\mathbb{F}_{q}[\varepsilon]\). Discussiones Mathematicae. General Algebra and Applications, Tome 41 (2021) no. 2, pp. 283-298. http://geodesic.mathdoc.fr/item/DMGAA_2021_41_2_a5/