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/

[1] W. Bosma and H.W. Lenstra, Complete System of Two Addition Laws for Elliptic Curves, J. Number Theory 53 (1995) 229–240. https://doi.org/10.1006/jnth.1995.1088

[2] A. Boulbot, A. Chillali and A. Mouhib, Elliptic curves over the ring \(\mathbb{F}_{q}[e], e^3 = e^2\), Gulf J. Math. 4 (2016) 123–129.

[3] A. Boulbot, A. Chillali and A. Mouhib, Elliptic curves over the ring R, Boletim da Sociedade Paranaense de Matematica 38 (2017) 193–201. https://doi.org/10.5269/bspm.v38i3.39868

[4] A. Boulbot, A. Chillali and A. Mouhib, Elliptic curve over a finite ring generated by 1 and an idempotent element ɛ with coefficients in the finite field \(\mathbb{F}_{3^d}\), Boletim da Sociedade Paranaense de Matematica (2018) 1–19. https://doi.org/10.5269/bspm.43654

[5] A. Chillali, Elliptic Curves of the Ring \(\mathbb{F}_{q}[\varepsilon], \varepsilon^n = 0\), Internat. Math. 6 (2011) 1501–1505.

[6] H.W. Lenstra, Jr., Elliptic curves and number-theoretic algorithms (Proceedings of the International Congress of Mathematicians, Berkely, California, USA, 1986).

[7] N. Koblitz, Elliptic curve cryptosystems, Math. Comp. 48 (1987) 203–209. https://doi.org/10.1090/S0025-5718-1987-0866109-5

[8] V. Miller, Use of elliptic curves in cryptography, Advanced cryptology-CRYPTO’85 218 (1986) 417–426. https://doi.org/10.1007/3-540-39799-X_−31

[9] J.H. Silverman, Advanced topics in the arithmetic of elliptic curves (Springer-Verlag, 1994). https://doi.org/10.1007/978-1-4612-0851-8

[10] M. Virat, Courbe elliptique sur un anneau et applications cryptographiques, Doctoral thesis (Universite Nice-Sophia Antipolis, Nice, France, 2009).