- Tytuł:
- Classification of Elements in Elliptic Curve Over the Ring \(\mathbb{F}_{q}[\varepsilon]\)
- Autorzy:
-
Selikh, Bilel
Mihoubi, Douadi
Ghadbane, Nacer - Tematy:
-
elliptic curves
finite ring
finite field
projective space - Pokaż więcej
- Wydawca:
- Uniwersytet Zielonogórski. Wydział Matematyki, Informatyki i Ekonometrii
- Powiązania:
- https://bibliotekanauki.pl/articles/55795855.pdf Link otwiera się w nowym oknie
- Opis:
- Let \(\mathbb{F}_{q}[\varepsilon] := \mathbb{F}_{q}[X]/(X^4 − X^3)\) be a finite quotient ring where \(\varepsilon^4 = \varepsilon^3\), with \(\mathbb{F}_{q}\) 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 \(\mathbb{F}_{q}[\varepsilon], \varepsilon^4 = \varepsilon^3\) 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 \(\mathbb{F}_{q}[\varepsilon]\). Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve \(E_{a,b}(\mathbb{F}_{q}[\varepsilon])\) and we will show that \(E_{π_{0}(a),π_{0}(b)}(\mathbb{F}_{q})\) and \(E_{π_{1}(a),π_{1}(b)}(\mathbb{F}_{q})\) are two elliptic curves over the finite field \(\mathbb{F}_{q}\), such that $π_0$ is a canonical projection and $π_1$ is a sum projection of coordinate of element in \(\mathbb{F}_{q}[\varepsilon]\). Precisely, we give a classification of elements in elliptic curve over the finite ring \(\mathbb{F}_{q}[\varepsilon]\).
- Dostawca treści:
- Biblioteka Nauki
Artykuł