Let Fq be a finite field of q elements, where q is a power of a prime number p greater than or equal to 5. In this paper, we study the elliptic curve denoted Ea,b(Fq[e]) over the ring Fq[e], where e2 = e and (a,b) ∈ (Fq[e])2. In a first time, we study the arithmetic of this ring. In addition, using the Weierstrass equation, we define the elliptic curve Ea,b(Fq[e]) and we will show that Eπ0(a),π0(b)(Fq) and Eπ1(a),π1(b)(Fq) are two elliptic curves over the field Fq, where π0 and π1 are respectively the canonical projection and the sum projection of coordinates of X ∈Fq[e]. Precisely, we give a bijection between the sets Ea,b(Fq[e]) and Eπ0(a),π0(b)(Fq)×Eπ1(a),π1(b)(Fq).

This paper suggests an efficient image encryption technique based on double permutation/diffusion process. The aim is to secure a color and gray scale images using an improved 2D logistic map. First, since chaotic properties are significantly related to cryptographic properties in permutation and diffusion, we enhanced the chaotic behavior of the two-dimensional logistic map to create a good pseudorandom number generator that is more suitable for image scrambling. Then, the traditional encryption techniques permutation and diffusion are applied on the plain image. Thus, we generate a first entire pseudo-random sequence coming from the enhanced 2D logistic map that will be used to swap the rows of the plain image and to apply a strong diffusion. Then, a second sequence will be generated to swap the columns and apply a second diffusion to reinforce the security. The results of the simulation concerning histogram, key space, correlation coefficient among neighboring pixels, entropy, NPCR and UACI show the efficiency and reliability of the suggested algorithm.

The Fibonacci sequence is a well-known sequence of numbers with numerous applications in mathematics, computer science, and other fields. In recent years, there has been growing interest in studying Fibonacci-like sequences on elliptic curves. These sequences have a number of exciting properties and can be used to build new encryption systems. This paper presents a further generalization of the Fibonacci sequence defined on elliptic curves. We also describe an encryption system using this sequence which is based on the discrete logarithm problem on elliptic curves.

In [17], A. chillali et al introduce a new cryptographic method based on matrices over a finite field Fpn , where p is a prime number. In this paper, we will generate this method in a new group of square block matrices based on an elliptic curve, called "elliptic" matrices.

Let Fq be a finite field, where q is a power of a prime p such that p>5. Let \alpha be a root of a monic polynomial of a minimal degree over Fq. In this paper, we will study elliptic curves over (Fq[\alpha],+,*), where + is the usual addition and * represent a non-standard product law over Fq[\alpha]. Elliptic curves over this ring can be used to create a new type of cryptography. The method is fast, simple, and secure.

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}). In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.