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.

Let L = Q(i) or Q(√−3) be a quadratic cyclotomic number field, and let K be a Galois extension of L of prime degree q. This paper examines the behavior of p-Sylow subgroups of elliptic curves defined over K, focusing on their growth under base change from L. The study uncovers distinctive patterns in subgroup growth, shaped by both the arithmetic nature of the base field and the intrinsic properties of the elliptic curves.