Matrix continued fractions and Expansions of the Error Function

In this paper we recall some results and some criteria on the convergence of matrix continued fractions. The aim of this paper is to give some properties and results of continued fractions with matrix arguments. Then we give continued fraction expansions of the error function erf(A) where A is a matrix. At the end, some numerical examples illustrating the theoretical results are discussed

Elliptic Curve over a Local Finite Ring <em>R_n</em>

The goal of this chapter is to study some arithmetic proprieties of an elliptic curve defined by a Weierstrass equation on the local ring Rn=FqX/Xn, where n≥1 is an integer. It consists of, an introduction, four sections, and a conclusion. In the first section, we review some fundamental arithmetic proprieties of finite local rings Rn, which will be used in the remainder of the chapter. The second section is devoted to a study the above mentioned elliptic curve on these finite local rings for arbitrary characteristics. A restriction to some specific characteristic cases will then be considered in the third section. Using these studies, we give in the fourth section some cryptography applications, and we give in the conclusion some current research perspectives concerning the use of this kind of curves in cryptography. We can see in the conclusion of research in perspectives on these types of curves

Elliptic curve and k-Fibonacci-like sequence

In this paper, we will introduce a modified k-Fibonacci-like sequence defined on an elliptic curve and prove Binet’s formula for this sequence. Moreover, we give a new encryption scheme using this sequence

Generalized Fibonacci Sequences for Elliptic Curve Cryptography

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

Cryptography based on the Matrices

In this work we introduce a new method of cryptography based on the matrices over a finite field $\mathbb{F}_{q}$, were $q$ is a power of a prime number $p$. The first time we construct the matrix $M=\left( \begin{array}{cc} A_{1} & A_{2} \\ 0 & A_{3} \\ \end{array} \right)$ were \ $A_{i}$ \ with $i \in \{1, 2, 3 \}$ is the matrix of order $n$ \ in \ $\mathcal{M}(\mathbb{F}_{q})$ - the set of matrices with coefficients in $\mathbb{F}_{q}$ - and $0$ is the zero matrix of order $n$. We prove that $M^{l}=\left( \begin{array}{cc} A_{1}^{l} & (A_{2})_{l} \\ 0 & A_{3}^{l} \\ \end{array} \right)$ were $(A_{2})_{l}=\sum\limits_{k=0}^{l-1} A_{1}^{l-1-k}A_{2}A_{3}^{k}$ for all $l\in \mathbb{N}^{\ast}$. After we will make a cryptographic scheme between the two traditional entities Alice and Bob