Cryptography based on the Matrices

Abstract

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.