Building matrixes of higher order to achieve the special commutative multiplication and its applications in cryptography

Abstract

In this paper, we introduce a method for building matrices that verify the commutative property of multiplication on the basis of circular matrices, as each of these matrices can be divided into four circular matrices, and we can also build matrices that verify the commutative property of multiplication from higher order and are not necessarily divided into circular matrices.

Using these matrixes, we provide a way to securely exchange a secret encryption key, which is a square matrix, over open communication channels, and then use this key to exchange encrypted messages between two sides or two parties. Moreover, using these matrixes we also offer a public-key encryption method, whereby the two parties exchange encrypted messages without previously agreeing on a common secret key between them.