Primitive Elements in the Finite Field of Square Matrices of Order 2 for Cryptographic Applications

This paper is based on the previous studies that allowed to define a family of square matrices of order 2 over the field of integers modulo prime. This family forms a finite field with the usual operations of matrix multiplication and addition. This paper aims at developing and applying an approach to determine primitive elements of such a finite field of square matrices. The relevance of this topic is explained by the fact that finding primitive elements is an integral component for applying a finite field in cryptographic transformation tasks, in particular, the Diffie-Hellman key agreement protocol. This study provides an answer to the conditions under which a matrix is a primitive element of a finite field, as well as the method for finding all primitive elements of a matrix field. It has been determined the number of different primitive elements of the field. Examples of searching for primitive elements for the finite field various parameters are provided. An example of using a finite field of square matrices of order 2 in the Diffie-Hellman key agreement protocol is demonstrated.