Abstract
This paper introduces a novel method for selecting parameters of finite fields formed by 2 × 2 matrices over a finite field of integers modulo a prime p. The method aims to simultaneously determine both the field parameters and primitive elements, thereby optimizing the construction of cryptographic algorithms. The proposed approach leverages the properties of quadratic residues and non-residues, simplifying the process of finding matrix field parameters while maintaining computational efficiency. The method is particularly effective when the prime number p is either a Mersenne prime or (p + 1)/2 is also a prime. This study demonstrates that the resulting matrix fields can be practically computed, offering a high degree of flexibility for cryptographic protocols such as key agreement and secure data transmission. Compared to previous methods, the new method reduces the parameter search space and provides a structured way to identify primitive elements without the need for a separate search procedure. The findings have significant implications for the development of efficient cryptographic systems using matrix-based finite fields.