Research and Development of Principles for Creating Digital Signatures in Matrix Fields

The article presents an innovative method for generating electronic digital signatures (EDS) based on operations in finite fields of matrices. The proposed approach utilizes the group of invertible matrices to construct base matrices, where secret exponents are represented as vectors, and the public key is computed as the product of matrix powers. The signature is formed by decomposing the hash of the message using a selected basis, which allows the integration of factorial coding ideas without requiring a secure channel for secret data exchange. The method is mathematically grounded, taking into account the study of primitive elements in the field of 2nd-order square matrices and is adapted for 3×3 matrices. The algorithm has been implemented in MATLAB, with experimental results presented in the form of graphs. The scientific novelty lies in introducing non-commutative matrix operations and utilizing factorial coding to enhance cryptographic resilience, which is particularly relevant for post-quantum information protection systems.