A Color Image Encryption Model Based on a System of Quaternion Matrix Equations

In the era of big data and multimedia communication, securing color images against unauthorized access and attacks is a pressing challenge. While quaternion-based models provide a unified representation for color images, most existing encryption schemes rely on single-image frameworks or lack the mathematical rigor to ensure both security and feasibility. To bridge this gap, this paper introduces a system of generalized Sylvester-type quaternion matrix equations as a novel encryption model. By using the equivalence canonical forms of five matrices arranged in a specific array, we provide necessary and sufficient conditions for the solvability of the generalized Sylvester-type quaternion matrix equation system, depending on the rank of the coefficient matrix. Numerical examples are provided to validate the obtained results. As an example of applications, we develop an encryption scheme for color images based on the proposed quaternion matrix equation system. Experimental results confirm the high feasibility of the proposed scheme. Notably, the proposed model supports dynamic key updates and multi-image secure transmission, making it highly adaptable for real-world applications. By integrating advanced quaternion matrix theory with practical image encryption, this work offers a scalable, secure, and mathematically sound approach to color image protection.