QGI Algorithm for Solving Quaternion Equation Arising in Image Encryption

ABSTRACT In view of the advantages of the solution of the quaternion matrix equation, which was originally employed to solve the color image encryption problem, the quaternion gradient‐based algorithm (QGI) is proposed by presenting a new real representation of quaternion matrix for solving the quaternion solution, pure imaginary solution, and real solution of the equation . The necessary and sufficient conditions for the above solutions existing are provided. It is demonstrated that our QGI method can also obtain the least squares quaternion solution, least squares pure imaginary solution, and least squares real solution when it is inconsistent. Finally, numerical examples are provided to illustrate the feasibility and validity of the QGI algorithm when different parameters are chosen. Experimental results demonstrate that the proposed method is feasible and much more effective for color image encryption and its encryption performance is evaluated from four different aspects. All parameters are found to be close to the ideal values, confirming the effectiveness of QGI encryption scheme and the accuracy of the theoretical results obtained.