Many recent color image encryption models are mathematically characterized by a numerical solution problem of quaternion matrix equation. In this paper, the quaternion biconjugate residual (QBCR) algorithm is firstly provided by means of a new real representation of quaternion matrix for solving the numerical solution of the equation \(AY = E\) . The necessary and sufficient conditions for the above solutions existing is provided. It is demonstrated that our QBCR method can achieve to converge to the exact solution within a finite number of iteration steps in the absence of round-off errors when it is consistent. Finally, three numerical examples are provided to show the feasibility and validity of our method in comparison with the QGI method and QRGI method, especially in terms of computing time and error. Moreover, the proposed method has been used to solve color image encryption problem 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 QBCR encryption scheme and the accuracy of the theoretical results obtained.
