Weighted Fractional-Order Transform Based on Periodic Matrix

Abstract

Tao et al. proposed the definition of the linear summation of fractional-order matrices based on the theory of Yeh and Pei. This definition was further extended and applied to image encryption. In this paper, we propose a reformulation of the definitions of Yeh et al. and Tao et al. and analyze them theoretically. The results show that many weighted terms are invalid. Therefore, we use the proposed reformulation to prove that the effective weighted terms depend on the period of the matrix. This also shows that the image encryption methods based on the weighted fractional-order transform will lead to the security risk of key invalidation. Finally, our hypothesis is verified by the unified theoretical framework of multiple-parameter discrete fractional-order transforms.