Fractional Riesz potentials based on standardized Euclidean distance and their application in image encryption

Within the framework of the standardized Euclidean distance, in this paper, we define a Riesz potential associated with the fractional Fourier transform and establish its intrinsic relationship with the Laplace operator. By introducing a scale-based metric to standardize the Euclidean distance, the proposed potential operator is made more adaptable to the anisotropic structures of images. Building upon this, we develop a double random phase encryption method for color images based on the fractional standardized Euclidean distance Riesz potential. Compared with existing fractional Riesz potential-based encryption methods, the proposed approach exhibits superior security performance, as measured by the mean squared error (MSE), and demonstrates enhanced flexibility in key parameter configuration, improved noise robustness, and higher image reconstruction quality. These advantages provide a new pathway for secure and efficient image encryption.