Quasi-symmetry and the half-cycle phenomenon in scrambling degrees for images with pixel locations scrambled by Arnold transformation.

*(Chinese. English summary)*Zbl 1349.68306Summary: By referring to the periodicity of the general Arnold transformation and the relationship between standard Arnold transformation and Fibonacci transformation, we deduce the equivalent one-step transformation matrix for \(k\) times of Arnold transformation with pixel position scrambled, especially the one at the half-cycle of the scrambling period. We analyze their characteristics and provide a proof of the quasi-symmetry in scrambling degrees for images in one cycle. We discuss the half-cycle effect in scrambling degrees in scrambled images with even and odd scrambling periods respectively. Results show that there exists a quasi-symmetry in scrambling performance between the two half cycles regardless of the period being even or odd. In a standard Arnold transformation with a commonly even period, the one-step transform is equivalent to a simple scaling matrix transform which leads to the scrambled image at the half period with an obvious lower scrambling degree, where the minus unitary matrix as a special case results in the scrambled image being the horizontal mirror image with an overlying vertical mirror image of the original image. For any general Arnold transformation with an even scrambling period, the one-step transformation at half cycle may be the same as the one-step transform for standard Arnold transform or with a translation of half of the image dimension superimposed, thus leading to a little less salient half-cycle phenomenon. For an Arnold transformation with an odd scrambling period, no such situation happens in general unless for images with very special contents and structure. The results can be applied in choice of scrambling time for the pre-processing in image encryption and the evaluation and comparison of image scrambling degree criteria.