A new method to investigate the CCZ-equivalence between functions with low differential uniformity

Recently, many new classes of differentially 4-uniform permutations have been constructed. However, it is difficult to decide whether they are CCZ-inequivalent or not. In this paper, we propose a new notion called “Projected Differential Spectrum”. By considering the properties of the projected differential spectrum, we find several relations that should be satisfied by CCZ-equivalent functions. Based on these results, we mathematically prove that any differentially 4-uniform permutation constructed in [11] by C. Carlet, D. Tang, X. Tang, et al., is CCZ-inequivalent to the inverse function. We also get two interesting results with the help of computer experiments. The first one is a proof that any permutation constructed in [11] is CCZ-inequivalent to a function which is the summation of the inverse function and any Boolean function on F22kF22k when 4≤k≤74≤k≤7. The second one is a differentially 4-uniform permutation on F26F26 which is CCZ-inequivalent to any function in the aforementioned two classes.