Paper 2016/744

A New Method to Investigate the CCZ-Equivalence between Functions with Low Differential Uniformity

Xi Chen, Longjiang Qu, Chao Li, and Jiao Du

Abstract

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 \cite{CTTL} 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 \cite{CTTL} is CCZ-inequivalent to a function which is the summation of the inverse function and any Boolean function on $\gf_{2^{2k}}$ when $4\le k\le 7$. The second one is a differentially $4$-uniform permutation on $\gf_{2^6}$ which is CCZ-inequivalent to any function in the aforementioned two classes.