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Title:
A New Method to Investigate the CCZ-Equivalence between Functions with Low Differential Uniformity
Authors: Xi Chen, Longjiang Qu, Chao Li, 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.
ePrint: https://eprint.iacr.org/2016/744
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