Constructing differential 4-uniform permutations from know ones

It is observed that exchanging two values of a function over ${\mathbb F}_{2^n}$, its differential uniformity and nonlinearity change only a little. Using this idea, we find permutations of differential $4$-uniform over ${\mathbb F}_{2^6}$ whose number of the pairs of input and output differences with differential $4$-uniform is $54$, less than $63$, which provides a solution for an open problem proposed by Berger et al. \cite{ber}. Moreover, for the inverse function over $\mathbb{F}_{2^n}$ ($n$ even), various possible differential uniformities are completely determined after its two values are exchanged. As a consequence, we get some highly nonlinear permutations with differential uniformity $4$ which are CCZ-inequivalent to the inverse function on $\mathbb{F}_{2^n}$

An Equivalent Condition on the Switching Construction of Differentially 44-uniform Permutations on \gf_{2^{2k}} from the Inverse Function

Differentially $4$-uniform permutations on \gf_{2^{2k}} with high nonlinearity are often chosen as substitution boxes in block ciphers. Recently, Qu et al. used the powerful switching method to construct permutations with low differential uniformity from the inverse function \cite{QTTL, QTLG} and proposed a sufficient but not necessary condition for these permutations to be differentially $4$-uniform. In this paper, a sufficient and necessary condition is presented. We also give a compact estimation for the number of constructed differentially $4$-uniform permutations. Comparing with those constructions in \cite{QTTL, QTLG}, the number of functions constructed here is much bigger. As an application, a new class of differentially $4$-uniform permutations is constructed. The obtained functions in this paper may provide more choices for the design of substitution boxes

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 \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