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Differentially 4-uniform permutations on $\gf_{2^{2k}}$ with highnonlinearity are often chosen as Substitution boxes in both block and streamciphers. Recently, Qu et al. introduced a class of functions, which are calledpreferred functions, to construct a lot of infinite families of suchpermutations \cite{QTTL}. In this paper, we propose a particular type ofBoolean functions to characterize the preferred functions. On the one hand,such Boolean functions can be determined by solving linear equations, and theygive rise to a huge number of differentially 4-uniform permutations over$\gf_{2^{2k}}$. Hence they may provide more choices for the design ofSubstitution boxes. On the other hand, by investigating the number of theseBoolean functions, we show that the number of CCZ-inequivalent differentially4-uniform permutations over $\gf_{2^{2k}}$ grows exponentially when $k$increases, which gives a positive answer to an open problem proposed in\cite{QTTL}.

More Constructions of Differentially 4-uniform Permutations on $\gf_{2^{2k}}$