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More constructions of differentially 4-uniform permutations on \(\mathbb {F}_{2^{2k}}\). (English) Zbl 1401.94239
Summary: Differentially 4-uniform permutations on \(\mathbb {F}_{2^{2k}}\) with high nonlinearity are chosen as Substitution boxes in many block ciphers and some stream ciphers. Recently, L. Qu et al. [IEEE Trans. Inf. Theory 59, No. 7, 4675–4686 (2013; Zbl 1364.94565)] introduced a class of functions, which are called preferred functions, to construct a lot of infinite families of such permutations. In this paper, we propose a particular type of Boolean functions to characterize the preferred functions. On the one hand, such Boolean functions can be determined by solving linear equations, and they give rise to a huge number of differentially 4-uniform permutations over \(\mathbb {F}_{2^{2k}}\). Hence, they may provide more choices for the design of Substitution boxes. On the other hand, by investigating the number of these Boolean functions, we show that the number of CCZ-inequivalent differentially 4-uniform permutations over \(\mathbb {F}_{2^{2k}}\) grows exponentially when \(k\) increases, which gives a positive answer to an open problem proposed in Qu et al. (loc. cit.).

MSC:
94B25 Combinatorial codes
94A60 Cryptography
Software:
Camellia; eSTREAM
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