More constructions of differentially 4-uniform permutations on $${\mathbb {F}}_{2^{2k}}$$ F 2 2 k

Abstract

Differentially $$4$$4-uniform permutations on $${\mathbb {F}}_{2^{2k}}$$F22k with high nonlinearity are chosen as Substitution boxes in many block ciphers and some stream ciphers. Recently, Qu et al. (IEEE Trans Inf Theory, 59(7), 4675---4686, 2013) 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 $$4$$4-uniform permutations over $${\mathbb {F}}_{2^{2k}}$$F22k. 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 $$4$$4-uniform permutations over $${\mathbb {F}}_{2^{2k}}$$F22k grows exponentially when $$k$$k increases, which gives a positive answer to an open problem proposed in Qu et al.(IEEE Trans Inf Theory, 59(7), 4675---4686, 2013).