An equivalent condition on the switching construction of differentially 4-uniform permutations on from the inverse function
Xi Chen, Yazhi Deng, Zhu Min, Longjiang Qu · International Journal of Computer Mathematics · 2016
Differentially 4-uniform permutations on F22k 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 [Qu et al., Constructing differentially 4-uniform permutations over F22k via the switching method, IEEE Trans. Inform. Theory 59(7) (2013), pp. 4675–4686, Qu et al., More constructions of differentially 4-uniform permutations on F22k, Des. Codes Cryptogr. DOI 10.1007/s10623-014-0006-x] 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 [11 L.J. Qu, Y. Tan, C. Tan, and C. Li, Constructing differentially 4-uniform permutations over F22k via the switching method, IEEE Trans. Inform. Theory 59(7) (2013), pp. 4675–4686. doi: 10.1109/TIT.2013.2252420[Crossref], [Web of Science ®] , [Google Scholar],12 L.J. Qu, Y. Tan, C. Li, and G. Gong, More constructions of differentially 4-uniform permutations on F22k, Des. Codes Cryptogr. Design Code Cryptogr. 78(2) (2016), pp. 391–408.[Web of Science ®] , [Google Scholar]], 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.