An equivalent condition on the switching construction of differentially 4-uniform permutations on from the inverse function

ABSTRACT

Differentially 4-uniform permutations on ${\mathbb{F}}_{2^{2k}}$ 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 [Qu et al., Constructing differentially 4-uniform permutations over ${\mathbb{F}}_{2^{2k}}$ 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 ${\mathbb{F}}_{2^{2k}}$, 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 [11,12], 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.

KEYWORDS:

• Differentially 4-uniform permutation
• substitution box
• 4-uniform BFI
• preferred Boolean function
• APN function

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 11T06
• 11T71
• 14G50
• 94A60

Acknowledgments

We are greatly indebted to Claude Carlet, who has instructed and helped us a lot in revising the last part of the paper. We would also like to thank the editor and the anonymous reviewers for their valuable comments and insightful observations, which allowed to significantly improve the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Basic Research Program of China [grant number 2013CB338002], the Nature Science Foundation of China (NSFC) [grant number 61272484], the Program for New Century Excellent Talents in University (NCET) and the Basic Research Fund of National University of Defense Technology [grant number CJ 13-02-01].