**An Equivalent Condition on the Switching Construction of Differentially 4-uniform Permutations on $\gf_{2^{2k}}$ from the Inverse Function**

*Xi Chen, Yazhi Deng, Min Zhu and Longjiang Qu*

**Abstract: **Differentially 4-uniform permutations on $\gf_{2^{2k}}$ with high nonlinearity are often chosen as Substitution boxes in block ciphers. Recently, Qu et al. used the powerful switching method to construct such permutations from the inverse function [9],[10]. More precisely, they studied the functions of the form G(x)=1/x+f(1/x),
where f is a Boolean function. They proved that if f is a preferred Boolean function (PBF), then G is a permutation polynomial over $\gf_{2^n}$ whose differential uniformity is at most 4. However, as pointed out in [9],f is a PBF is a sufficient but not necessary condition. In this paper, a sufficient and necessary condition for G to be a differentially 4-uniform permutation is presented. We also show that G can not be an almost perfect nonlinear (APN) function. As an application, a new class of differentially 4-uniform permutations where f are not PBFs are constructed. By comparing this family with previous constructions, the number of permutations here is much bigger. The obtained functions in this paper may provide more choices for the design of Substitution boxes.

**Category / Keywords: **foundations / Differentially 4-uniform function, Substitution box, 4-Uniform BFI, Preferred Boolean function, Permutation function

**Date: **received 21 Aug 2014

**Contact author: **1138470214 at qq com;cxcxcxcx007@126 com

**Available format(s): **PDF | BibTeX Citation

**Note: **This paper have been submitted to The Institute of Electronics, Information and Communication Engineers (IEIEC) in May 10, 2014.

**Version: **20140827:073359 (All versions of this report)

**Short URL: **ia.cr/2014/648

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