**Differentially 4-Uniform Permutations with the Best Known Nonlinearity from Butterflies**

*Shihui Fu and Xiutao Feng and Baofeng Wu*

**Abstract: **Many block ciphers use permutations defined over the finite field $\mathbb{F}_{2^{2k}}$ with low differential uniformity, high nonlinearity, and high algebraic degree to provide confusion. Due to the lack of knowledge about the existence of almost perfect nonlinear (APN) permutations over $\mathbb{F}_{2^{2k}}$, which have lowest possible differential uniformity, when $k>3$, constructions of differentially 4-uniform permutations are usually considered. However, it is also very difficult to construct such permutations together with high nonlinearity; there are very few known families of such functions, which can have the best known nonlinearity and a high algebraic degree. At Crypto'16, Perrin et al. introduced a structure named butterfly, which leads to permutations over $\mathbb{F}_{2^{2k}}$ with differential uniformity at most 4 and very high algebraic degree when $k$ is odd. It is posed as an open problem in Perrin et al.'s paper and solved by Canteaut et al. that the nonlinearity is equal to $2^{2k-1}-2^k$. In this paper, we extend Perrin et al.'s work and study the functions constructed from butterflies with exponent $e=2^i+1$. It turns out that these functions over $\mathbb{F}_{2^{2k}}$ with odd $k$ have differential uniformity at most 4 and algebraic degree $k+1$. Moreover, we prove that for any integer $i$ and odd $k$ such that $\gcd(i,k)=1$, the nonlinearity equality holds, which also gives another solution to the open problem proposed by Perrin et al. This greatly expands the list of differentially 4-uniform permutations with good nonlinearity and hence provides more candidates for the design of block ciphers.

**Category / Keywords: **foundations/S-boxes,APN,butterfly structure,permutation,differential uniformity,nonlinearity

**Original Publication**** (in the same form): **IACR-TOSC ISSUE 2-2017

**Date: **received 16 May 2017, last revised 22 May 2017

**Contact author: **fushihui at amss ac cn

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

**Version: **20170523:131100 (All versions of this report)

**Short URL: **ia.cr/2017/449

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