Paper 2017/449

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

Shihui Fu, Xiutao Feng, and Baofeng Wu


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.

Available format(s)
Publication info
Published by the IACR in TOSC 2017 ISSUE 2
S-boxesAPNbutterfly structurepermutationdifferential uniformitynonlinearity
Contact author(s)
fushihui @ amss ac cn
2017-05-23: received
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Creative Commons Attribution


      author = {Shihui Fu and Xiutao Feng and Baofeng Wu},
      title = {Differentially 4-Uniform Permutations with the Best Known Nonlinearity from Butterflies},
      howpublished = {Cryptology ePrint Archive, Paper 2017/449},
      year = {2017},
      note = {\url{}},
      url = {}
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