Paper 2017/449

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

Shihui Fu, 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.

Metadata
Available format(s)
PDF
Category
Foundations
Publication info
Published by the IACR in TOSC 2017 ISSUE 2
Keywords
S-boxesAPNbutterfly structurepermutationdifferential uniformitynonlinearity
Contact author(s)
fushihui @ amss ac cn
History
2017-05-23: received
Short URL
https://ia.cr/2017/449
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2017/449,
      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{https://eprint.iacr.org/2017/449}},
      url = {https://eprint.iacr.org/2017/449}
}
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