Open AccessDifferentially 4-Uniform Permutations with the Best Known Nonlinearity from Butterflies
Author(s) -
Shihui Fu,
XunLi Feng,
Bing-Fei Wu
Publication year - 2017
Publication title -
iacr transaction on symmetric cryptology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.715
H-Index - 10
ISSN - 2519-173X
DOI - 10.46586/tosc.v2017.i2.228-249
Subject(s) - degree (music) , mathematics , nonlinear system , algebraic number , integer (computer science) , exponent , differential (mechanical device) , finite field , block cipher , block (permutation group theory) , combinatorics , discrete mathematics , permutation (music) , pure mathematics , algorithm , mathematical analysis , cryptography , computer science , physics , quantum mechanics , programming language , linguistics , philosophy , acoustics , thermodynamics
Many block ciphers use permutations defined over the finite field F22k 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 F22k, 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 F22k 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 22k−1−2k. In this paper, we extend Perrin et al.’s work and study the functions constructed from butterflies with exponent e = 2i + 1. It turns out that these functions over F22k 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.