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Qu, Longjiang; Xiong, Hai; Li, Chao

doi:10.1016/j.ffa.2013.05.007

Cited by 3 publications

(7 citation statements)

References 7 publications

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“…It is conjectured that this table is complete [8], up to equivalence and including the compositional inverse of the functions. For multinomial functions satisfying the same properties see [9,19,33]. Such functions are rare and generally quadratic or derived from APN functions.…”

Section: Remarkmentioning

confidence: 99%

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New links between nonlinearity and differential uniformity

Charpin

Peng

2019

Finite Fields and Their Applications

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In this paper some new links between the nonlinearity and differential uniformity of some large classes of functions are established. Differentially two-valued functions and quadratic functions are mainly treated. A lower bound for the nonlinearity of monomial δ-uniform permutations is obtained, for any δ, as well as an upper bound for differentially two-valued functions. Concerning quadratic functions, significant relations between nonlinearity and differential uniformity are exhibited. In particular, we show that the quadratic differentially 4-uniform permutations should be differentially two-valued and possess the best known nonlinearity.

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Section: Remarkmentioning

confidence: 99%

“…It was proved later in [33] that for t = 2, the function F is not always bijective and at the same time that δ = 4. Moreover the authors suggest that the proofs given in [6,11], for the APN case (t = 1), can be slightly modified to solve Problem 5.1.…”

mentioning

confidence: 99%

New links between nonlinearity and differential uniformity

Charpin

Peng

2019

Finite Fields and Their Applications

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“…Recently, there has been significant progress in finding functions with low differential uniformity [5,8,10,13,16,17,20,24]. In [3,4] Bracken et al firstly studied highly nonlinear monomials and binomials which are differentially 4-uniform permutations over F 2 2m .…”

Section: Introductionmentioning

confidence: 99%

Further results on differentially 4-uniform permutations over $\mathbb{F}_{2^{2m} } $

Zha

Sun

et al. 2015

Sci. China Math.

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“…The reasons are as follows. On the one hand, thanks to Theorem 3, it can be shown that there are 2 22 preferred Boolean functions on F 2 6 . Hence by Corollary 1 we may construct 2 22 differentially 4-uniform permutations on F 2 6 .…”

Section: Introductionmentioning

confidence: 97%

“…However, due to the lack of knowledge of the APN permutation in even dimension, constructions of differentially 4uniform permutations with high algebraic degree and high nonlinearity over F 2 2k have attracted many researchers' interest. One may refer to [3,4,8,9,[19][20][21][22][23]28] for recent progress on this problem. To the best of our knowledge, before [21], not many infinite families of differentially 4-uniform permutations are known.…”

Section: Introductionmentioning

confidence: 99%

More constructions of differentially 4-uniform permutations on $${\mathbb {F}}_{2^{2k}}$$ F 2 2 k

Tan

et al. 2014

Des. Codes Cryptogr.

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Differentially 4-uniform permutations on F 2 2k with high nonlinearity are often chosen as Substitution boxes in both block and stream ciphers. Recently, Qu et al. introduced a class of functions, which are called preferred functions, to construct a lot of infinite families of such permutations [21]. In this paper, we propose a particular type of Boolean functions to characterize the preferred functions. On the one hand, such Boolean functions can be determined by solving linear equations, and they give rise to a huge number of differentially 4-uniform permutations over F 2 2k . Hence they may provide more choices for the design of Substitution boxes. On the other hand, by investigating the number of these Boolean functions, we show that the number of CCZinequivalent differentially 4-uniform permutations over F 2 2k grows exponentially when k increases, which gives a positive answer to an open problem proposed in [21].

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A negative answer to Bracken–Tan–Tan's problem on differentially 4-uniform permutations overF2n

Cited by 3 publications

References 7 publications

New links between nonlinearity and differential uniformity

New links between nonlinearity and differential uniformity

Further results on differentially 4-uniform permutations over $\mathbb{F}_{2^{2m} } $

More constructions of differentially 4-uniform permutations on $${\mathbb {F}}_{2^{2k}}$$ F 2 2 k

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