### 4-Uniform Permutations with Null Nonlinearity

Christof Beierle and Gregor Leander

##### Abstract

We consider $n$-bit permutations with differential uniformity of 4 and null nonlinearity. We first show that the inverses of Gold functions have the interesting property that one component can be replaced by a linear function such that it still remains a permutation. This directly yields a construction of 4-uniform permutations with trivial nonlinearity in odd dimension. We further show their existence for all $n = 3$ and $n \geq 5$ based on a construction in [1]. In this context, we also show that 4-uniform 2-1 functions obtained from admissible sequences, as defined by Idrisova in [8], exist in every dimension $n = 3$ and $n \geq 5$. Such functions fulfill some necessary properties for being subfunctions of APN permutations. Finally, we use the 4-uniform permutations with null nonlinearity to construct some 4-uniform 2-1 functions from $\mathbb{F}_2^n$ to $\mathbb{F}_2^{n-1}$ which are not obtained from admissible sequences. This disproves a conjecture raised by Idrisova.

Available format(s)
Category
Foundations
Publication info
Published elsewhere. Cryptography and Communications
DOI
10.1007/s12095-020-00434-2
Keywords
Boolean functionCryptographic S-boxesAPN permutationsGold functions
Contact author(s)
christof beierle @ rub de
gregor leander @ rub de
History
2020-04-20: revised
See all versions
Short URL
https://ia.cr/2020/334

CC BY

BibTeX

@misc{cryptoeprint:2020/334,
author = {Christof Beierle and Gregor Leander},
title = {4-Uniform Permutations with Null Nonlinearity},
howpublished = {Cryptology ePrint Archive, Paper 2020/334},
year = {2020},
doi = {10.1007/s12095-020-00434-2},
note = {\url{https://eprint.iacr.org/2020/334}},
url = {https://eprint.iacr.org/2020/334}
}

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