Two classes of differentially 4-uniform permutations over $ \mathbb{F}_{2^{n}} $ with $ n $ even

Guangkui Xu; Longjiang Qu

doi:10.3934/amc.2020008

Article Contents

2020, Volume 14, Issue 1: 97-110. Doi: 10.3934/amc.2020008

This issue Previous Article New doubly even self-dual codes having minimum weight 20 Next Article A note on the fast algebraic immunity and its consequences on modified majority functions

Two classes of differentially 4-uniform permutations over $ \mathbb{F}_{2^{n}} $ with $ n $ even

Guangkui Xu^1,2, and
Longjiang Qu^1, ,

1.
College of Liberal Arts and Science, National University of Defense Technology, Changsha 410073, China
2.
Department of Applied Mathematics, Huainan Normal University, Huainan 232038, China

^* Corresponding author: Longjiang Qu
^* Corresponding author: Longjiang Qu

Received: October 2018

Revised: January 2019

Published: August 2019

This research is supported by the National Natural Science Foundation of China (11601177, 61722213, 11771007 and 11701336), Anhui Provincial Natural Science Foundation (1608085QA05) and the Foundation for Distinguished Young Talents in Higher Education of Anhui Province of China (gxyqZD2016258)

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Abstract

A construction of differentially 4-uniform permutations by modifying the values of the inverse function on a union of some cosets of a multiplication subgroup of $ \mathbb{F}_{2^n}^* $ was given by Peng et al. in [15]. In this paper, we extend their results to differentially 4-uniform permutations whose values are different from the values of the inverse function on some subsets of the unit circle of $ \mathbb{F}_{2^n} $ or on the multiplication group of some subfield of $ \mathbb{F}_{2^n} $. Moreover, it has been checked by the Magma software that some permutations in the resulted differentially 4-uniform permutations are CCZ-inequivalent to the known functions for small $ n $.

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Mathematics Subject Classification: Primary: 94A60, 11T71; Secondary: 06E30.

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Table 1. Differential spectrum and nonlinearity of functions in Theorem 3.2 over $ \mathbb{F}_{2^{10}} $

$ U $	Differential spectrum	Nonlinearity	Ref.
$ U_0 $	[526419, 518490, 2643]	478	[15]
$ U_1 $	[528021, 515286, 4245]	476	This paper
$ U_2 $	[529605, 512118, 5829]	474	This paper
$ U_3 $	[531171, 508986, 7395]	474	This paper
$ U_4 $	[532719, 505890, 8943]	472	This paper
$ U_5 $	[534249, 502830, 10473]	472	This paper
$ U_6 $	[535761, 499806, 11985]	470	This paper
$ U_7 $	[537255, 496818, 13479]	468	This paper
$ U_8 $	[538731, 493866, 14955]	466	This paper
$ U_9 $	[540189, 490950, 16413]	464	This paper
$ \mathcal{U} $	[541629, 488070, 17853]	462	This paper

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Table 2. Differential spectrum and nonlinearity of functions in Section 4 over $ \mathbb{F}_{2^{n}} $

$ n $	$ m $	$ f $	$ \gamma $	Differential spectrum	Nonlinearity
12	4	Theorem 4.4	$ \xi^{273} $	[8420895, 8317890, 34335]	1978
12	4	Theorem 4.4	$ \xi^{819} $	[8422155, 8315370, 35595]	1978
12	4	Theorem 4.4	$ \omega $	[8421255, 8317170, 34695]	1980
12	4	Theorem 4.4	$ \xi^{1638} $	[8422155, 8315370, 35595]	1980
12	4	Theorem 4.4	$ \xi^{1911} $	[8420895, 8317890, 34335]	1980
6	3	Theorem 4.8	$ \eta^{9} $	[2226, 1596,210]	22
8	4	Theorem 4.8	$ \omega $	[34515, 28890, 1875]	108

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