Possible values for the nonlinearity of de bruijn feedback functions

Ming Li; Yufan Liu; Yupeng Jiang; Xiaofang Xu

doi:10.1007/s10623-026-01823-z

Possible values for the nonlinearity of de bruijn feedback functions

Ming Li
, Yufan Liu
, Yupeng Jiang^*
, Xiaofang Xu

^*此作品的通讯作者

网络空间安全学院

CAS - Institute of Information Engineering
University of Chinese Academy of Sciences
Hubei Polytechnic University

科研成果: 期刊稿件 › 文章 › 同行评审

摘要

We study the nonlinearities of de Bruijn feedback functions and present two results regarding their possible values. First, we show that the maximal nonlinearity of de Bruijn feedback functions is nearly as large as that of bent functions. Specifically, for de Bruijn feedback functions of order n, their maximal nonlinearity is shown to be at least 2n-1-2n+12n, compared to 2n-1-2n2-1 for bent functions. Second, we show that the nonlinearities cover all numbers of the form 4r+2 between 2 and the maximum value.

源语言	英语
文章编号	66
期刊	Designs, Codes, and Cryptography
卷	94
期	3
DOI	https://doi.org/10.1007/s10623-026-01823-z
出版状态	已出版 - 3月 2026

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title = "Possible values for the nonlinearity of de bruijn feedback functions",

abstract = "We study the nonlinearities of de Bruijn feedback functions and present two results regarding their possible values. First, we show that the maximal nonlinearity of de Bruijn feedback functions is nearly as large as that of bent functions. Specifically, for de Bruijn feedback functions of order n, their maximal nonlinearity is shown to be at least 2n-1-2n+12n, compared to 2n-1-2n2-1 for bent functions. Second, we show that the nonlinearities cover all numbers of the form 4r+2 between 2 and the maximum value.",

keywords = "Bent function, Entropy function, Feedback shift register, Nonlinearity, de Bruijn sequence",

author = "Ming Li and Yufan Liu and Yupeng Jiang and Xiaofang Xu",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2026.",

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doi = "10.1007/s10623-026-01823-z",

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T1 - Possible values for the nonlinearity of de bruijn feedback functions

AU - Li, Ming

AU - Liu, Yufan

AU - Jiang, Yupeng

AU - Xu, Xiaofang

N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2026.

PY - 2026/3

Y1 - 2026/3

N2 - We study the nonlinearities of de Bruijn feedback functions and present two results regarding their possible values. First, we show that the maximal nonlinearity of de Bruijn feedback functions is nearly as large as that of bent functions. Specifically, for de Bruijn feedback functions of order n, their maximal nonlinearity is shown to be at least 2n-1-2n+12n, compared to 2n-1-2n2-1 for bent functions. Second, we show that the nonlinearities cover all numbers of the form 4r+2 between 2 and the maximum value.

AB - We study the nonlinearities of de Bruijn feedback functions and present two results regarding their possible values. First, we show that the maximal nonlinearity of de Bruijn feedback functions is nearly as large as that of bent functions. Specifically, for de Bruijn feedback functions of order n, their maximal nonlinearity is shown to be at least 2n-1-2n+12n, compared to 2n-1-2n2-1 for bent functions. Second, we show that the nonlinearities cover all numbers of the form 4r+2 between 2 and the maximum value.

KW - Bent function

KW - Entropy function

KW - Feedback shift register

KW - Nonlinearity

KW - de Bruijn sequence

UR - https://www.scopus.com/pages/publications/105033807635

U2 - 10.1007/s10623-026-01823-z

DO - 10.1007/s10623-026-01823-z

M3 - 文章

AN - SCOPUS:105033807635

SN - 0925-1022

VL - 94

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

IS - 3

M1 - 66

ER -

Possible values for the nonlinearity of de bruijn feedback functions

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