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

^*Corresponding author for this work

School of Cyber Science and Technology

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

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Article number	66
Journal	Designs, Codes, and Cryptography
Volume	94
Issue number	3
DOIs	https://doi.org/10.1007/s10623-026-01823-z
State	Published - Mar 2026

Keywords

Bent function
Entropy function
Feedback shift register
Nonlinearity
de Bruijn sequence

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10.1007/s10623-026-01823-z

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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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Keywords

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