Article Contents

2023, Volume 17, Issue 4: 757-770. Doi: 10.3934/amc.2021020

This issue Previous Article Next Article Designing tweakable enciphering schemes using public permutations

A new construction of weightwise perfectly balanced Boolean functions

Rui Zhang^1, and
Sihong Su^1,2, ,

1.
School of Mathematics and Statistics, Henan University, Kaifeng, 475004, China
2.
Henan Engineering Research Center for Artificial Intelligence Theory and Algorithms, Henan University, Kaifeng, 475004, China

^* Corresponding author: Sihong Su (E-mail: sush@henu.edu.cn)
^* Corresponding author: Sihong Su (E-mail: sush@henu.edu.cn)

Received: January 2021

Early access: June 2021

Published: August 2023

The second author is supported by the Key Scientific Research Project of Colleges and Universities in Henan Province (Grant No. 21A413003) and the National Natural Science Foundation of China (Grant No. 61502147)

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Abstract

In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on $ 2^m $ variables are given by modifying the support of the quartic functions, where $ m $ is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Table 1. The $ k $-weight nonlinearity of $ g_3 $ in (11) for $ k = 2,3,4,5,6 $

k	$ k $-weight nonlinearity of $ g_3 $ in (11)	$ \lfloor {8\choose k}/{2}-\sqrt{8\choose k}/{2}\rfloor $
2	$ \mathrm{NL}_{2}(g_{3})=2 $	11
3	$ \mathrm{NL}_{3}(g_3)=12 $	24
4	$ \mathrm{NL}_{4}(g_3)=19 $	30
5	$ \mathrm{NL}_{5}(g_3)=12 $	24
6	$ \mathrm{NL}_{6}(g_3)=6 $	11

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Table 2. The algebraic immunity of $ g_{m} $ for $ m = 2,3,4 $

$ m $	algebraic immunity of $ g_{m} $ in (11)	optimal algebraic immunity
2	$ AI(g_2)=2 $	2
3	$ AI(g_3)=3 $	4
4	$ AI(g_4)=3 $	8

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