Balanced Boolean Functions of σƒ>22n+2n+3(n≥4)

Yu ZHOU; Lin WANG; Weiqiong WANG; Xiaoni DU

doi:10.1587/transfun.E98.A.1313

Balanced Boolean Functions of σ_ƒ>2²ⁿ+2ⁿ⁺³(n≥4)

Yu ZHOU, Lin WANG, Weiqiong WANG, Xiaoni DU

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The global avalanche characteristics measure the overall avalanche properties of Boolean functions, an n-variable balanced Boolean function of the sum-of-square indicator reaching σ_ƒ=2²ⁿ+2ⁿ⁺³ is an open problem. In this paper, we prove that there does not exist a balanced Boolean function with σ_ƒ=2²ⁿ+2ⁿ⁺³ for n≥4, if the hamming weight of one decomposition function belongs to the interval Q^*. Some upper bounds on the order of propagation criterion of balanced Boolean functions with n (3≤n≤100) variables are given, if the number of vectors of propagation criterion is equal and less than 7·2^n-3-1. Two lower bounds on the sum-of-square indicator for balanced Boolean functions with optimal autocorrelation distribution are obtained. Furthermore, the relationship between the sum-of-squares indicator and nonlinearity of balanced Boolean functions is deduced, the new nonlinearity improves the previously known nonlinearity.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E98-A No.6 pp.1313-1319

Publication Date: 2015/06/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.E98.A.1313

Type of Manuscript: LETTER

Category: Cryptography and Information Security

Authors

Yu ZHOU
  Science and Technology on Communication Security Laboratory
Lin WANG
  Science and Technology on Communication Security Laboratory
Weiqiong WANG
  Chang'an University
Xiaoni DU
  Northwest Normal University

Keyword

stream cipher, Boolean functions, global avalanche characteristics, algebraic immunity, balanced

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Yu ZHOU, Lin WANG, Weiqiong WANG, Xiaoni DU, "Balanced Boolean Functions of σƒ>22n+2n+3(n≥4)" in IEICE TRANSACTIONS on Fundamentals, vol. E98-A, no. 6, pp. 1313-1319, June 2015, doi: 10.1587/transfun.E98.A.1313.
Abstract: The global avalanche characteristics measure the overall avalanche properties of Boolean functions, an n-variable balanced Boolean function of the sum-of-square indicator reaching σ_ƒ=2²ⁿ+2ⁿ⁺³ is an open problem. In this paper, we prove that there does not exist a balanced Boolean function with σ_ƒ=2²ⁿ+2ⁿ⁺³ for n≥4, if the hamming weight of one decomposition function belongs to the interval Q^*. Some upper bounds on the order of propagation criterion of balanced Boolean functions with n (3≤n≤100) variables are given, if the number of vectors of propagation criterion is equal and less than 7·2^n-3-1. Two lower bounds on the sum-of-square indicator for balanced Boolean functions with optimal autocorrelation distribution are obtained. Furthermore, the relationship between the sum-of-squares indicator and nonlinearity of balanced Boolean functions is deduced, the new nonlinearity improves the previously known nonlinearity.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1587/transfun.E98.A.1313/_p

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@ARTICLE{e98-a_6_1313,
author={Yu ZHOU, Lin WANG, Weiqiong WANG, Xiaoni DU, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Balanced Boolean Functions of σƒ>22n+2n+3(n≥4)},
year={2015},
volume={E98-A},
number={6},
pages={1313-1319},
abstract={The global avalanche characteristics measure the overall avalanche properties of Boolean functions, an n-variable balanced Boolean function of the sum-of-square indicator reaching σ_ƒ=2²ⁿ+2ⁿ⁺³ is an open problem. In this paper, we prove that there does not exist a balanced Boolean function with σ_ƒ=2²ⁿ+2ⁿ⁺³ for n≥4, if the hamming weight of one decomposition function belongs to the interval Q^*. Some upper bounds on the order of propagation criterion of balanced Boolean functions with n (3≤n≤100) variables are given, if the number of vectors of propagation criterion is equal and less than 7·2^n-3-1. Two lower bounds on the sum-of-square indicator for balanced Boolean functions with optimal autocorrelation distribution are obtained. Furthermore, the relationship between the sum-of-squares indicator and nonlinearity of balanced Boolean functions is deduced, the new nonlinearity improves the previously known nonlinearity.},
keywords={},
doi={10.1587/transfun.E98.A.1313},
ISSN={1745-1337},
month={June},}

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TY - JOUR
TI - Balanced Boolean Functions of σƒ>22n+2n+3(n≥4)
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1313
EP - 1319
AU - Yu ZHOU
AU - Lin WANG
AU - Weiqiong WANG
AU - Xiaoni DU
PY - 2015
DO - 10.1587/transfun.E98.A.1313
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E98-A
IS - 6
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - June 2015
AB - The global avalanche characteristics measure the overall avalanche properties of Boolean functions, an n-variable balanced Boolean function of the sum-of-square indicator reaching σ_ƒ=2²ⁿ+2ⁿ⁺³ is an open problem. In this paper, we prove that there does not exist a balanced Boolean function with σ_ƒ=2²ⁿ+2ⁿ⁺³ for n≥4, if the hamming weight of one decomposition function belongs to the interval Q^*. Some upper bounds on the order of propagation criterion of balanced Boolean functions with n (3≤n≤100) variables are given, if the number of vectors of propagation criterion is equal and less than 7·2^n-3-1. Two lower bounds on the sum-of-square indicator for balanced Boolean functions with optimal autocorrelation distribution are obtained. Furthermore, the relationship between the sum-of-squares indicator and nonlinearity of balanced Boolean functions is deduced, the new nonlinearity improves the previously known nonlinearity.
ER -