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

Yu ZHOU  Lin WANG  Weiqiong WANG  Xiaoni DU  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E98-A   No.6   pp.1313-1319
Publication Date: 2015/06/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E98.A.1313
Type of Manuscript: LETTER
Category: Cryptography and Information Security
stream cipher,  Boolean functions,  global avalanche characteristics,  algebraic immunity,  balanced,  

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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 σƒ=22n+2n+3 is an open problem. In this paper, we prove that there does not exist a balanced Boolean function with σƒ=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 7·2n-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.