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Propagation Characteristics of Boolean Functions and Their Balancedness
Shouichi HIROSE Katsuo IKEDA
Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Vol.E78A
No.1
pp.1118 Publication Date: 1995/01/25 Online ISSN:
DOI: Print ISSN: 09168508 Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security) Category: Keyword: Boolean functions, nonlinearity criteria, propagation criterion, balancedness, symmetric cryptosystems,
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Summary:
This paper discusses Boolean functions satisfying the propagation criterion (PC) and their balancedness. Firstly, we discuss Boolean functions with n variables that satisfy the PC with respect to all but three elements in {0,1}^{n}{(0,...,0)}. For even n4, a necessary and sufficient condition is presented for Boolean functions with n variables to satisfy the PC with respect to all but three elements in {0,1}^{n}{(0,...,0)}. From this condition, it is proved that all of these Boolean functions are constructed from all perfectly nonlinear Boolean functions with n2 variables. For odd n3, it is shown that Boolean functions with n variables satisfying the PC with respect to all but three elements in {0,1}^{n}{(0,...,0)} satisfy the PC with respect to all but one elements in it. Secondly, Boolean functions satisfying the PC of degree n2 and their balancedness are considered. For even n4, it is proved that an upper bound on the degree of the PC is n3 for balanced Boolean functions with n variables. This bound is optimal for n=4,6. It is also proved that, for odd n3, balanced Boolean functions with n variables satisfying the PC of degree n2 satisfy the PC with respect to all but one elements in {0,1}^{n}{(0,...,0)}.

