Perfectly Balanced Boolean Functions and Golić Conjecture

In the current paper we consider the following properties of filters: perfect balancedness of a filter function (i.e. preserving pure randomness of the input sequence) and linearity of a filter function in the first or the last essential variable. Previous results on this subject are discussed, incl...

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Veröffentlicht in:Journal of cryptology Jg. 25; H. 3; S. 464 - 483
1. Verfasser: Smyshlyaev, Stanislav V.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer-Verlag 01.07.2012
Springer Nature B.V
Schlagworte:
Boolean algebra
Boolean functions
Coding and Information Theory
Combinatorics
Communications Engineering
Computational Mathematics and Numerical Analysis
Computer Science
Linearity
Lower bounds
Networks
Probability Theory and Stochastic Processes
Keystream generator
Golić conjecture
Boolean function
Filter
Perfectly balanced function
ISSN:0933-2790, 1432-1378
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Zusammenfassung:In the current paper we consider the following properties of filters: perfect balancedness of a filter function (i.e. preserving pure randomness of the input sequence) and linearity of a filter function in the first or the last essential variable. Previous results on this subject are discussed, including misleading statements in Gouget and Sibert (LNCS, vol. 4876, 2007 ) about the connection between perfect balancedness and resistance to Anderson conditional correlation attack; the incorrectness of two known results, the sufficient condition of perfect balancedness in Golić (LNCS, vol. 1039, 1996 ) and the necessary condition of perfect balancedness in Dichtl (LNCS, vol. 1267, 1997 ), is demonstrated by providing counterexamples. We present a novel method of constructing large classes of perfectly balanced functions that are nonlinear in the first and the last essential variable and obtain a new lower bound of the number of such functions. Golić conjecture (LNCS, vol. 1039, 1996 ) states that the necessary and sufficient condition for a function to be perfectly balanced for any choice of a tapping sequence is linearity of a function in the first or the last essential variable. In the second part of the current paper we prove the Golić conjecture.
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ISSN:0933-2790
1432-1378
DOI:10.1007/s00145-011-9100-7