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On correlation-immune and stable Boolean functions. (Russian) Zbl 1280.94128
The weight \(\operatorname{wt}(f)\) of a Boolean function \(f(x)=f(x_1,\ldots,x_n)\) is the number of binary tuples \((\alpha_1,\ldots,\alpha_n)\) such that \(f(\alpha_1,\ldots,\alpha_n)=1\). The function \(f\) is called balanced if \(\operatorname{wt}(f)=\operatorname{wt}(f\oplus 1)=2^{n-1}\). A subfunction \(f'\) of \(f\) is a Boolean function obtained from \(f\) by substituting constants 0 or 1 for some variables of \(f\). The function \(f\) is called correlation-immune of order \(m\) \((1\le m\le n)\) if \(\operatorname{wt}(f')=\operatorname{wt}(f)/2^m\) for each of its subfunctions \(f'\) of \(n-m\) variables. A balanced correlation-immune function of order \(m\) is called \(m\)-stable. In other words, the function \(f\) is \(m\)-stable if \(\operatorname{wt}(f')=2^{n-m-1}\) for each of its subfunctions \(f'\) of \(n-m\) variables.
This paper reviews most of the author’s results obtained since 2001 concerning properties of \(m\)-stable correlation-immune functions.

94D10 Boolean functions
94C11 Switching theory, applications of Boolean algebras to circuits and networks
62H20 Measures of association (correlation, canonical correlation, etc.)