On the inheritance of properties of Boolean functions under restrictions

O. A. Logachev ; A. A. Sal'nikov ; V. V. Yashchenko

Diskretnaya Matematika, Tome 14 (2002) no. 2, pp. 9-19

Résumé

For a property $\mathcal P$ of Boolean functions, a Boolean function $f(x)$, $x\in V_n$, and a subspace $H$ of the space $V_n$ of all $n$-tuples of zeros and ones, we consider the set of all restrictions of the Boolean function $f(x)$ onto the cosets of $V_n$ with respect to $H$. If the function $f(x)$ itself and all its $2^{n-\dim H}$ restrictions possess the property $\mathcal P$, we say that the property $\mathcal P$ is inherited under the restrictions of the Boolean function $f(x)$ and consider it as a new derived property. In this paper, this approach is applied to the following property of Boolean functions: the value $\hat f(\alpha)/2^n$, where $\hat f(\alpha)$ is the Walsh–Hadamard coefficient, is fixed; the corresponding derived property is called the $(H,\alpha)$-stability. We give convenient criteria for $(H,\alpha)$-stability in terms of zeros of the Walsh–Hadamard coefficients, and establish relations between the $(H,\alpha)$-stability, correlation immunity, and $m$-resiliency. This research was supported by the Russian Foundation for Basic Research, grants 99–01–00929 and 99–01–00941.