On the inheritance of properties of Boolean functions under restrictions
O. A. Logachev,
A. A. Sal'nikov,
V. V. Yashchenko
Abstract:
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.
UDC:
519.7 Received: 02.07.2001
Revised: 11.04.2002
DOI:
10.4213/dm237
Fulltext:
PDF file (851 kB)