Finding periods of Zhegalkin polynomials
Diskretnaya Matematika, Tome 33 (2021) no. 3, pp. 107-120
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A period of a Boolean function $f(x_1, \ldots, x_n)$ is a binary $n$-tuple $a = (a_1, \ldots, a_n)$ that satisfies the identity $f(x_1+a_1, \ldots, x_n+a_n) = f(x_1, \ldots, x_n)$. A Boolean function is periodic if it admits a nonzero period. We propose an algorithm that takes the Zhegalkin polynomial of a Boolean function $f(x_1, \ldots, x_n)$ as the input and finds a basis of the space of all periods of $f(x_1, \ldots, x_n)$. The complexity of this algorithm is $n^{O(d)}$, where $d$ is the degree of the function $f$. As a corollary we show that a basis of the space of all periods of a Boolean function specified by the Zhegalkin polynomial of a bounded degree may be found with complexity which is polynomial in the number of variables.
Keywords:
Boolean function, Zhegalkin polynomial, periodicity, linear structure, complexity.
@article{DM_2021_33_3_a7,
author = {S. N. Selezneva},
title = {Finding periods of {Zhegalkin} polynomials},
journal = {Diskretnaya Matematika},
pages = {107--120},
publisher = {mathdoc},
volume = {33},
number = {3},
year = {2021},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/DM_2021_33_3_a7/}
}
S. N. Selezneva. Finding periods of Zhegalkin polynomials. Diskretnaya Matematika, Tome 33 (2021) no. 3, pp. 107-120. http://geodesic.mathdoc.fr/item/DM_2021_33_3_a7/