Geodesic
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. 
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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/

[1] Logachev O. A., Salnikov A. A., Smyshlyaev S. V., Yaschenko V. V., Bulevy funktsii v teorii kodirovaniya i kriptologii, MTsNMO, M., 2012, 584 pp.

[2] Selezneva S. N., “O slozhnosti raspoznavaniya polnoty mnozhestv bulevykh funktsii, realizovannykh polinomami Zhegalkina”, Diskretnaya matematika, 9:4 (1997), 24–31 | DOI | MR | Zbl

[3] Selezneva S. N., Bukhman A. V., “Polynomial-time algorithms for checking some properties of Boolean functions given by polynomials”, Theory Computer Systems, 58:3 (2016), 383–391 | DOI | MR | Zbl

[4] Dawson E., Wu C.-K., “On the linear structure of symmetric Boolean functions”, Australasian J. Combinatorics, 16 (1997), 239–243 | Zbl

[5] Leontev V. K., “O nekotorykh zadachakh, svyazannykh s bulevymi polinomami”, Zhurnal vychisl. matem. i matem. fiziki, 39:6 (1999), 1045–1054 | MR | Zbl

[6] Charpin P., Kyureghyan G. M., “On a class of permutation polynomials over $F_{2^n}$”, Lect. Notes Comput. Sci., 5203 (2008), 368–376 | DOI | MR | Zbl

[7] Charpin P., Sarkar S., “Polynomials with linear structure and Maiorana-McFarland construction”, IEEE Trans. Inf. Theory., 57:6 (2011), 3796–3804 | DOI | MR | Zbl

[8] Bukhman A. V., “O svoistvakh polinomov periodicheskikh funktsii i slozhnosti raspoznavaniya periodichnosti po polinomu bulevoi funktsii”, Diskretnaya matematika, 26:1 (2014), 21–31 | MR | Zbl

[9] Yang L., Li H.-W., “Investigating the linear structure of Boolean functions based on Simon's period-finding quantum algorithm”, arXiv: https://arxiv.org/pdf/1306.2008.pdf

[10] Lidl R., Niderraiter G., Konechnye polya, V 2-kh t., v. 1,2, Mir, M., 1988, 822 pp. ; Lidl R., Niederreiter H., Finite Fields, Addison-Wesley Publ. Inc., 1983, 755 pp. | MR | MR | Zbl

[11] Yablonskii S. V., Vvedenie v diskretnuyu matematiku, Vysshaya shkola, M., 2001, 384 pp.

[12] Geri M., Dzhonson D., Vychislitelnye mashiny i trudnoreshaemye zadachi, Mir, M., 1982, 416 pp.; Garye M. R., Johnson D. S., Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman Co., San Francisco, 1979 | MR