Geodesic
On some invariants under the action of an extension of $GA(n,2)$ on the set of Boolean functions
Diskretnaya Matematika, Tome 33 (2021) no. 2, pp. 66-85.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $G$ be the extension of a general affine group by the group of affine functions. We study the action of $G$ on the set of Boolean functions. The action consists in nondegenerate affine transformations of variables and addition of affine Boolean functions. We introduce and examine some parameters of Boolean functions which are invariant with respect to the action of $G$. These are the amplitude (which is closely related to the nonlinearity), the dimension of a function, and some others. The invariants, together with some additionally proposed notions, could be used to obtain new bounds on cryptographic parameters of Boolean functions, including the maximum nonlinearity of functions in an odd number of variables.
Keywords: Boolean function, discrete Fourier (Walsh–Hadamard) transform, maximum nonlinearity, amplitude, dimension of a Boolean function, extension of general affine group. 
@article{DM_2021_33_2_a5,
     author = {O. A. Logachev and S. N. Fedorov and V. V. Yashchenko},
     title = {On some invariants under the action of an extension of $GA(n,2)$ on the set of {Boolean} functions},
     journal = {Diskretnaya Matematika},
     pages = {66--85},
     publisher = {mathdoc},
     volume = {33},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2021_33_2_a5/}
}
TY  - JOUR
AU  - O. A. Logachev
AU  - S. N. Fedorov
AU  - V. V. Yashchenko
TI  - On some invariants under the action of an extension of $GA(n,2)$ on the set of Boolean functions
JO  - Diskretnaya Matematika
PY  - 2021
SP  - 66
EP  - 85
VL  - 33
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DM_2021_33_2_a5/
LA  - ru
ID  - DM_2021_33_2_a5
ER  - 
%0 Journal Article
%A O. A. Logachev
%A S. N. Fedorov
%A V. V. Yashchenko
%T On some invariants under the action of an extension of $GA(n,2)$ on the set of Boolean functions
%J Diskretnaya Matematika
%D 2021
%P 66-85
%V 33
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DM_2021_33_2_a5/
%G ru
%F DM_2021_33_2_a5
O. A. Logachev; S. N. Fedorov; V. V. Yashchenko. On some invariants under the action of an extension of $GA(n,2)$ on the set of Boolean functions. Diskretnaya Matematika, Tome 33 (2021) no. 2, pp. 66-85. http://geodesic.mathdoc.fr/item/DM_2021_33_2_a5/

[1] Cheremushkin A. V., Dekompozitsiya i klassifikatsiya diskretnykh funktsii, KURS, Moskva, 2018, 288 pp.

[2] Logachev O. A., Salnikov A. A., Smyshlyaev S. V., Yaschenko V. V., Bulevy funktsii v teorii kodirovaniya i kriptologii, LENAND, Moskva, 2015, 576 pp.

[3] Logachev O. A., Fedorov S. N., Yashchenko V. V., “Boolean functions as points on the hypersphere in the Euclidean space”, Discrete Math. Appl., 29:2 (2019), 89–101 | DOI | MR | MR | Zbl

[4] Lechner R., “Harmonic analysis of switching functions”, Recent developments in switching theory, ed. A. Mukhopadhyay, Acad. Press, New York, 1971, 121–228 | DOI | MR

[5] Wu Ch.-K., Dawson E., “Construction of correlation immune Boolean functions”, Australasian J. Combinatorics, 21 (1997), 141–166 | MR

[6] Gopalan P., O'Donnell R., Servedio R. A., Shpilka A., Wimmer K., “Testing Fourier dimensionality and sparsity”, ICALP 2009, Lect. Notes Comput. Sci., 5555, 2009, 500–512 | DOI | MR | Zbl

[7] Carlet C. , Charpin P., “Cubic Boolean functions with highest resiliency”, IEEE Trans. Inf. Theory, 51:2 (2005), 562–571 | DOI | MR | Zbl

[8] Tarannikov Yu. V., “O znacheniyakh affinnogo ranga nositelya spektra platovidnykh funktsii”, Matematika i bezopasnost informatsionnykh tekhnologii (Mater. konf. v MGU 28–29 oktyabrya 2004 g.), MTsNMO, M., 2005, 226–231

[9] Logachev O. A., Sal'nikov A. A., Yashchenko V. V., “Nondegenerate normal forms of Boolean functions”, Dokl. Math., 62:1 (2000), 35–38 | MR | Zbl

[10] Yaschenko V. V., “O kriterii rasprostraneniya dlya bulevykh funktsii i o bent-funktsiyakh”, Problemy peredachi informatsii, 33:1 (1997), 75–86 | MR

[11] Kavut S., Yücel M. D., “Generalized rotation symmetric and dihedral symmetric Boolean functions – 9 variable Boolean functions with nonlinearity 242”, AAECC–17, Lect. Notes Comput. Sci., 4851, 2007, 321–329 | DOI | MR | Zbl

[12] Maitra S., Balanced Boolean function on 13-variables having nonlinearity strictly greater than the bent concatenation bound, IACR eprint archive, 2007 http://eprint.iacr.org/2007/309.pdf | MR

[13] Patterson N. J., Wiedemann D. H., “The covering radius of the $(2^{15}, 16)$ Reed–Muller code is at least 16276”, IEEE Trans. Inf. Theory, IT-29:3 (1983), 354–356 | DOI | MR | Zbl

[14] Kavut S., Boolean functions with excellent cryptographic properties in autocorrelation and Walsh spectra, Ph. D. thesis, Dept of Electrical and Electronics Engineering, Middle East Techn. Univ., 2008, 78 pp.