Geodesic
On symmetric boolean functions invariant under the Möbius transform
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 224 (2023), pp. 71-79
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The work is devoted to the study of the class of Boolean functions that are invariant under the Möbius transform. In the first part of the paper, we systematize general information on the Möbius transform and its fixed points. In the second part, we consider a class of symmetric Boolean functions that are invariant under the Möbius transform. The relationship of these functions with columns of the Sierpinski triangle is shown. We propose a method for obtaining masks of all such functions as sums of columns of the Sierpinski triangle. For the case $n=2^m-1$, we proved that a symmetric function is invariant if and only if its mask is invariant.
Mots-clés : algebraic normal form, Möbius transform
Keywords: coincident functions, symmetric Boolean functions, weight of a binary set. 
@article{INTO_2023_224_a8,
     author = {O. V. Zubkov},
     title = {On symmetric boolean functions invariant under the {M\"obius} transform},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {71--79},
     year = {2023},
     volume = {224},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2023_224_a8/}
}
TY  - JOUR
AU  - O. V. Zubkov
TI  - On symmetric boolean functions invariant under the Möbius transform
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2023
SP  - 71
EP  - 79
VL  - 224
UR  - http://geodesic.mathdoc.fr/item/INTO_2023_224_a8/
LA  - ru
ID  - INTO_2023_224_a8
ER  - 
%0 Journal Article
%A O. V. Zubkov
%T On symmetric boolean functions invariant under the Möbius transform
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2023
%P 71-79
%V 224
%U http://geodesic.mathdoc.fr/item/INTO_2023_224_a8/
%G ru
%F INTO_2023_224_a8
O. V. Zubkov. On symmetric boolean functions invariant under the Möbius transform. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 224 (2023), pp. 71-79. http://geodesic.mathdoc.fr/item/INTO_2023_224_a8/

[1] Bukhman A. V., “O raspoznavanii funktsii, invariantnykh otnositelno preobrazovaniya Mebiusa, i chetnykh funktsii, zadannykh v forme polinomov”, Prikladnaya matematika i informatika / Tr. f-ta VMK MGU im. M. V. Lomonosova, MAKS Press, M., 2012, 105–112

[2] Zubkov O. V., “Predstavlenie polinomialno ustoichivykh funktsii summami bespovtornykh v elementarnom bazise slagaemykh”, Mat. 6 Mezhdunar. shkoly-seminara «Sintaksis i semantika logicheskikh sistem» (Mongoliya, Khankh, 11-16 avgusta 2019 g.), Izd-vo IGU, Irkutsk, 2019, 48–52

[3] Zubkov O. V., “O klasse polinomialno ustoichivykh bulevykh funktsii”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obzory., 214 (2022), 37–43 | DOI

[4] Logachev O. A., Salnikov A. A., Yaschenko V. V., Bulevy funktsii v teorii kodirovaniya i kriptologii, MTsNMO, M., 2004

[5] Yablonskii S. V., Vvedenie v diskretnuyu matematiku, Vysshaya shkola, M., 2001

[6] MacWilliams F. J., Sloane N. J. A., The theory of Error-Correcting Codes, Noth-Holland, Amsterdam–New York–Oxford, 1978 | MR

[7] Pieprzyk J., Zhang X.-M., “Computing Möbius transform of boolean functions and characterising coincident boolean functions”, Boolean Functions: Cryptography and Applications, Publications des Univercites de Rouen et du Havre, Rouen, France, 2007, 135–151

[8] Sloane N. J. A., “Rows of Sierpinski's triangle”, http://oeis.org/A006943/b006943.txt