Geodesic
On the number of perfectly balanced Boolean functions with barrier of length~$3$
Prikladnaâ diskretnaâ matematika, no. 1 (2011), pp. 26-33.

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

Some lower and upper bounds are obtained for the logarithm of the number of Boolean functions with the right barrier of length $3$ essentially depended on the last variable. Also, the following new lower bound for the logarithm of the number of perfectly balanced Boolean functions of $n$ variables with nonlinear dependence on the first and on the last variable is obtained: $2^{n-2}\left(1+\dfrac{\log_25}4-\mathrm O(1/\sqrt n)\right)$.
Keywords: perfectly balanced functions, barriers of Boolean functions, cryptography. 
@article{PDM_2011_1_a2,
     author = {S. V. Smyshlyaev},
     title = {On the number of perfectly balanced {Boolean} functions with barrier of length~$3$},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {26--33},
     publisher = {mathdoc},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2011_1_a2/}
}
TY  - JOUR
AU  - S. V. Smyshlyaev
TI  - On the number of perfectly balanced Boolean functions with barrier of length~$3$
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2011
SP  - 26
EP  - 33
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2011_1_a2/
LA  - ru
ID  - PDM_2011_1_a2
ER  - 
%0 Journal Article
%A S. V. Smyshlyaev
%T On the number of perfectly balanced Boolean functions with barrier of length~$3$
%J Prikladnaâ diskretnaâ matematika
%D 2011
%P 26-33
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2011_1_a2/
%G ru
%F PDM_2011_1_a2
S. V. Smyshlyaev. On the number of perfectly balanced Boolean functions with barrier of length~$3$. Prikladnaâ diskretnaâ matematika, no. 1 (2011), pp. 26-33. http://geodesic.mathdoc.fr/item/PDM_2011_1_a2/

[1] Hedlund G. A., “Endomorphisms and automorphisms of the shift dynamical system”, Math. Sys. Theory, 3:4 (1969), 320–375 | DOI | MR | Zbl

[2] Sumarokov S. N., “Zaprety dvoichnykh funktsii i obratimost dlya odnogo klassa kodiruyuschikh ustroistv”, Obozrenie prikladnoi i promyshlennoi matematiki, 1:1 (1994), 33–55 | MR | Zbl

[3] Anderson R. J., “Searching for the Optimum Correlation Attack”, LNCS, 1008, 1995, 137–143 | Zbl

[4] Golic Dj. J., “On the Security of Nonlinear Filter Generators”, LNCS, 1039, 1996, 173–188

[5] Smyshlyaev S. V., “O nekotorykh svoistvakh sovershenno uravnoveshennykh bulevykh funktsii”, Materialy Chetvertoi Mezhdunar. nauch. konf. po problemam bezopasnosti i protivodeistviya terrorizmu (MGU im. M. V. Lomonosova, Moskva, 30–31 oktyabrya 2008), MTsNMO, M., 2009, 57–64

[6] Logachev O. A., Smyshlyaev S. V., Yaschenko V. V., “Novye metody izucheniya sovershenno uravnoveshennykh bulevykh funktsii”, Diskretnaya matematika, 21:2 (2009), 51–74 | DOI | MR

[7] Logachev O. A., “Ob odnom klasse sovershenno uravnoveshennykh bulevykh funktsii”, Materialy Tretei Mezhdunar. nauch. konf. po problemam bezopasnosti i protivodeistviya terrorizmu (MGU im. M. V. Lomonosova, Moskva, 25–27 oktyabrya 2007), MTsNMO, M., 2008, 137–141

[8] Smyshlyaev S. V., “Barery sovershenno uravnoveshennykh bulevykh funktsii”, Diskretnaya matematika, 22:2 (2010), 66–79 | DOI | MR | Zbl

[9] Smyshlyaev S. V., “O preobrazovanii dvoichnykh posledovatelnostei s pomoschyu sovershenno uravnoveshennykh bulevykh funktsii”, Materialy Pyatoi Mezhdunar. nauch. konf. po problemam bezopasnosti i protivodeistviya terrorizmu (MGU im. M. V. Lomonosova, Moskva, 29–30 oktyabrya 2009), MTsNMO, M., 2010, 31–41

[10] Smyshlyaev S. V., “O kriptograficheskikh slabostyakh nekotorykh klassov preobrazovanii dvoichnykh posledovatelnostei”, Prikladnaya diskretnaya matematika, 2010, no. 1(7), 5–15

[11] Smyshlyaev S. V., “O sovershenno uravnoveshennykh bulevykh funktsiyakh bez barera”, Materialy Vosmoi Mezhdunar. nauch. konf. “Diskretnye modeli v teorii upravlyayuschikh sistem” (MGU im. M. V. Lomonosova, Moskva, 6–9 aprelya 2009), MAKS Press, M., 2009, 278–284

[12] Smyshlyaev S. V., “Postroenie klassov sovershenno uravnoveshennykh bulevykh funktsii bez barera”, Prikladnaya diskretnaya matematika, 2010, no. 3(9), 41–50

[13] Kholl M., Kombinatorika, Mir, M., 1970 | MR

[14] Lichiardopol N., “Independence number of de Bruijn graphs”, Dicrete Mathematics, 306:12 (2006), 1145–1160 | DOI | MR | Zbl