Geodesic
The class of balanced algebraic threshold functions
Prikladnaâ diskretnaâ matematika, no. 2 (2018), pp. 5-9.

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

The paper proposes an approach to the construction of a class of balanced algebraic threshold functions (ATF). The function $f$ of $k$-valued logic is called ATF if there are sequences $\mathbf c=(c_0,c_1,\dots,c_n)$, $\mathbf b=(b_0,b_1,\dots,b_k)$ of integers and the natural modulus $m$ such that $f(x_1,x_2,\dots,x_n)=\alpha\Leftrightarrow b_\alpha\leq(c_0+c_1x_1+c_2x_2+\dots+c_n x_n)\mod m$ for any $\alpha\in\Omega_k=\{0,1,\dots,k-1\}$. The triple $(\mathbf c;\mathbf b;m)$ is called the structure of the function $f$. The central result of the paper is a class of balanced ATF constructed in the following way: if an ATF $f$ has a structure $(\mathbf c,\mathbf b,m)=((c_0,c_1,c_2,\dots,c_n);(0,p,2p,\dots,kp);kp)$ where $c_i=pq$ and $(q,k)=1$, then this function is balanced. Such functions can be used as coordinate functions of substitutions.
Keywords: algebraic threshold functions, balanced functions. 
@article{PDM_2018_2_a0,
     author = {D. A. Soshin},
     title = {The class of balanced algebraic threshold functions},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {5--9},
     publisher = {mathdoc},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDM_2018_2_a0/}
}
TY  - JOUR
AU  - D. A. Soshin
TI  - The class of balanced algebraic threshold functions
JO  - Prikladnaâ diskretnaâ matematika
PY  - 2018
SP  - 5
EP  - 9
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/PDM_2018_2_a0/
LA  - ru
ID  - PDM_2018_2_a0
ER  - 
%0 Journal Article
%A D. A. Soshin
%T The class of balanced algebraic threshold functions
%J Prikladnaâ diskretnaâ matematika
%D 2018
%P 5-9
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/PDM_2018_2_a0/
%G ru
%F PDM_2018_2_a0
D. A. Soshin. The class of balanced algebraic threshold functions. Prikladnaâ diskretnaâ matematika, no. 2 (2018), pp. 5-9. http://geodesic.mathdoc.fr/item/PDM_2018_2_a0/

[1] Soshin D. A., “Constructing substitutions on the basis of threshold functions of multivalued logic”, Prikladnaya Diskretnaya Matematika, 2016, no. 2(32), 20–32 (in Russian) | DOI | MR

[2] Soshin D. A., “The implementation of Magma and 2-GOST block cipher substitutions by algebraic threshold functions”, Prikladnaya Diskretnaya Matematika, 2016, no. 3(33), 53–66 (in Russian) | DOI | MR

[3] Soshin D. A., “The constructive method for synthesis of balanced $k$-valued algebraic threshold functions”, Comp. Nanotechnol., 2015, no. 4, 31–36 (in Russian)

[4] Glukhov M. M., Elizarov V. P., Nechaev A. A., “Algebra”, Lan' Publ., St. Petersburg, 2015, 608 pp. (in Russian)

[5] Nikonov V. G., Nikonov N. V., “Peculiarities of threshold representations of $k$-valued functions”, Proc. Discr. Math., 11, no. 1, Fizmatlit Publ., Moscow, 2008, 60–85 (in Russian)

[6] Knuth D. E., The Art of Computer Programming, v. 2, Seminumerical Algorithms, 3rd Ed., Addison-Wesley, Massachusetts, 1998, 762 pp. | MR | MR | Zbl