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Prikl. Diskr. Mat. Suppl., 2018, Issue 11, Pages 6–9 (Mi pdma384)  

Theoretical Foundations of Applied Discrete Mathematics

On mixing digraphs of nonlinear substitutions for binary shift registers

V. S. Grigorievab

a Financial University under the Government of the Russian Federation, Moscow
b "Positive Technologies", Moscow

Abstract: In this paper, we research the class $R(n,m)$ of substitutions on $n$-dimensional vector space produced by the binary left-shift registers of the length $n$ with one feedback $f(x_1,…,x_n)=x_1\oplus\psi(x_2,…,x_n)$ essentially depending on $m$ variables, $3\le m\le n$. We have obtained the following double-ended estimate for the exponent of the mixing digraphs $\Gamma(g)$ for nonlinear substitutions $g\in R(n,m)$:
$$ n+\lceil\frac{n-1}{m-1}\rceil-1\le\exp{\Gamma(g)}\le\Delta(D)+n+\lfloor\frac{(n-2)^2}2\rfloor-1, $$
where $D(g)=\{i_1,…,i_m\}$ is the set of indexes of essential variables of the shift register feedback function $f$, $1=i_1<…<i_m\le n$, $m\le n$; $\Delta(D)=\max\{i_2-i_1,…,i_m-i_{m-1},n-i_m\}$. We have also obtained some upper-bound estimates for the sum and for the ratio of exponents of mixing digraphs of substitution $g\in R(n,m)$ and its inverse substitution $g^{-1}$:
\begin{gather*} \exp{\Gamma(g)}+\exp{\Gamma(g^{-1})}\le2(\Delta(D)+\lfloor\frac{n^2}m\rfloor)+i_m,
\frac{\exp{\Gamma(g)}}{\exp{\Gamma(g^{-1})}}\le\frac{\Delta(D)+n+\lfloor\frac{(n-2)^2}2\rfloor-1}{n+\lceil\frac{n-1}{m-1}\rceil-1}. \end{gather*}


Keywords: mixing digraph approach, primitive digraph, exponent of graph, shift register, Frobenius number.

DOI: https://doi.org/10.17223/2226308X/11/1

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UDC: 519.1

Citation: V. S. Grigoriev, “On mixing digraphs of nonlinear substitutions for binary shift registers”, Prikl. Diskr. Mat. Suppl., 2018, no. 11, 6–9

Citation in format AMSBIB
\Bibitem{Gri18}
\by V.~S.~Grigoriev
\paper On mixing digraphs of nonlinear substitutions for binary shift registers
\jour Prikl. Diskr. Mat. Suppl.
\yr 2018
\issue 11
\pages 6--9
\mathnet{http://mi.mathnet.ru/pdma384}
\crossref{https://doi.org/10.17223/2226308X/11/1}


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