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Prikl. Diskr. Mat., 2018, Number 42, Pages 6–17 (Mi pdm639)  

This article is cited in 1 scientific paper (total in 1 paper)

Theoretical Backgrounds of Applied Discrete Mathematics

On estimations of distribution of the length of aperiodicity segment in the graph of $k$-fold iteration of uniform random mapping

V. O. Mironkin

National Research University Higher School of Economics, Moscow, Russia

Abstract: Given $k,n\in\mathbb{N}$, $x_0\in S=\{1,\ldots,n\}$, and $ f:S\to S$, define $x_{i+1}=f^k(x_i)$ for every $i\in\{0,1,\ldots\}$ and $\tau_{f^k}(x_0)$ as the least integer $i$ such that $f^k(x_i)=x_j$ for some $j$, $j<i$. For the local probability $\mathsf{P}\{\tau_{f^k}(x_0)=z \}$ and for the distribution function $F_{\tau_{f^k}(x_0)}( z )$, the following estimates are obtained. If $kz<n$, then
\begin{gather}\notag \mathsf{P}\{\tau_{f^k}(x_0){=}z \}>\frac 1n{\textstyle\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}=z
\end{smallmatrix}}}{{e^{-( 1+\frac{m}{n} )\frac{{{m}^{2}}}{2n}}}}\;{+}{\textstyle\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}<z
\end{smallmatrix}}}{\frac1{r+k}e^{-( 1+\frac{r}{n} )\frac{r^2}{2n}}( 1{-}{( 1{-}\frac{r+k}{n} )}^k )},
\notag \mathsf{P}\{\tau_{f^k}(x_0)=z \}<\frac1n{\textstyle\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}=z
\end{smallmatrix}}}{e^{-\frac{{( m-1 )}^2}{2n}}}+{\textstyle\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}<z
\end{smallmatrix}}}{\frac{1}{r}{e^{-\frac{{{( r-1 )}^{2}}}{2n}}}( 1-{{( 1-\frac{r}{n} )}^k} )}, \end{gather}
where $r=m+( z-\dfrac{m}{(m,k)}-1 )k$. If $k^2z\leq n$, then
\begin{equation}\notag \begin{gathered} \frac1n\textstyle\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}=z
\end{smallmatrix}}{e^{-( 1+\frac{m}{n} )\frac{m^2}{2n}}}+( 1-\dfrac{k^2z}{2n} )\dfrac{k}{n}\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}<z
\end{smallmatrix}}{e^{-( 1+\frac{r}{n} )\frac{r^2}{2n}}}<
<\mathsf{P}\{\tau_{f^k}(x_0)=z \}<\frac1n\textstyle\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}=z
\end{smallmatrix}}{e^{-\frac{{( m-1 )}^2}{2n}}}+\dfrac{k}{n}\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}<z
\end{smallmatrix}}{e^{-\frac{{( r-1 )}^2}{2n}}}, \end{gathered} \end{equation}
which, for a prime $k$, is expressed in elementary functions and efficiently computable for used in practice values of $n$ ($2^{256}$ and more). Also, if $ kz\leq\sqrt{n}$, then
$$\textstyle\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}\leq z
\end{smallmatrix}}{\dfrac{r}{n}( 1-\dfrac{r( m+r )}{2n} ){e^{-( 1+\frac{m}{n} )\frac{m^2}{2n}}}}<F_{\tau_{f^k}(x_0)}(z)<\sum\limits_{\begin{smallmatrix} m\geq1,
\frac{m}{(m,k)}\leq z
\end{smallmatrix}}{\dfrac{r+1}{n}{e^{-\frac{{( m-1 )}^2}{2n}}}},$$
where $r=m+( z-\dfrac{m}{(m,k)} )k$. In some cases, the obtained results allow to estimate the allowable period of usage of the encryption keys generated by iterative algorithms and to build criteria for quality assessment of random sequences.

Keywords: equiprobable random mapping, iteration of random mapping, graph of a mapping, aperiodicity segment, local probability, distribution.

DOI: https://doi.org/10.17223/20710410/42/1

Full text: PDF file (880 kB)
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Document Type: Article
UDC: 519.212.2+519.719.2

Citation: V. O. Mironkin, “On estimations of distribution of the length of aperiodicity segment in the graph of $k$-fold iteration of uniform random mapping”, Prikl. Diskr. Mat., 2018, no. 42, 6–17

Citation in format AMSBIB
\Bibitem{Mir18}
\by V.~O.~Mironkin
\paper On estimations of distribution of the length of~aperiodicity segment in the graph of $k$-fold iteration of~uniform random mapping
\jour Prikl. Diskr. Mat.
\yr 2018
\issue 42
\pages 6--17
\mathnet{http://mi.mathnet.ru/pdm639}
\crossref{https://doi.org/10.17223/20710410/42/1}


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    This publication is cited in the following articles:
    1. V. O. Mironkin, “Sloi v grafe $k$-kratnoi iteratsii ravnoveroyatnogo sluchainogo otobrazheniya”, Matem. vopr. kriptogr., 10:1 (2019), 73–82  mathnet  crossref
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