# A Generic Model of the Pseudo-Random Generator Based on Permutations Suitable for Security Solutions in Computationally-Constrained Environments

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## Abstract

**:**

## 1. Introduction

## 2. Notation and Generating Algorithm Description

- For any ${\Pi}_{j}\in {\mathcal{P}}_{k},\phantom{\rule{0.166667em}{0ex}}j=1,2,\dots ,k!,$ exists a number $l>0$ and set of integers ${i}_{1},{i}_{2},\dots ,{i}_{l}\in {I}_{m}$ such that ${\Pi}_{j}={f}_{{i}_{l}}\circ {f}_{{i}_{l-1}}\circ \dots \circ \phantom{\rule{0.166667em}{0ex}}{f}_{{i}_{1}}$, where ∘ is composition of functions.
- For all $p,q\in {I}_{m}$ if $p\ne q$ then ${f}_{p}\ne {f}_{q}$.
- ${\Pi}_{1}=I\in S$ where I is identical permutation.

## 3. Analysis of the Generator

#### 3.1. Distribution of the Generated Sequence

**Theorem**

**1.**

- $\sum _{i=1}^{k}}{a}_{i}=1$and${a}_{i}>0$for all$i\in {I}_{k},$
- $\sum _{i=0}^{m-1}}{c}_{i}=1$and${c}_{i}>0$for all$i\in {I}_{m},$

**Proof.**

**Remark**

**1.**

**Theorem**

**2.**

**Proof.**

#### Correlation Properties

**Theorem**

**3.**

- $P\left\{{Z}_{n+l}=b\wedge {Z}_{n}=a\right\}=\frac{1}{{k}^{2}}$
- $P\left\{{Z}_{n+l}=b|{Z}_{n}=a\right\}=\frac{1}{k}$

**Proof.**

- By the generator definition we have$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& P\left\{{Z}_{n+l}=b\wedge {Z}_{n}=a\right\}=P\left\{{g}_{n+k}\left({A}_{n+k}\right)=b\wedge {g}_{n}\left({A}_{n}\right)=a\right\}=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}P\left\{{g}_{n+k}\left({A}_{n+k}\right)=b\wedge {g}_{n}\left({A}_{n}\right)=a\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{g}_{n+k}={\Pi}_{i}\wedge {g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{52.03448pt}{0ex}}\phantom{\rule{52.03448pt}{0ex}}\phantom{\rule{52.03448pt}{0ex}}\phantom{\rule{52.03448pt}{0ex}}\xb7P\left\{{g}_{n+k}={\Pi}_{i}\wedge {g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}P\left\{{\Pi}_{i}\left({A}_{n+k}\right)=b\wedge {\Pi}_{j}\left({A}_{n}\right)=a\phantom{\rule{0.166667em}{0ex}}\right\}\xb7P\left\{{g}_{n+k}={\Pi}_{i}\wedge {g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}P\left\{{\Pi}_{i}\left({A}_{n+k}\right)=b\wedge {\Pi}_{j}\left({A}_{n}\right)=a\phantom{\rule{0.166667em}{0ex}}\right\}\xb7P\left\{{g}_{n+k}={\Pi}_{i}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{52.03448pt}{0ex}}\phantom{\rule{52.03448pt}{0ex}}\phantom{\rule{52.03448pt}{0ex}}\phantom{\rule{52.03448pt}{0ex}}\xb7P\left\{{g}_{n}={\Pi}_{j}\right\}.\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& P\left\{{Z}_{n+l}=b\wedge {Z}_{n}=a\right\}=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}P\left\{{\Pi}_{i}\left({A}_{n+k}\right)=b\wedge {\Pi}_{j}\left({A}_{n}\right)=a\phantom{\rule{0.166667em}{0ex}}\right\}\xb7P\left\{{g}_{n+k}={\Pi}_{i}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phantom{\rule{52.03448pt}{0ex}}\phantom{\rule{52.03448pt}{0ex}}\phantom{\rule{52.03448pt}{0ex}}\phantom{\rule{52.03448pt}{0ex}}\xb7P\left\{{g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}P\left\{{\Pi}_{i}\left({A}_{n+k}\right)=b\wedge {\Pi}_{j}\left({A}_{n}\right)=a\phantom{\rule{0.166667em}{0ex}}\right\}\xb7{t}_{i,j}^{k}\xb7P\left\{{g}_{n}={\Pi}_{j}\right\}.\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& P\left\{{Z}_{n+l}=b\wedge {Z}_{n}=a\right\}=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}P\left\{{\Pi}_{i}\left({A}_{n+k}\right)=b\wedge {\Pi}_{j}\left({A}_{n}\right)=a\phantom{\rule{0.166667em}{0ex}}\right\}\xb7{t}_{i,j}^{k}\xb7P\left\{{g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}P\left\{{A}_{n+k}={\Pi}_{i}^{-1}\left(b\right)\wedge {A}_{n}={\Pi}_{j}^{-1}\left(a\phantom{\rule{0.166667em}{0ex}}\right)\right\}\xb7{t}_{i,j}^{k}\xb7P\left\{{g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}P\left\{{A}_{n+k}={\Pi}_{i}^{-1}\left(b\right)\right\}\xb7P\left\{{A}_{n}={\Pi}_{j}^{-1}\left(a\phantom{\rule{0.166667em}{0ex}}\right)\right\}\xb7{t}_{i,j}^{k}\xb7P\left\{{g}_{n}={\Pi}_{j}\right\}\hfill \end{array}$$$$\begin{array}{cc}\hfill P& \left\{{Z}_{n+l}=b\wedge {Z}_{n}=a\right\}=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}P\left\{{A}_{n+k}={\Pi}_{i}^{-1}\left(b\right)\right\}\xb7P\left\{{A}_{n}={\Pi}_{j}^{-1}\left(a\phantom{\rule{0.166667em}{0ex}}\right)\right\}\xb7{t}_{i,j}^{k}\xb7P\left\{{g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\sum _{j=1}^{k!}\frac{1}{k}\xb7\frac{1}{k}\xb7{t}_{i,j}^{k}\xb7P\left\{{g}_{n}={\Pi}_{j}\right\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{{k}^{2}}\sum _{i=1}^{k!}P\left\{{g}_{n}={\Pi}_{i}\right\}\sum _{j=1}^{k!}{t}_{i,j}^{k}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{{k}^{2}}\hfill \end{array}$$
- Using statement of the Theorem 2 that $\phantom{\rule{4pt}{0ex}}P\left\{{Z}_{n}=a\right\}=\frac{1}{k}$ by the definition of conditional probability it follows that$$P\left\{{Z}_{n+l}=b|{Z}_{n}=a\right\}=\frac{P\left\{{Z}_{n+k}=b\wedge {Z}_{n}=a\right\}}{P\left\{{Z}_{n}=a\right\}}=\frac{\frac{1}{{k}^{2}}}{\frac{1}{k}}=\frac{1}{k}$$

#### 3.2. Information Leakage

**Theorem**

**4.**

- $\underset{n\to \infty}{\mathrm{lim}}\phantom{\rule{0.166667em}{0ex}}P\left({Z}_{n}=z|{C}_{n}=c\right)=\frac{1}{k}$
- $\underset{n\to \infty}{\mathrm{lim}}\phantom{\rule{0.166667em}{0ex}}I\left({Z}_{n},{C}_{n}\right)=0$

**Proof.**

- By the definition$$\begin{array}{cc}\hfill P& \left({Z}_{n}=z|{C}_{n}=c\right)=\phantom{\rule{91.0598pt}{0ex}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{P\left({Z}_{n}=z\wedge {C}_{n}=c\right)}{P\left({C}_{n}=c\right)}=\frac{P\left({g}_{n}\left({A}_{n}\right)=z\wedge {C}_{n}=c\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{P\left(\left({\displaystyle \bigcup _{i=1}^{k!}}{g}_{n}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {C}_{n}=c\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{P\left(\left({\displaystyle \bigcup _{i=1}^{k!}}{f}_{{C}_{n}}\circ {g}_{n-1}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {C}_{n}=c\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{P\left({\displaystyle \bigcup _{i=1}^{k!}}\left(\left({f}_{{C}_{n}}\circ {g}_{n-1}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {C}_{n}=c\right)\right)}{P\left({C}_{n}=c\right)}.\hfill \end{array}$$$$\begin{array}{cc}\hfill P& \left({Z}_{n}=z|{C}_{n}=c\right)=\frac{P\left({\displaystyle \bigcup _{i=1}^{k!}}\left(\left({f}_{{C}_{n}}\circ {g}_{n-1}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {C}_{n}=c\right)\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}P\left(\left(\left({f}_{{C}_{n}}\circ {g}_{n-1}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {C}_{n}=c\right)\right)}{P\left({C}_{n}=c\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill P& \left({Z}_{n}=z|{C}_{n}=c\right)=\frac{{\displaystyle \sum _{i=1}^{k!}}P\left(\left(\left({f}_{{C}_{n}}\circ {g}_{n-1}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {C}_{n}=c\right)\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}{\displaystyle \sum _{j=0}^{k-1}}P\left(\left(\left({f}_{{C}_{n}}\circ {g}_{n-1}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {C}_{n}=c\right)|\left({A}_{n}=j\right)\right)\xb7P\left({A}_{n}=j\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}{\displaystyle \sum _{j=0}^{k-1}}P\left(\left(\left({f}_{{C}_{n}}\circ {g}_{n-1}={\Pi}_{i}\wedge {\Pi}_{i}\left(j\right)=z\right)\wedge {C}_{n}=c\right)\right)\xb7P\left({A}_{n}=j\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}\sum _{j=0}^{k-1}}P\left(\left({g}_{n-1}={f}_{{C}_{n}}^{-1}\circ {\Pi}_{i}\wedge {C}_{n}=c\right)\wedge {\Pi}_{i}\left(j\right)=z\right)\xb7P\left({A}_{n}=j\right)}{P\left({C}_{n}=c\right)}.\hfill \end{array}$$$$\begin{array}{cc}\hfill P& \left({Z}_{n}=z|{C}_{n}=c\right)=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}{\displaystyle \sum _{j=0}^{k-1}}P\left(\left({g}_{n-1}={f}_{{C}_{n}}^{-1}\circ {\Pi}_{i}\wedge {C}_{n}=c\right)\wedge {\Pi}_{i}\left(j\right)=z\right)\xb7P\left({A}_{n}=j\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}{\displaystyle \sum _{j=0}^{k-1}}P\left({g}_{n-1}={f}_{{C}_{n}}^{-1}\circ {\Pi}_{i}\wedge {C}_{n}=c\right)\xb7P\left({\Pi}_{i}\left(j\right)=z\right)\xb7P\left({A}_{n}=j\right)}{P\left({C}_{n}=c\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill P& \left({Z}_{n}=z|{C}_{n}=c\right)=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}{\displaystyle \sum _{j=0}^{k-1}}P\left({g}_{n-1}={f}_{{C}_{n}}^{-1}\circ {\Pi}_{i}\wedge {C}_{n}=c\right)\xb7P\left({\Pi}_{i}\left(j\right)=z\right)\xb7P\left({A}_{n}=j\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\frac{P\left({g}_{n-1}={f}_{{C}_{n}}^{-1}\circ {\Pi}_{i}\wedge {C}_{n}=c\right){\displaystyle \sum _{j=0}^{k-1}}P\left({A}_{n}=j\right)\xb7P\left({\Pi}_{i}\left(j\right)=z\right)}{P\left({C}_{n}=c\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\frac{P\left({g}_{n-1}={f}_{{C}_{n}}^{-1}\circ {\Pi}_{i}\wedge {C}_{n}=c\right)}{P\left({C}_{n}=c\right)}{\displaystyle \sum _{j=0}^{k-1}}P\left({A}_{n}=j\right)\xb7P\left({\Pi}_{i}\left(j\right)=z\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}P\left({g}_{n-1}={f}_{{C}_{n}}^{-1}\circ {\Pi}_{i}|{C}_{n}=c\right){\displaystyle \sum _{j=0}^{k-1}}P\left({A}_{n}=j\right)\xb7P\left({\Pi}_{i}\left(j\right)=z\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}P\left({g}_{n-1}={f}_{c}^{-1}\circ {\Pi}_{i}\right){\displaystyle \sum _{j=0}^{k-1}}P\left({A}_{n}=j\right)\xb7P\left({\Pi}_{i}\left(j\right)=z\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \underset{n\to \infty}{lim}& P\left({Z}_{n}=z|{C}_{n}=c\right)=\phantom{\rule{91.0598pt}{0ex}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\underset{n\to \infty}{lim}\sum _{j=0}^{k-1}P\left({A}_{n}=j\right)\sum _{i=1}^{k!}P\left({g}_{n-1}={f}_{c}^{-1}\circ {\Pi}_{i}\right)\xb7P\left({\Pi}_{i}\left(j\right)=z\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{j=0}^{k-1}P\left({A}_{n}=j\right)\sum _{i=1}^{k!}P\left({\Pi}_{i}\left(j\right)=z\right)\xb7\underset{n\to \infty}{lim}P\left({g}_{n-1}={f}_{c}^{-1}\circ {\Pi}_{i}\right)=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{j=0}^{k-1}P\left({A}_{n}=j\right)\sum _{i=1}^{k!}P\left({\Pi}_{i}\left(j\right)=z\right)\xb7\frac{1}{k!}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{k!}\sum _{j=0}^{k-1}P\left({A}_{n}=j\right)\sum _{i=1}^{k!}P\left({\Pi}_{i}\left(j\right)=z\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{k!}\sum _{j=0}^{k-1}P\left({A}_{n}=j\right)\xb7\left(k-1\right)!=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{\left(k-1\right)!}{k!}\sum _{j=0}^{k-1}P\left({A}_{n}=j\right)=\frac{1}{k}\hfill \end{array}$$
- By the definition of mutual information we have that$$I\left({Z}_{n},{C}_{n}\right)=H\left({Z}_{n}\right)-H\left({Z}_{n}|{C}_{n}\right)$$$$H\left({Z}_{n}\right)=\sum _{i=0}^{k-1}P\left({Z}_{n}=i\right){log}_{2}\frac{1}{P\left({Z}_{n}=i\right)}$$$$\underset{n\to \infty}{lim}H\left({Z}_{n}\right)={log}_{2}k.$$$$\begin{array}{cc}\hfill H\left({Z}_{n}|{C}_{n}\right)& =\sum _{i=0}^{m-1}P\left({C}_{n}=i\right)\xb7H\left({Z}_{n}|{C}_{n}=i\right)=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=0}^{m-1}P\left({C}_{n}=i\right)\xb7\sum _{j=0}^{k-1}P\left({Z}_{n}=j|{C}_{n}={c}_{i}\right){log}_{2}\frac{1}{P\left({Z}_{n}=j|{C}_{n}=i\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \underset{n\to \infty}{lim}H\left({Z}_{n}|{C}_{n}\right)=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=0}^{m-1}P\left({C}_{n}=i\right)\xb7\sum _{j=0}^{k-1}\underset{n\to \infty}{lim}P\left({Z}_{n}=j|{C}_{n}=i\right)\xb7\underset{n\to \infty}{lim}{log}_{2}\frac{1}{P\left({Z}_{n}=j|{C}_{n}=i\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=0}^{m-1}P\left({C}_{n}=i\right)\xb7\sum _{j=0}^{k-1}\frac{1}{k}{log}_{2}k={log}_{2}k\xb7\sum _{i=0}^{m-1}P\left({C}_{n}=i\right)={log}_{2}k.\hfill \end{array}$$$$\underset{n\to \infty}{lim}I\left({Z}_{n},{C}_{n}\right)=\underset{n\to \infty}{lim}H\left({Z}_{n}\right)-\underset{n\to \infty}{lim}H\left({Z}_{n}|{C}_{n}\right)={log}_{2}k-{log}_{2}k=0$$

**Theorem**

**5.**

- $\underset{n\to \infty}{\mathrm{lim}}P\left({Z}_{n}=z|{A}_{n}=a\right)=\frac{1}{k}$
- $\underset{n\to \infty}{\mathrm{lim}}I\left({Z}_{n},{A}_{n}\right)=0$

**Proof.**

- By the definition of conditional probability it follows that$$\begin{array}{cc}\hfill P& \left({Z}_{n}=z|{A}_{n}=a\right)=\phantom{\rule{91.0598pt}{0ex}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{P\left({Z}_{n}=z\wedge {A}_{n}=a\right)}{P\left({A}_{n}=a\right)}=\frac{P\left({g}_{n}\left({A}_{n}\right)=z\wedge {A}_{n}=a\right)}{P\left({A}_{n}=a\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{P\left(\left({\displaystyle \bigcup _{i=1}^{k!}}{g}_{n}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {A}_{n}=a\right)}{P\left({A}_{n}=a\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{P\left(\left({\displaystyle \bigcup _{i=1}^{k!}}{g}_{n}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {A}_{n}=a\right)}{P\left({A}_{n}=a\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{P\left({\displaystyle \bigcup _{i=1}^{k!}}\left(\left(\left({g}_{n}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {A}_{n}=a\right)\right)\right)}{P\left({A}_{n}=a\right)}.\hfill \end{array}$$$$\begin{array}{cc}\hfill P& \left({Z}_{n}=z|{A}_{n}=a\right)=\phantom{\rule{91.0598pt}{0ex}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{P\left({\displaystyle \bigcup _{i=1}^{k!}}\left(\left(\left({g}_{n}={\Pi}_{i}\wedge {\Pi}_{i}\left({A}_{n}\right)=z\right)\wedge {A}_{n}=a\right)\right)\right)}{P\left({A}_{n}=a\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}P\left(\left({g}_{n}={\Pi}_{i}\wedge \left({\Pi}_{i}\left({A}_{n}\right)=z\wedge {A}_{n}=a\right)\right)\right)}{P\left({A}_{n}=a\right)}.\hfill \end{array}$$$$\begin{array}{cc}\hfill P& \left({Z}_{n}=z|{A}_{n}=a\right)=\phantom{\rule{91.0598pt}{0ex}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}P\left(\left({g}_{n}={\Pi}_{i}\wedge \left({\Pi}_{i}\left({A}_{n}\right)=z\wedge {A}_{n}=a\right)\right)\right)}{P\left({A}_{n}=a\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}P\left({g}_{n}={\Pi}_{i}|\left({\Pi}_{i}\left({A}_{n}\right)=z\wedge {A}_{n}=a\right)\right)\xb7P\left({\Pi}_{i}\left(a\right)=z\wedge {A}_{n}=a\right)}{P\left({A}_{n}=a\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}P\left({g}_{n}={\Pi}_{i}|\left({\Pi}_{i}\left({A}_{n}\right)=z\wedge {A}_{n}=a\right)\right)\xb7\frac{P\left({\Pi}_{i}\left(a\right)=z\wedge {A}_{n}=a\right)}{P\left({A}_{n}=a\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}P\left({g}_{n}={\Pi}_{i}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\left({\Pi}_{i}\left({A}_{n}\right)=z\wedge {A}_{n}=a\right)\right)\xb7P\left({\Pi}_{i}\left({A}_{n}\right)=z\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{A}_{n}=a\right).\hfill \end{array}$$$$\begin{array}{cc}\hfill P& \left({Z}_{n}=z|{A}_{n}=a\right)=\phantom{\rule{91.0598pt}{0ex}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}P\left(\left({g}_{n}={\Pi}_{i}\wedge \left({\Pi}_{i}\left({A}_{n}\right)=z\wedge {A}_{n}=a\right)\right)\right)}{P\left({A}_{n}=a\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{\displaystyle \sum _{i=1}^{k!}}P\left({g}_{n}={\Pi}_{i}|\left({\Pi}_{i}\left({A}_{n}\right)=z\wedge {A}_{n}=a\right)\right)\xb7P\left({\Pi}_{i}\left(a\right)=z\wedge {A}_{n}=a\right)}{P\left({A}_{n}=a\right)}=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}P\left({g}_{n}={\Pi}_{i}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\left({\Pi}_{i}\left({A}_{n}\right)=z\wedge {A}_{n}=a\right)\right)\xb7P\left({\Pi}_{i}\left({A}_{n}\right)=z\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{A}_{n}=a\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}P\left({g}_{n}={\Pi}_{i}\right)\xb7P\left({\Pi}_{i}\left(a\right)=z\phantom{\rule{0.166667em}{0ex}}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \underset{n\to \infty}{lim}& P\left({Z}_{n}=z|{A}_{n}=a\right)=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\underset{n\to \infty}{lim}{\displaystyle \sum _{i=1}^{k!}}P\left({g}_{n}={\Pi}_{i}\right)\xb7P\left({\Pi}_{i}\left(a\right)=z\phantom{\rule{0.166667em}{0ex}}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\underset{n\to \infty}{lim}P\left({g}_{n}={\Pi}_{i}\right)\xb7P\left({\Pi}_{i}\left(a\right)=z\phantom{\rule{0.166667em}{0ex}}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=1}^{k!}\frac{1}{k!}\xb7P\left({\Pi}_{i}\left(a\right)=z\phantom{\rule{0.166667em}{0ex}}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{k!}\sum _{i=1}^{k!}P\left({\Pi}_{i}\left(a\right)=z\phantom{\rule{0.166667em}{0ex}}\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{k!}\xb7\left(k-1\right)!=\frac{1}{k}\hfill \end{array}$$
- In the same way as in the Theorem 4 it follows that$$\underset{n\to \infty}{lim}H\left({Z}_{n}\right)={log}_{2}k$$$$\begin{array}{cc}\hfill H\left({Z}_{n}|{A}_{n}\right)& =\sum _{i=0}^{k-1}P\left({A}_{n}=i\right)\xb7H\left({Z}_{n}|{A}_{n}=i\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=0}^{k-1}P\left({A}_{n}=i\right)\xb7\sum _{j=0}^{k-1}P\left({Z}_{n}=j|{A}_{n}=i\right){log}_{2}\frac{1}{P\left({Z}_{n}=j|{A}_{n}=i\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \underset{n\to \infty}{lim}H\left({Z}_{n}|{A}_{n}\right)=\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=0}^{k-1}P\left({A}_{n}=i\right)\xb7\sum _{j=0}^{k-1}\underset{n\to \infty}{lim}P\left({Z}_{n}=j|{A}_{n}=i\right)\xb7\underset{n\to \infty}{lim}{log}_{2}\frac{1}{P\left({Z}_{n}=j|{A}_{n}=i\right)}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\sum _{i=0}^{k-1}P\left({A}_{n}=i\right)\xb7\sum _{j=0}^{k-1}\frac{1}{k}{log}_{2}k={log}_{2}k\xb7\sum _{i=0}^{k-1}P\left({A}_{n}=i\right)={log}_{2}k\hfill \end{array}$$$$\underset{n\to \infty}{lim}I\left({Z}_{n},{A}_{n}\right)=\underset{n\to \infty}{lim}H\left({Z}_{n}\right)-\underset{n\to \infty}{lim}H\left({Z}_{n}|{A}_{n}\right)={log}_{2}k-{log}_{2}k=0$$

#### 3.3. Periodicity

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**6.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Unkašević, T.; Banjac, Z.; Milosavljević, M.
A Generic Model of the Pseudo-Random Generator Based on Permutations Suitable for Security Solutions in Computationally-Constrained Environments. *Sensors* **2019**, *19*, 5322.
https://doi.org/10.3390/s19235322

**AMA Style**

Unkašević T, Banjac Z, Milosavljević M.
A Generic Model of the Pseudo-Random Generator Based on Permutations Suitable for Security Solutions in Computationally-Constrained Environments. *Sensors*. 2019; 19(23):5322.
https://doi.org/10.3390/s19235322

**Chicago/Turabian Style**

Unkašević, Tomislav, Zoran Banjac, and Milan Milosavljević.
2019. "A Generic Model of the Pseudo-Random Generator Based on Permutations Suitable for Security Solutions in Computationally-Constrained Environments" *Sensors* 19, no. 23: 5322.
https://doi.org/10.3390/s19235322