
\documentclass[preprint,review,12pt]{elsarticle}

\usepackage{graphicx}
\usepackage{caption}
\usepackage{subcaption}

\usepackage{dsfont}
\usepackage{stmaryrd}
%\usepackage[font=footnotesize]{subfig}
\usepackage{ifthen}
\usepackage{color}
\usepackage{algorithm2e}
\usepackage{epstopdf}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc} 
\usepackage[english]{babel}
\usepackage{amsmath,amssymb,amsthm,latexsym,eufrak,euscript}
\usepackage{pstricks,pst-node,pst-coil}


\usepackage{url,tikz}
\usepackage{pgflibrarysnakes}

\usepackage{multicol}

\usetikzlibrary{arrows}
\usetikzlibrary{automata}
\usetikzlibrary{snakes}
\usetikzlibrary{shapes}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Définitions personnelles
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definecolor{bleuclair}{rgb}{0.75,0.75,1.0}
\newcommand{\danger}{!!}
\newcommand{\ANNOT}[1]{
  ~\linebreak
  \centerline{
    {\Huge{$\Rightarrow$}}
    \large\fcolorbox{black}{bleuclair}{
      \begin{minipage}[h]{.8\linewidth}
        #1
      \end{minipage}
    }
    {\Huge{$\Leftarrow$}}
  }
}



\newcommand {\tv}[1] {\lVert #1 \rVert_{\rm TV}}
\def \top {1.8}
\def \topt {2.3}
\def \P {\mathbb{P}}
\def \ov {\overline}
\def \ts {\tau_{\rm stop}}


\newtheorem*{xpl}{Running Example}

\newtheorem{definition}{Definition}
\newtheorem{prpstn}{Proposition}
\newtheorem{thrm}{Theorem}
\newtheorem{crllr}{Corollary}
\newtheorem{lmm}{Lemma}



\newcommand{\vectornorm}[1]{\ensuremath{\left|\left|#1\right|\right|_2}}
%\newcommand{\ie}{\textit{i.e.}}
\newcommand{\Nats}[0]{\ensuremath{\mathbb{N}}}
\newcommand{\R}[0]{\ensuremath{\mathbb{R}}}
\newcommand{\Z}[0]{\ensuremath{\mathbb{Z}}}
\newcommand{\Bool}[0]{\ensuremath{\mathds{B}}}
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\newcommand{\Gall}[0]{\ensuremath{\mathcal{G}}}


\newcommand{\JFC}[1]{\begin{color}{green}\textit{#1}\end{color}}
\newcommand{\CG}[1]{\begin{color}{blue}\textit{}\end{color}}
\newcommand{\og}[0]{``}
\newcommand{\fg}[1]{''}







% Some very useful LaTeX packages include:
% (uncomment the ones you want to load)


% *** MISC UTILITY PACKAGES ***
%
%\usepackage{ifpdf}
% Heiko Oberdiek's ifpdf.sty is very useful if you need conditional
% compilation based on whether the output is pdf or dvi.
% usage:
% \ifpdf
%   % pdf code
% \else
%   % dvi code
% \fi
% The latest version of ifpdf.sty can be obtained from:
% http://www.ctan.org/pkg/ifpdf
% Also, note that IEEEtran.cls V1.7 and later provides a builtin
% \ifCLASSINFOpdf conditional that works the same way.
% When switching from latex to pdflatex and vice-versa, the compiler may
% have to be run twice to clear warning/error messages.






% *** CITATION PACKAGES ***
%
%\usepackage{cite}
% cite.sty was written by Donald Arseneau
% V1.6 and later of IEEEtran pre-defines the format of the cite.sty package
% \cite{} output to follow that of the IEEE. Loading the cite package will
% result in citation numbers being automatically sorted and properly
% "compressed/ranged". e.g., [1], [9], [2], [7], [5], [6] without using
% cite.sty will become [1], [2], [5]--[7], [9] using cite.sty. cite.sty's
% \cite will automatically add leading space, if needed. Use cite.sty's
% noadjust option (cite.sty V3.8 and later) if you want to turn this off
% such as if a citation ever needs to be enclosed in parenthesis.
% cite.sty is already installed on most LaTeX systems. Be sure and use
% version 5.0 (2009-03-20) and later if using hyperref.sty.
% The latest version can be obtained at:
% http://www.ctan.org/pkg/cite
% The documentation is contained in the cite.sty file itself.






\begin{document}
%
% paper title
% Titles are generally capitalized except for words such as a, an, and, as,
% at, but, by, for, in, nor, of, on, or, the, to and up, which are usually
% not capitalized unless they are the first or last word of the title.
% Linebreaks \\ can be used within to get better formatting as desired.
% Do not put math or special symbols in the title.
\title{Random Walk in a N-cube Without Hamiltonian Cycle 
  to Chaotic Pseudorandom Number Generation: Theoretical and Practical 
  Considerations}

%
%
% author names and IEEE memberships
% note positions of commas and nonbreaking spaces ( ~ ) LaTeX will not break
% a structure at a ~ so this keeps an author's name from being broken across
% two lines.
% use \thanks{} to gain access to the first footnote area
% a separate \thanks must be used for each paragraph as LaTeX2e's \thanks
% was not built to handle multiple paragraphs
%
\author[label1]{Sylvain Contassot-Vivier}
\author[label2]{Jean-François Couchot}
\author[label2]{Christophe Guyeux}
\author[label2]{Pierre-Cyrille Heam}
\address[label1]{LORIA, Université de Lorraine, Nancy, France}
\address[label2]{FEMTO-ST Institute, University of Franche-Comté, Belfort, France}



% \author{Michael~Shell,~\IEEEmembership{Member,~IEEE,}
%         John~Doe,~\IEEEmembership{Fellow,~OSA,}
%         and~Jane~Doe,~\IEEEmembership{Life~Fellow,~IEEE}% <-this % stops a space
% \thanks{M. Shell was with the Department
% of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta,
% GA, 30332 USA e-mail: (see http://www.michaelshell.org/contact.html).}% <-this % stops a space
% \thanks{J. Doe and J. Doe are with Anonymous University.}% <-this % stops a space
% \thanks{Manuscript received April 19, 2005; revised August 26, 2015.}}

% note the % following the last \IEEEmembership and also \thanks - 
% these prevent an unwanted space from occurring between the last author name
% and the end of the author line. i.e., if you had this:
% 
% \author{....lastname \thanks{...} \thanks{...} }
%                     ^------------^------------^----Do not want these spaces!
%
% a space would be appended to the last name and could cause every name on that
% line to be shifted left slightly. This is one of those "LaTeX things". For
% instance, "\textbf{A} \textbf{B}" will typeset as "A B" not "AB". To get
% "AB" then you have to do: "\textbf{A}\textbf{B}"
% \thanks is no different in this regard, so shield the last } of each \thanks
% that ends a line with a % and do not let a space in before the next \thanks.
% Spaces after \IEEEmembership other than the last one are OK (and needed) as
% you are supposed to have spaces between the names. For what it is worth,
% this is a minor point as most people would not even notice if the said evil
% space somehow managed to creep in.



% The only time the second header will appear is for the odd numbered pages
% after the title page when using the twoside option.
% 
% *** Note that you probably will NOT want to include the author's ***
% *** name in the headers of peer review papers.                   ***
% You can use \ifCLASSOPTIONpeerreview for conditional compilation here if
% you desire.




% If you want to put a publisher's ID mark on the page you can do it like
% this:
%\IEEEpubid{0000--0000/00\$00.00~\copyright~2015 IEEE}
% Remember, if you use this you must call \IEEEpubidadjcol in the second
% column for its text to clear the IEEEpubid mark.



% use for special paper notices
%\IEEEspecialpapernotice{(Invited Paper)}




% make the title area
%\maketitle

% As a general rule, do not put math, special symbols or citations
% in the abstract or keywords.


% \begin{abstract}
% This paper is dedicated to the design of chaotic random generators
% and extends previous works proposed by some of the authors.
% We propose a theoretical framework proving both the chaotic properties and
% that the limit distribution is uniform.
% A theoretical bound on the stationary time is given and
% practical experiments show that the generators successfully pass
% the classical statistical tests.
% \end{abstract}

\begin{abstract}
Designing a pseudorandom number generator (PRNG) is a hard and complex task.
Many recent works have considered chaotic functions as the basis of built 
PRNGs:
the quality of the output would be an obvious consequence of some chaos 
properties.  
However, there is no direct reasoning that goes from chaotic functions to 
uniform distribution of the output. 
Moreover, it is not clear that embedding such kind of functions into a PRNG
allows to get a chaotic output, which could be required for simulating 
some chaotic behaviors.

In a previous work, some of the authors have proposed the idea of walking
into a $\mathsf{N}$-cube where a balanced Hamiltonian cycle has been
removed as the basis of a chaotic PRNG. In this article,  all the difficult
issues observed in the previous work have been tackled. The chaotic behavior
of the whole PRNG is proven. The construction of the balanced Hamiltonian
cycle is theoretically and practically solved. An upper bound of the
expected length of the walk to obtain a uniform distribution is calculated.
Finally practical experiments show that the generators successfully pass the
classical statistical tests.
\end{abstract}




% Note that keywords are not normally used for peerreview papers.
% \begin{IEEEkeywords}
% IEEE, IEEEtran, journal, \LaTeX, paper, template.
% \end{IEEEkeywords}






% For peer review papers, you can put extra information on the cover
% page as needed:
% \ifCLASSOPTIONpeerreview
% \begin{center} \bfseries EDICS Category: 3-BBND \end{center}
% \fi
%
% For peerreview papers, this IEEEtran command inserts a page break and
% creates the second title. It will be ignored for other modes.
\maketitle


\section{Introduction}
\input{intro}

\section{Preliminaries}\label{sec:preliminaries}
\input{preliminaries}

\section{Proof Of Chaos}\label{sec:proofOfChaos}
\input{chaos}

\section{Functions with Strongly Connected $\Gamma_{\{b\}}(f)$}\label{sec:SCCfunc}
\input{generating}

\section{Balanced Hamiltonian Cycle}\label{sec:hamilton}
\input{hamilton}


\section{Mixing Time}\label{sec:hypercube}
\input{stopping}

\section{Experiments}\label{sec:prng}
\input{prng}



\section{Conclusion}
\input{conclusion}

%\acknowledgements{...}

\section*{Acknowledgements}
This work is partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01). 
Computations presented in this article were realised on the supercomputing
facilities provided by the M\'esocentre de calcul de Franche-Comt\'e.




\bibliographystyle{elsarticle-num} 
\bibliography{biblio}


%\section{Introduction}
% The very first letter is a 2 line initial drop letter followed
% by the rest of the first word in caps.
% 
% form to use if the first word consists of a single letter:
% \IEEEPARstart{A}{demo} file is ....
% 
% form to use if you need the single drop letter followed by
% normal text (unknown if ever used by the IEEE):
% \IEEEPARstart{A}{}demo file is ....
% 
% Some journals put the first two words in caps:
% \IEEEPARstart{T}{his demo} file is ....
% 
% Here we have the typical use of a "T" for an initial drop letter
% and "HIS" in caps to complete the first word.
% \IEEEPARstart{T}{his} demo file is intended to serve as a ``starter file''
% for IEEE journal papers produced under \LaTeX\ using
% IEEEtran.cls version 1.8b and later.
% % You must have at least 2 lines in the paragraph with the drop letter
% % (should never be an issue)
% I wish you the best of success.

% \hfill mds
 
% \hfill August 26, 2015

% \subsection{Subsection Heading Here}
% Subsection text here.

% % needed in second column of first page if using \IEEEpubid
% %\IEEEpubidadjcol

% \subsubsection{Subsubsection Heading Here}
% Subsubsection text here.


% % An example of a floating figure using the graphicx package.
% % Note that \label must occur AFTER (or within) \caption.
% % For figures, \caption should occur after the \includegraphics.
% % Note that IEEEtran v1.7 and later has special internal code that
% % is designed to preserve the operation of \label within \caption
% % even when the captionsoff option is in effect. However, because
% % of issues like this, it may be the safest practice to put all your
% % \label just after \caption rather than within \caption{}.
% %
% % Reminder: the "draftcls" or "draftclsnofoot", not "draft", class
% % option should be used if it is desired that the figures are to be
% % displayed while in draft mode.
% %
% %\begin{figure}[!t]
% %\centering
% %\includegraphics[width=2.5in]{myfigure}
% % where an .eps filename suffix will be assumed under latex, 
% % and a .pdf suffix will be assumed for pdflatex; or what has been declared
% % via \DeclareGraphicsExtensions.
% %\caption{Simulation results for the network.}
% %\label{fig_sim}
% %\end{figure}

% % Note that the IEEE typically puts floats only at the top, even when this
% % results in a large percentage of a column being occupied by floats.


% % An example of a double column floating figure using two subfigures.
% % (The subfig.sty package must be loaded for this to work.)
% % The subfigure \label commands are set within each subfloat command,
% % and the \label for the overall figure must come after \caption.
% % \hfil is used as a separator to get equal spacing.
% % Watch out that the combined width of all the subfigures on a 
% % line do not exceed the text width or a line break will occur.
% %
% %\begin{figure*}[!t]
% %\centering
% %\subfloat[Case I]{\includegraphics[width=2.5in]{box}%
% %\label{fig_first_case}}
% %\hfil
% %\subfloat[Case II]{\includegraphics[width=2.5in]{box}%
% %\label{fig_second_case}}
% %\caption{Simulation results for the network.}
% %\label{fig_sim}
% %\end{figure*}
% %
% % Note that often IEEE papers with subfigures do not employ subfigure
% % captions (using the optional argument to \subfloat[]), but instead will
% % reference/describe all of them (a), (b), etc., within the main caption.
% % Be aware that for subfig.sty to generate the (a), (b), etc., subfigure
% % labels, the optional argument to \subfloat must be present. If a
% % subcaption is not desired, just leave its contents blank,
% % e.g., \subfloat[].


% % An example of a floating table. Note that, for IEEE style tables, the
% % \caption command should come BEFORE the table and, given that table
% % captions serve much like titles, are usually capitalized except for words
% % such as a, an, and, as, at, but, by, for, in, nor, of, on, or, the, to
% % and up, which are usually not capitalized unless they are the first or
% % last word of the caption. Table text will default to \footnotesize as
% % the IEEE normally uses this smaller font for tables.
% % The \label must come after \caption as always.
% %
% %\begin{table}[!t]
% %% increase table row spacing, adjust to taste
% %\renewcommand{\arraystretch}{1.3}
% % if using array.sty, it might be a good idea to tweak the value of
% % \extrarowheight as needed to properly center the text within the cells
% %\caption{An Example of a Table}
% %\label{table_example}
% %\centering
% %% Some packages, such as MDW tools, offer better commands for making tables
% %% than the plain LaTeX2e tabular which is used here.
% %\begin{tabular}{|c||c|}
% %\hline
% %One & Two\\
% %\hline
% %Three & Four\\
% %\hline
% %\end{tabular}
% %\end{table}


% % Note that the IEEE does not put floats in the very first column
% % - or typically anywhere on the first page for that matter. Also,
% % in-text middle ("here") positioning is typically not used, but it
% % is allowed and encouraged for Computer Society conferences (but
% % not Computer Society journals). Most IEEE journals/conferences use
% % top floats exclusively. 
% % Note that, LaTeX2e, unlike IEEE journals/conferences, places
% % footnotes above bottom floats. This can be corrected via the
% % \fnbelowfloat command of the stfloats package.




% \section{Conclusion}
% The conclusion goes here.





% % if have a single appendix:
% %\appendix[Proof of the Zonklar Equations]
% % or
% %\appendix  % for no appendix heading
% % do not use \section anymore after \appendix, only \section*
% % is possibly needed

% % use appendices with more than one appendix
% % then use \section to start each appendix
% % you must declare a \section before using any
% % \subsection or using \label (\appendices by itself
% % starts a section numbered zero.)
% %


% \appendices
% \section{Proof of the First Zonklar Equation}
% Appendix one text goes here.

% % you can choose not to have a title for an appendix
% % if you want by leaving the argument blank
% \section{}
% Appendix two text goes here.


% % use section* for acknowledgment
% \section*{Acknowledgment}


% The authors would like to thank...


% Can use something like this to put references on a page
% by themselves when using endfloat and the captionsoff option.



% trigger a \newpage just before the given reference
% number - used to balance the columns on the last page
% adjust value as needed - may need to be readjusted if
% the document is modified later
%\IEEEtriggeratref{8}
% The "triggered" command can be changed if desired:
%\IEEEtriggercmd{\enlargethispage{-5in}}

% references section

% can use a bibliography generated by BibTeX as a .bbl file
% BibTeX documentation can be easily obtained at:
% http://mirror.ctan.org/biblio/bibtex/contrib/doc/
% The IEEEtran BibTeX style support page is at:
% http://www.michaelshell.org/tex/ieeetran/bibtex/
%\bibliographystyle{IEEEtran}
% argument is your BibTeX string definitions and bibliography database(s)
%\bibliography{IEEEabrv,../bib/paper}
%
% <OR> manually copy in the resultant .bbl file
% set second argument of \begin to the number of references
% (used to reserve space for the reference number labels box)
% \begin{thebibliography}{1}

% \bibitem{IEEEhowto:kopka}
% H.~Kopka and P.~W. Daly, \emph{A Guide to \LaTeX}, 3rd~ed.\hskip 1em plus
%   0.5em minus 0.4em\relax Harlow, England: Addison-Wesley, 1999.

% \end{thebibliography}

% biography section
% 
% If you have an EPS/PDF photo (graphicx package needed) extra braces are
% needed around the contents of the optional argument to biography to prevent
% the LaTeX parser from getting confused when it sees the complicated
% \includegraphics command within an optional argument. (You could create
% your own custom macro containing the \includegraphics command to make things
% simpler here.)
%\begin{IEEEbiography}[{\includegraphics[width=1in,height=1.25in,clip,keepaspectratio]{mshell}}]{Michael Shell}
% or if you just want to reserve a space for a photo:

% \begin{IEEEbiography}{Michael Shell}
% Biography text here.
% \end{IEEEbiography}

% % if you will not have a photo at all:
% \begin{IEEEbiographynophoto}{John Doe}
% Biography text here.
% \end{IEEEbiographynophoto}

% % insert where needed to balance the two columns on the last page with
% % biographies
% %\newpage

% \begin{IEEEbiographynophoto}{Jane Doe}
% Biography text here.
% \end{IEEEbiographynophoto}

% You can push biographies down or up by placing
% a \vfill before or after them. The appropriate
% use of \vfill depends on what kind of text is
% on the last page and whether or not the columns
% are being equalized.

%\vfill

% Can be used to pull up biographies so that the bottom of the last one
% is flush with the other column.
%\enlargethispage{-5in}



% that's all folks
\end{document}
