quelques corrections après remarques de Sylvain

[16dcc.git] / main.tex

%% bare_jrnl.tex
%% V1.4b
%% 2015/08/26
%% by Michael Shell
%% see http://www.michaelshell.org/
%% for current contact information.
%%
%% This is a skeleton file demonstrating the use of IEEEtran.cls
%% (requires IEEEtran.cls version 1.8b or later) with an IEEE
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%% and
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% *** Authors should verify (and, if needed, correct) their LaTeX system  ***
% *** with the testflow diagnostic prior to trusting their LaTeX platform ***
% *** with production work. The IEEE's font choices and paper sizes can   ***
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% The testflow support page is at:
% http://www.michaelshell.org/tex/testflow/



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% Some very useful LaTeX packages include:
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% *** MISC UTILITY PACKAGES ***
%
%\usepackage{ifpdf}
% Heiko Oberdiek's ifpdf.sty is very useful if you need conditional
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% usage:
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% You can find documentation about the pdfTeX application at:
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% *** MATH PACKAGES ***
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%\usepackage{amsmath}
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% *** SUBFIGURE PACKAGES ***
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% subfig.sty, written by Steven Douglas Cochran, is the modern replacement
% for subfigure.sty, the latter of which is no longer maintained and is
% incompatible with some LaTeX packages including fixltx2e. However,
% subfig.sty requires and automatically loads Axel Sommerfeldt's caption.sty
% which will override IEEEtran.cls' handling of captions and this will result
% in non-IEEE style figure/table captions. To prevent this problem, be sure
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%\ifCLASSOPTIONcaptionsoff
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% \renewcommand{\caption}[2][\relax]{\MYoriglatexcaption[#2]{#2}}
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% endfloat.sty was written by James Darrell McCauley, Jeff Goldberg and 
% Axel Sommerfeldt. This package may be useful when used in conjunction with 
% IEEEtran.cls'  captionsoff option. Some IEEE journals/societies require that
% submissions have lists of figures/tables at the end of the paper and that
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% IEEE papers do not typically make use of \caption[]'s optional argument,
% so this should not be an issue. A similar trick can be used to disable
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% \let\MYorigsubfloat\subfloat
% \renewcommand{\subfloat}[2][\relax]{\MYorigsubfloat[]{#2}}
% However, the above trick will not work if both optional arguments of
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% *** PDF, URL AND HYPERLINK PACKAGES ***
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%\usepackage{url}
% url.sty was written by Donald Arseneau. It provides better support for
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% Basically, \url{my_url_here}.




% *** Do not adjust lengths that control margins, column widths, etc. ***
% *** Do not use packages that alter fonts (such as pslatex).         ***
% There should be no need to do such things with IEEEtran.cls V1.6 and later.
% (Unless specifically asked to do so by the journal or conference you plan
% to submit to, of course. )


% correct bad hyphenation here
\hyphenation{op-tical net-works semi-conduc-tor}


\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{Sylvain Contassot-Vivier, Jean-François Couchot, Christophe Guyeux, Pierre-Cyrille Heam
\thanks{LORIA, Université de Lorraine, Nancy, France}
\thanks{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 - 
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% The paper headers
\markboth{Journal of \LaTeX\ Class Files,~Vol.~14, No.~8, August~2015}%
{Shell \MakeLowercase{\textit{et al.}}: Bare Demo of IEEEtran.cls for IEEE Journals}
% 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 consider 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 behaviours.

In a previous work, some of the authors have proposed the idea of walking
into a $\mathsf{N}$-cube where a balanced Hamiltonian cycle have 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.
\IEEEpeerreviewmaketitle


\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{...}

\bibliographystyle{IEEEtran}
\bibliography{biblio}


%\section{Introduction}
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% 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

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% % 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
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% % and a .pdf suffix will be assumed for pdflatex; or what has been declared
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% %\caption{Simulation results for the network.}
% %\label{fig_sim}
% %\end{figure}

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% % results in a large percentage of a column being occupied by floats.


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% %\caption{Simulation results for the network.}
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% %\end{figure*}
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% % last word of the caption. Table text will default to \footnotesize as
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% %
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% %\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.
\ifCLASSOPTIONcaptionsoff
  \newpage
\fi



% 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}