maj avant prise en main par sylvain

[16dcc.git] / main.tex

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% 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
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% *** CITATION PACKAGES ***
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%\usepackage{cite}
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% noadjust option (cite.sty V3.8 and later) if you want to turn this off
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\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}
\address{LORIA, Université de Lorraine, Nancy, France\\
sylvain.contassotvivier@loria.fr}

\author{Jean-François Couchot}
\address{FEMTO-ST Institute,  CNRS, Univ. Bourgogne Franche-Comté (UBFC), France\\
jean-francois.couchot@univ-fcomte.fr}

\author{Christophe Guyeux}
\address{FEMTO-ST Institute,  CNRS, Univ. Bourgogne Franche-Comté (UBFC), France\\
christophe.guyeux@univ-fcomte.fr}

\author{Pierre-Cyrille Heam}
\address{FEMTO-ST Institute,  CNRS, Univ. Bourgogne Franche-Comté (UBFC), France\\
pierre-cyrille.heam@univ-fcomte.fr}



% \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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% 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.



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% *** 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}


% Note that keywords are not normally used for peerreview papers.
% \begin{IEEEkeywords}
% IEEE, IEEEtran, journal, \LaTeX, paper, template.
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% For peer review papers, you can put extra information on the cover
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\maketitle

\begin{abstract}
  Designing a pseudorandom number generator (PRNG) is a
difficult  and complex  task.  Many  recent works  have considered  chaotic
functions as the basis of built PRNGs: the quality of the output would
indeed
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,  
embedding such kind of functions into a PRNG does not necessarily
allow 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}


\keywords{Pseudorandom Numbers Generator, Chaotic iterations, Random Walk}


\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{ws-ijbc} 
\bibliography{biblio}


%\section{Introduction}
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% that's all folks
\end{document}