Ajout diapos présentation PRNG

[16dcc.git] / main.tex

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\begin{document}
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\title{Random Walk in a N-cube Without Hamiltonian Cycle 
  to Chaotic Pseudorandom Number Generation: Theoretical and Practical 
  Considerations}

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\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}



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% \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.
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% practical experiments show that the generators successfully pass
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% \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}




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\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.




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