-%% 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
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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 ***
-% *** trigger bugs that do not appear when using other class files. *** ***
-% The testflow support page is at:
-% http://www.michaelshell.org/tex/testflow/
-
-
-
-\documentclass[journal]{IEEEtran}
-%
-% If IEEEtran.cls has not been installed into the LaTeX system files,
-% manually specify the path to it like:
-% \documentclass[journal]{../sty/IEEEtran}
-
-
+\documentclass{ws-ijbc}
\usepackage{graphicx}
-\usepackage{caption}
-\usepackage{subcaption}
+%\usepackage{amsthm}
+%\usepackage{subcaption}
+\usepackage{subfigure}
\usepackage{dsfont}
\usepackage{stmaryrd}
%\usepackage[font=footnotesize]{subfig}
\usepackage{ifthen}
\usepackage{color}
+%\usepackage{subfigure}
\usepackage{algorithm2e}
\usepackage{epstopdf}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage[english]{babel}
-\usepackage{amsmath,amssymb,amsthm,latexsym,eufrak,euscript}
+%\usepackage{amsmath,amssymb,amsthm,latexsym,eufrak,euscript}
\usepackage{pstricks,pst-node,pst-coil}
\def \ts {\tau_{\rm stop}}
-\newtheorem*{xpl}{Running Example}
+%\newtheorem*{xpl}{Running Example}
+\newenvironment{xpl}[1][Running Example]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
-\newtheorem{definition}{Definition}
+%\newtheorem{definition}{Definition}
\newtheorem{prpstn}{Proposition}
\newtheorem{thrm}{Theorem}
\newtheorem{crllr}{Corollary}
-% *** GRAPHICS RELATED PACKAGES ***
-%
-\ifCLASSINFOpdf
- % \usepackage[pdftex]{graphicx}
- % declare the path(s) where your graphic files are
- % \graphicspath{{../pdf/}{../jpeg/}}
- % and their extensions so you won't have to specify these with
- % every instance of \includegraphics
- % \DeclareGraphicsExtensions{.pdf,.jpeg,.png}
-\else
- % or other class option (dvipsone, dvipdf, if not using dvips). graphicx
- % will default to the driver specified in the system graphics.cfg if no
- % driver is specified.
- % \usepackage[dvips]{graphicx}
- % declare the path(s) where your graphic files are
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- % and their extensions so you won't have to specify these with
- % every instance of \includegraphics
- % \DeclareGraphicsExtensions{.eps}
-\fi
-% graphicx was written by David Carlisle and Sebastian Rahtz. It is
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-% which can result in "jaggedy"/blurry rendering of lines and letters as
-% well as large increases in file sizes.
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-% You can find documentation about the pdfTeX application at:
-% http://www.tug.org/applications/pdftex
-
-
-
-
-
-% *** MATH PACKAGES ***
-%
-%\usepackage{amsmath}
-% A popular package from the American Mathematical Society that provides
-% many useful and powerful commands for dealing with mathematics.
-%
-% Note that the amsmath package sets \interdisplaylinepenalty to 10000
-% thus preventing page breaks from occurring within multiline equations. Use:
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-% *** ALIGNMENT PACKAGES ***
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-% Frank Mittelbach's and David Carlisle's array.sty patches and improves
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-
-% *** SUBFIGURE PACKAGES ***
-%\ifCLASSOPTIONcompsoc
-% \usepackage[caption=false,font=normalsize,labelfont=sf,textfont=sf]{subfig}
-%\else
-% \usepackage[caption=false,font=footnotesize]{subfig}
-%\fi
-% 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
-% and invoke subfig.sty's "caption=false" package option (available since
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-% Note that the Computer Society format requires a larger sans serif font
-% than the serif footnote size font used in traditional IEEE formatting
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-% http://www.ctan.org/pkg/subfig
-
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-% *** FLOAT PACKAGES ***
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-%\usepackage{fixltx2e}
-% fixltx2e, the successor to the earlier fix2col.sty, was written by
-% Frank Mittelbach and David Carlisle. This package corrects a few problems
-% in the LaTeX2e kernel, the most notable of which is that in current
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-% Be aware that LaTeX2e kernels dated 2015 and later have fixltx2e.sty's
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-% needed.
-% The latest version and documentation can be found at:
-% http://www.ctan.org/pkg/fixltx2e
-
-
-%\usepackage{stfloats}
-% stfloats.sty was written by Sigitas Tolusis. This package gives LaTeX2e
-% the ability to do double column floats at the bottom of the page as well
-% as the top. (e.g., "\begin{figure*}[!b]" is not normally possible in
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-% to enable the placement of footnotes below bottom floats (the standard
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-% packages (also by Sigitas Tolusis) as the IEEE does not format its papers in
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-%\ifCLASSOPTIONcaptionsoff
-% \usepackage[nomarkers]{endfloat}
-% \let\MYoriglatexcaption\caption
-% \renewcommand{\caption}[2][\relax]{\MYoriglatexcaption[#2]{#2}}
-%\fi
-% 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
-% figures/tables without any captions are placed on a page by themselves at
-% the end of the document. If needed, the draftcls IEEEtran class option or
-% \CLASSINPUTbaselinestretch interface can be used to increase the line
-% spacing as well. Be sure and use the nomarkers option of endfloat to
-% prevent endfloat from "marking" where the figures would have been placed
-% in the text. The two hack lines of code above are a slight modification of
-% that suggested by in the endfloat docs (section 8.4.1) to ensure that
-% the full captions always appear in the list of figures/tables - even if
-% the user used the short optional argument of \caption[]{}.
-% 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
-% captions of packages such as subfig.sty that lack options to turn off
-% the subcaptions:
-% For subfig.sty:
-% \let\MYorigsubfloat\subfloat
-% \renewcommand{\subfloat}[2][\relax]{\MYorigsubfloat[]{#2}}
-% However, the above trick will not work if both optional arguments of
-% the \subfloat command are used. Furthermore, there needs to be a
-% description of each subfigure *somewhere* and endfloat does not add
-% subfigure captions to its list of figures. Thus, the best approach is to
-% avoid the use of subfigure captions (many IEEE journals avoid them anyway)
-% and instead reference/explain all the subfigures within the main caption.
-% The latest version of endfloat.sty and its documentation can obtained at:
-% http://www.ctan.org/pkg/endfloat
-%
-% The IEEEtran \ifCLASSOPTIONcaptionsoff conditional can also be used
-% later in the document, say, to conditionally put the References on a
-% page by themselves.
-
-
-
-
-% *** PDF, URL AND HYPERLINK PACKAGES ***
-%
-%\usepackage{url}
-% url.sty was written by Donald Arseneau. It provides better support for
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-% http://www.ctan.org/pkg/url
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-
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-
-% *** 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
% 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{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}
-% 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.
%
% make the title area
-\maketitle
+%\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}
+% 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}
%
% For peerreview papers, this IEEEtran command inserts a page break and
% creates the second title. It will be ignored for other modes.
-\IEEEpeerreviewmaketitle
+\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}
\section{Preliminaries}\label{sec:preliminaries}
\input{preliminaries}
-\section{Proof Of Chaos}\label{sec:proofOfChaos}
+\section{Proof of Chaos}\label{sec:proofOfChaos}
\input{chaos}
\section{Functions with Strongly Connected $\Gamma_{\{b\}}(f)$}\label{sec:SCCfunc}
%\acknowledgements{...}
-\bibliographystyle{IEEEtran}
+\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}
% Can use something like this to put references on a page
% by themselves when using endfloat and the captionsoff option.
-\ifCLASSOPTIONcaptionsoff
- \newpage
-\fi
% that's all folks
\end{document}
-
-
