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\author{
  Jean-Fran\c cois Couchot, Christophe Guyeux, and 
  Jacques M. Bahi,~\IEEEmembership{Senior Member,~IEEE}\\
  FEMTO-ST Institute, UMR 6174 CNRS\\
  DISC Department, University of Franche-Comt\'{e}\\
  Belfort, France\\
  \{jean-francois.couchot, christophe.guyeux, jacques.bahi\}@femto-st.fr\\
}


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\title{Mathematical topology: a new practicable framework for
studying information-hiding security. 
Application to Spread-Spectrum schemes.}




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\begin{abstract}
Information  hiding security is often expressed as a probability problem.
However, various classes of attacks cannot currently be addressed, 
due to strong hypotheses not compatible with  a probabilistic approach.   
In this work, a complementary theoretical framework is presented to
improve security. Contrary to existing ones, it is not based
on probability theory, but on mathematical topology.
It addresses thus security issues in classes of
attacks that are not currently  studied.
It can also  be used to  reinforce the
confidence  in   a  new scheme.
In this paper, first the theoretical framework  of the study is presented,
then some concrete examples are detailed  in order to show how our approach
can be applied.
\end{abstract}



\begin{IEEEkeywords}
Information Hiding Security,
 Mathematical Theory of Chaos, 
Spread-Spectrum, Discrete Dynamical Systems,
Chaotic Iterations
\end{IEEEkeywords}








\section{Introduction}  
\input{intro}




\section{Related Work and Contributions}
\label{Refs}
\input{refs}





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\section{Chaos for Data Hiding Security}
\label{section:Chaos}

This section starts with a state of the art in chaos-based information hiding 
(Sec.~\ref{subsection:ChaosInComputerScience}). 
It reminds the readers of the theory of chaos as introduced by Devaney 
(Sec.~\ref{subsection:Devaney}).
Other qualitative and quantitative properties are next introduced 
(Sec.~\ref{subsection:properties}).
Their application to information hiding concludes this section (Sec.~\ref{subsection:links}).  

\subsection{State of the Art}
\label{subsection:ChaosInComputerScience} 
\input{art.tex}

\subsection{Devaney's Chaotic Dynamical Systems}
\label{subsection:Devaney}
\input{devaney}

\subsection{Qualitative and Quantitative 
Properties of Discrete Dynamical  Systems}
\label{subsection:properties}
\input{properties}

\subsection{Chaos Properties and Information Hiding Security}
\label{subsection:links}
\input{relations}


\section{Chaos-Security of two Data Hiding Schemes}
\label{CS}

To check whether an existing data hiding scheme is chaos-secure, we 
first  write  it as an iterate process $X^{n+1}=f(X^n)$
defined  on   the  set  $\mathcal{X}$,  with  $X^0$   as  the  initial
configuration of the machine.  
Let then $\mathcal{T}(S)$  be the iterative process of a data hiding scheme $S$ 
and $\tau$  be a  topology on the topological space $\mathcal{X}$.   
If  $\mathcal{T}(S)$ has a chaotic  behavior on $\mathcal{X}$,
as  defined by  Devaney, $S$ is said
to be  \emph{chaos-secure} on  $(\mathcal{X},\tau)$.

This section studies two classes of
data hiding schemes in the perspective
of chaos theory.  




\subsection{Spread-Spectrum Data Hiding Schemes}
\label{SS}
\input{spreadspectrum}


\subsection{dhCI: Chaos-based  Expansive Data Hiding Schemes}
\label{sec:Algo}
\input{dhci}



\section{Discussion and Future Work}
\input{conclusion}


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