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\begin{abstract}
-Information hiding schemes are studied
-Spread-spectrum data-hiding techniques have been widely studied in recent years
-under the scope of security. These techniques encompass several schemes, such as
-Improved Spread Spectrum~(ISS), Circular Watermarking~(CW), and Natural
-Watermarking~(NW). Some of these schemes have revealed various
-security issues. On the contrary, it has
-been proven in~\cite{Cayre2008} that the Natural Watermarking technique is
-stego-secure. This stego-security is one of the security classes defined
-in~\cite{Cayre2008}, where probabilistic models are used to
-categorize the security of data hiding algorithms in the Watermark Only
-Attack~(WOA) framework.
-
-We have explained in our previous research works~\cite{guyeux13:bc}
-that any algorithm can be rewritten as an iterative
-process, leading to the possibility to study its
-topological behavior. As a concrete example,
-we have shown that the security level of some information hiding algorithms
-(of the spread-spectrum kind) can be studied into a
-novel framework based on unpredictability, as it is understood in the mathematical
-theory of chaos~\cite{guyeux13:bc}.
-The key idea motivating our research works is that: \emph{if artificial intelligence (AI)
-tools seem to have difficulties to deal with chaos, then steganalyzers (software based
-on AI that try to separate original from stego-contents) may
-be proven defective against chaotic information hiding schemes}. Our work
-has thus constituted in showing theoretically that such chaotic schemes can be constructed.
-We are not looking to struggle with best available information hiding techniques and
-we do not focus on effective and operational aspects, as
-our questioning are more locating in a conceptual domain. Among other things, we do
-not specify how to chose embedding coefficients, but the way to insert the hidden
-message in a selection of these ``least significant coefficient'' in an unpredictable
-manner. To say this another
-way, our intention is not to realize an hidden channel that does not appear as sleazy
-to a steganalyzer, but to construct an information hiding scheme whose behavior cannot
-be predicted: supposing that the adversary has anything (algorithm, possible embedding
-coefficient, etc.) but the secret key, we want to determine if he can predict which coefficients
-will be finally used, and in which order. To do so, a new class of security has been
-introduced in~\cite{bg10b:ip}, namely the topological security. This new class can be used to study
-some categories of attacks that are difficult to investigate in the existing
-security approach. It also enriches the variety of qualitative and quantitative
-tools that evaluate how strong the security is, thus reinforcing the confidence
-that can be added in a given scheme.
-
-In addition of being stego-secure, we have proven in~\cite{gfb10:ip} that
-Natural Watermarking (NW) technique is topologically secure. Moreover, this technique
-possesses additional properties of unpredictability, namely, strong transitivity,
-topological mixing, and a constant of sensitivity equal to $\frac{N}{2}$~\cite{Guyeux2012}.
-However NW are not expansive, which is in our opinion problematic in the Constant-Message
-Attack (CMA) and Known Message Attack (KMA) setups, when we
-consider that the attacker has all but the embedding key~\cite{Guyeux2012}.
-Since these initial investigations, our research works in that information hiding field have
-thus consisted in searching more secure schemes than NW, regarding the
-concerns presented in the first paragraph of this introduction. The objective
-of this review paper is to list the results obtained by following such an approach.
-
-This article is organized as follows. Notations and terminologies are firstly recalled in
-the next section. Then the formerly published $\mathcal{CIW}_1$ chaotic iteration based one-bit watermarking
-process is recalled in detail in Section~\ref{sec:ciw1}. Its steganographic version $\mathcal{CIS}_2$
-is then explained in Section~\ref{sec:secrypt11}, while Section~\ref{di3sec} presents the
-$\mathcal{DI}_3$ process, whose aims is to merge the two previous approaches.
-This review article of chaotic iterations based information hiding algorithms
-ends by a conclusion section containing intended future works.
-
+Steganographic Information Hiding (IH) schemes
+are in competition with Artificial Intelligence based
+Steganalysers.
+However, such kind of latter approaches suffer for dealing with
+unpredictable behaviors.
+The open problem is how defective they can be
+proven against IH schemes based on mathematical chaos,
+since this framework is theoretically and practically unpredictable.
+This article addresses this problem and presents a review of IH schemes
+based on Devaney's mathematical chaos.
\end{abstract}
\input{IH/intro}
-\contentsline {Def}{{Definition}{1}{}}{3}
-\contentsline {Def}{{Definition}{2}{}}{3}
-\contentsline {Def}{{Definition}{3}{}}{3}
-\contentsline {Def}{{Definition}{4}{Devaney's chaos}}{3}
-\contentsline {Def}{{Definition}{5}{}}{3}
-\contentsline {Def}{{Definition}{6}{}}{3}
-\contentsline {Th}{{Theorem}{1}{}}{4}
-\contentsline {Def}{{Definition}{7}{}}{5}
-\contentsline {Ex}{{Example}{1}{}}{5}
-\contentsline {Def}{{Definition}{8}{}}{5}
-\contentsline {Def}{{Definition}{9}{PLCM}}{7}
-\contentsline {Def}{{Definition}{10}{Security~\cite {Kalker2001}}}{8}
-\contentsline {Def}{{Definition}{11}{Robustness~\cite {Kalker2001}}}{8}
-\contentsline {Def}{{Definition}{12}{}}{8}
-\contentsline {Def}{{Definition}{13}{}}{8}
-\contentsline {Def}{{Definition}{14}{}}{8}
-\contentsline {Def}{{Definition}{15}{}}{8}
-\contentsline {Def}{{Definition}{16}{Stego-Security~\cite {Cayre2008}}}{9}
-\contentsline {Prop}{{Proposition}{1}{}}{9}
-\contentsline {Def}{{Definition}{17}{}}{10}
-\contentsline {Prop}{{Proposition}{2}{}}{10}
-\contentsline {Def}{{Definition}{18}{}}{11}
-\contentsline {Def}{{Definition}{19}{}}{11}
-\contentsline {Def}{{Definition}{20}{}}{11}
-\contentsline {Prop}{{Proposition}{3}{}}{12}
-\contentsline {Pre}{{Proof}{1}{}}{12}
-\contentsline {Prop}{{Proposition}{4}{}}{13}
-\contentsline {Prop}{{Proposition}{5}{}}{13}
-\contentsline {Th}{{Theorem}{2}{}}{13}
-\contentsline {Prop}{{Proposition}{6}{}}{14}
-\contentsline {Def}{{Definition}{21}{}}{18}
-\contentsline {Def}{{Definition}{22}{}}{18}
-\contentsline {Def}{{Definition}{23}{}}{19}
-\contentsline {Def}{{Definition}{24}{}}{19}
-\contentsline {Def}{{Definition}{25}{}}{20}
-\contentsline {Prop}{{Proposition}{7}{}}{20}
-\contentsline {Th}{{Theorem}{3}{}}{20}
-\contentsline {Prop}{{Proposition}{8}{}}{21}
-\contentsline {Th}{{Theorem}{4}{}}{21}
-\contentsline {Th}{{Theorem}{5}{}}{21}
-\contentsline {Th}{{Theorem}{6}{}}{22}
-\contentsline {Rem}{{Rem}{1}{}}{22}
-\contentsline {Def}{{Definition}{26}{}}{23}
-\contentsline {Def}{{Definition}{27}{}}{23}
-\contentsline {Rem}{{Rem}{2}{}}{23}
-\contentsline {Def}{{Definition}{28}{$\mathcal {DI}_3$ Data hiding scheme}}{23}
-\contentsline {Prop}{{Proposition}{9}{}}{23}
-\contentsline {Rem}{{Rem}{3}{}}{25}
+\contentsline {Def}{{Definition}{1}{}}{4}
+\contentsline {Def}{{Definition}{2}{}}{4}
+\contentsline {Def}{{Definition}{3}{}}{4}
+\contentsline {Def}{{Definition}{4}{Devaney's chaos}}{4}
+\contentsline {Def}{{Definition}{5}{}}{4}
+\contentsline {Def}{{Definition}{6}{}}{5}
+\contentsline {Th}{{Theorem}{1}{}}{5}
+\contentsline {Def}{{Definition}{7}{}}{6}
+\contentsline {Ex}{{Example}{1}{}}{6}
+\contentsline {Def}{{Definition}{8}{}}{6}
+\contentsline {Def}{{Definition}{9}{PLCM}}{8}
+\contentsline {Def}{{Definition}{10}{Security~\cite {Kalker2001}}}{9}
+\contentsline {Def}{{Definition}{11}{Robustness~\cite {Kalker2001}}}{9}
+\contentsline {Def}{{Definition}{12}{}}{9}
+\contentsline {Def}{{Definition}{13}{}}{10}
+\contentsline {Def}{{Definition}{14}{}}{10}
+\contentsline {Def}{{Definition}{15}{}}{10}
+\contentsline {Def}{{Definition}{16}{Stego-Security~\cite {Cayre2008}}}{10}
+\contentsline {Prop}{{Proposition}{1}{}}{11}
+\contentsline {Def}{{Definition}{17}{}}{11}
+\contentsline {Prop}{{Proposition}{2}{}}{11}
+\contentsline {Def}{{Definition}{18}{}}{12}
+\contentsline {Def}{{Definition}{19}{}}{13}
+\contentsline {Def}{{Definition}{20}{}}{13}
+\contentsline {Prop}{{Proposition}{3}{}}{13}
+\contentsline {Pre}{{Proof}{1}{}}{13}
+\contentsline {Prop}{{Proposition}{4}{}}{14}
+\contentsline {Prop}{{Proposition}{5}{}}{14}
+\contentsline {Th}{{Theorem}{2}{}}{14}
+\contentsline {Prop}{{Proposition}{6}{}}{15}
+\contentsline {Def}{{Definition}{21}{}}{19}
+\contentsline {Def}{{Definition}{22}{}}{19}
+\contentsline {Def}{{Definition}{23}{}}{20}
+\contentsline {Def}{{Definition}{24}{}}{20}
+\contentsline {Def}{{Definition}{25}{}}{21}
+\contentsline {Prop}{{Proposition}{7}{}}{21}
+\contentsline {Th}{{Theorem}{3}{}}{21}
+\contentsline {Prop}{{Proposition}{8}{}}{22}
+\contentsline {Th}{{Theorem}{4}{}}{22}
+\contentsline {Th}{{Theorem}{5}{}}{22}
+\contentsline {Th}{{Theorem}{6}{}}{23}
+\contentsline {Rem}{{Rem}{1}{}}{23}
+\contentsline {Def}{{Definition}{26}{}}{24}
+\contentsline {Def}{{Definition}{27}{}}{24}
+\contentsline {Rem}{{Rem}{2}{}}{24}
+\contentsline {Def}{{Definition}{28}{$\mathcal {DI}_3$ Data hiding scheme}}{24}
+\contentsline {Prop}{{Proposition}{9}{}}{25}
+\contentsline {Rem}{{Rem}{3}{}}{26}
