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2 %\documentclass{comjnl}
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7 % Added packages (Nicolas Friot)
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11 \usepackage{amsfonts}
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12 \usepackage{dsfont} %Pour mathds
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14 \usepackage{stmaryrd}
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16 \usepackage{multirow}
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17 %\usepackage{graphicx}
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23 \usepackage[english]{varioref}
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25 %%%%%%%%%%%% ATTENTION : A VIRER POUR LA SOUMISSION
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26 \usepackage[linkcolor=blue,colorlinks=true,citecolor=blue]{hyperref}
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27 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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32 %\copyrightyear{2009} \vol{00} \issue{0} \DOI{000}
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38 %\title[Modelling Bidders in Sequential Automated Auctions]{Modelling Bidders in
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39 %Sequential Automated Auctions}
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41 \title[Information Hiding by Chaotic Iterations, Security and
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42 Robustness]{Information Hiding by Chaotic Iterations: Security and Robustness}
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44 %\author{Kumaara Velan}
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46 % \author{Jacques M. Bahi\textsuperscript{1}, Nicolas Friot\textsuperscript{1},
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47 % Christophe Guyeux\textsuperscript{1}, and Kamel
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48 % Mazouzi\textsuperscript{2}~*\\\tiny * Authors in alphabetic order.}
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50 \author{* Jacques M. Bahi}
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51 \email{\{jacques.bahi,christophe.guyeux,kamel.mazouzi\}@univ-fcomte.fr\\nicolas.friot@lifc.univ-fcomte.fr\\\vspace{0.5cm}*
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52 Authors are cited in alphabetic order.}
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54 \author{Nicolas Friot}
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55 \author{Christophe Guyeux}
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57 \affiliation{Computer Science Laboratory LIFC}
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58 \author{Kamel Mazouzi}
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59 \affiliation{M\'esocentre de calcul de Franche-Comt\'e\\University of
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60 Franche-Comt\'e, 16 route de Gray, Besan\c con, France\\
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63 %\\\tiny Authors in alphabetic order.
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65 % \affiliation{Intelligent Systems and Networks Group, Department of
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66 % Electrical and Electronic Engineering, Imperial College, London
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67 % SW7 2BT, UK} \email{kumaara.velan@imperial.ac.uk}
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69 % \affiliation{\textsuperscript{1}Computer Science Laboratory LIFC\\
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70 % %, University of Franche-Comt\'e\\
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71 % %16 route de Gray, Besan\c con, France\\
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73 % \textsuperscript{2}M\'esocentre de calcul de Franche-Comt\'e\\
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74 % University of Franche-Comt\'e, 16 route de Gray, Besan\c con, France
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77 %\email{\{jacques.bahi,christophe.guyeux,kamel.mazouzi\}@univ-fcomte.fr\\nicolas.friot@lifc.univ-fcomte.fr}
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80 %\shortauthors{K. Velan}
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81 \shortauthors{J. M. Bahi, N. Friot, C. Guyeux, and K. Mazouzi}
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82 % \shortauthors{N. Friot}
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83 % \shortauthors{C. Guyeux}
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84 % \shortauthors{K. Mazouzi}
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86 % \received{00 January 2009}
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87 % \revised{00 Month 2009}
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92 %\category{C.2}{Computer Communication Networks}{Computer Networks}
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93 %\category{C.4}{Performance of Systems}{Analytical Models}
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94 %\category{G.3}{Stochastic Processes}{Queueing Systems}
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95 %\terms{Internet Technologies, E-Commerce}
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96 % \keywords{Automated Auctions; Analytical Models; Autonomic
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97 % Systems; Internet Technologies; E-Commerce; Queueing Systems}
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99 \keywords{Information hiding; Chaos; Security; Robustness.
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105 In this paper, recent contributions in the field of chaos-based information hiding security is summed up and deepened.
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106 Additionally, two variants of an information hiding scheme based on chaotic iterations are recalled and their security proofs are completed.
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107 Among other things, new evaluations of qualitative and quantitative properties of security concerning the improved version of our algorithm are proposed.
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108 The security notions used here encompass the probabilistic stego-security against watermark-only attacks, and the topological chaos-security to face known message, known original, and constant-message attacks.
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109 Finally, a complete and original evaluation of the robustness of the first variant in spatial domain is given, encompassing several kinds of attacks against a set of hundreds of watermarked medias.
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115 %%%%%%%%%%%% ATTENTION : A VIRER POUR LA SOUMISSION
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121 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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123 \section{Introduction}\label{sec:introduction}
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125 \textcolor{red}{\textbf{REMPLACER chaos par topologie si possible}}
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127 \textcolor{red}{\textbf{A FAIRE: dans le tableau des smileys, évaluer la
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128 complexité des algorithmes et l'évaluation des temps d'exécution}}
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130 \textcolor{red}{\textbf{ENVISAGER DE PARLER DE $\mathcal{DI}_1$ POUR LE
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131 COMPARER AUX AUTRES}}
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135 \textbf{Begin TODO}
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137 \item La robustesse est relative au type de filigrane considéré
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138 \item Regarder la fonction d'extraction
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139 \item Comparaison avec des algorithmes connus comme étant robustes (ou
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140 l'étalement de spectre)
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141 \item Mettre en évidence une échèlle de comparaison (comme pour les images)
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149 Information hiding has recently become a major security technology, especially with the increasing importance and widespread distribution of digital media through the Internet.
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150 It includes several techniques, among which are digital watermarking and steganography. On the one hand, the aim of digital watermarking is to embed a piece of information into digital documents, like pictures or movies, for a large panel of reasons, such as: copyright protection, control utilization, data description, integrity checking, or content authentication. Digital watermarking must have an essential characteristic called robustness against attacks. On the other hand, steganography consisting in the deployment of an hidden channel in public exchanges, requires security properties guaranteeing that the channel is undetectable.
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152 Robustness and security are thus two major concerns in information hiding.
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153 Even if security and robustness are neighboring concepts without clearly established definitions~\cite{Perez-Freire06}, robustness is often considered to be mostly concerned with blind elementary attacks, whereas security is not limited to certain specific attacks.
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154 Indeed, security encompasses robustness and intentional attacks~\cite{Kalker2001,ComesanaPP05bis}.
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155 The best attempt to give an elegant and concise definition for each of these two terms was proposed in \cite{Kalker2001}.
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156 Following Kalker, we will consider in this research work that:
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158 watermarking is a mechanism to create a communication channel that is
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159 multiplexed into original content [...]. It is required that, firstly, the
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160 perceptual degradation of the marked content [...] is minimal and, secondly,
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161 that the capacity of the watermark channel degrades as a smooth function of the
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162 degradation of the marked content. [...]. Watermarking security refers to the
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163 inability by unauthorized users to have access to the raw watermarking channel.
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164 [...] to remove, detect and estimate, write or modify the raw watermarking
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168 In the framework of watermarking and steganography, security has seen several important developments since the last decade~\cite{BarniBF03,Cayre2005,Ker06}.
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169 The first fundamental work in security was made by Cachin in the context of steganography~\cite{Cachin2004}.
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170 Cachin interprets the attempts of an attacker to distinguish between an innocent image and a stego-content as a hypothesis testing problem.
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171 In this document, the basic properties of a stegosystem are defined using the notions of entropy, mutual information, and relative entropy.
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172 Mittelholzer, inspired by the work of Cachin, proposed the first theoretical framework for analyzing the security of a watermarking scheme~\cite{Mittelholzer99}.
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173 These efforts to bring a theoretical framework for security in steganography and watermarking have been followed up by Kalker, who tries to clarify the concepts (robustness versus security), and the classifications of watermarking attacks~\cite{Kalker2001}.
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174 This work has been deepened by Furon \emph{et al.}, who have translated Kerckhoffs' principle (Alice and Bob shall only rely on some previously shared secret for privacy), from cryptography to data hiding~\cite{Furon2002}.
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175 They used Diffie and Hellman methodology, and Shannon's cryptographic framework~\cite{Shannon49}, to classify the watermarking attacks into categories, according to the type of information Eve has access to~\cite{Cayre2005,Perez06}, namely: Watermarked Only Attack (WOA), Known Message Attack (KMA), Known Original Attack (KOA), and Constant-Message Attack (CMA).
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176 Levels of security have been recently defined in these setups.
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177 The highest level of security in WOA is called stego-security \cite{Cayre2008}, whereas chaos-security tends to improve the ability to withstand attacks in KMA, KOA, and CMA setups \cite{gfb10:ip}.
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180 This paper is both an overview of our contributions in the field of information hiding~\cite{gfb10:ip,guyeuxThese10,guyeux10ter}, an extension of our researches concerning the security of chaotic iterations based schemes, and a complete study of their robustness. In \cite{gfb10:ip,guyeuxThese10}, we have shown that the security level of such algorithms can be studied into a novel framework based on unpredictability, as it is understood in the mathematical theory of chaos~\cite{Devaney}. To do so, a new class of security has been introduced, namely the chaos-security. This new class can be used to study some categories of attacks that are difficult to investigate in the existing security approach: namely, the KMA, KOA, and CMA setups. It also enriches the variety of qualitative and quantitative tools that evaluate how strong the security is, thus reinforcing the confidence that can be had in a given scheme.
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182 We have applied this framework in concrete examples. For instance, we have proven in~\cite{guyeuxThese10,ih10} that Natural Watermarking (NW) technique is chaos-secure. Moreover, this technique possesses additional properties of unpredictability, namely, strong transitivity, topological mixing, and a constant of sensitivity equal to $\frac{N}{2}$. However NW are not expansive, which is problematic in the CMA and KMA setups~\cite{guyeuxThese10,ih10}.
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183 This is why a new information hiding scheme has been introduced in~\cite{guyeux10ter}, which is based on the so-called chaotic iterations. Its security has been established in \cite{gfb10:ip}. However, this former algorithm allow to embed only one bit. So this scheme has been improved in \cite{fgb11:ip}, in a version that allows to embed $n$ bits, while preserving the highest levels of security.
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185 In this research work, all of these contributions are regrouped together. This summary is the first contribution of this paper. The security study of the new version of our information hiding scheme is deepened, by giving qualitative and quantitative measures of its level of chaos-security. This second contribution encompasses the evaluation of the constant of sensibility, and the qualitative properties of strong transitivity and indecomposability. Finally, a complete and rigorous evaluation of the robustness of the algorithm is proposed as a third contribution. This study has been initiated in \cite{guyeux10ter}, but only on one given image and for a small number of attacks. We have extend this work by using hundreds of images and a large variety of parameters and attacks.
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187 The remainder of this document is organized as follows.
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188 In Section \ref{sec:basic-recalls}, some basic recalls concerning both chaotic iterations and Devaney's chaos are given, and the relation unifying them is recalled.
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189 In Section \ref{section:IH based on CIs} is presented the first version, denoted by $CI_1$, of our information hiding scheme.
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190 The next section is devoted to the security study of $CI_1$, encompassing both the stego-security and the chaos-security.
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191 Then, in Section \ref{section:robustness-ci-1}, a complete and rigorous evaluation of the robustness of $CI_1$, realized by using the ``M\'{e}socentre de calcul de Franche-Comt\'{e}'', is given.
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193 This research work ends by a conclusion section where our contribution is
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194 summarized and intended future researches are presented.
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200 \section{Basic Recalls}
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201 \label{sec:basic-recalls}
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204 \subsection{Chaotic Iterations}
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205 \label{sec:chaotic-iterations}
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207 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and
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208 $V_{i}$ is for the $i^{th}$ component of a vector $V$. Finally, the following notation
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209 is used: $\llbracket0;N\rrbracket=\{0,1,\hdots,N\}$.\newline
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211 Let us consider a \emph{system} of a finite number $\mathsf{N}$ of elements (or
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212 \emph{cells}), so that each cell has a boolean \emph{state}. A sequence of length
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213 $\mathsf{N}$ of boolean states of the cells corresponds to a particular
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214 \emph{state of the system}. A sequence that elements belong into $\llbracket
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215 0;\mathsf{N-1} \rrbracket $, \emph{i.e.}, an element of $\llbracket 0; \mathsf{N-1}\rrbracket^\mathds{N}$, is called a \emph{strategy}. The set of all
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216 strategies is denoted by $\mathbb{S}.$
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219 \label{Def:chaotic iterations}
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220 The set $\mathds{B}$ denoting $\{0,1\}$, let
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221 $f:\mathds{B}^{\mathsf{N}}\longrightarrow \mathds{B}^{\mathsf{N}}$ be a function
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222 and $S\in \mathbb{S}$ be a strategy.
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223 The so-called \emph{chaotic iterations} are
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224 defined by $x^0\in \mathds{B}^{\mathsf{N}}$ and $\forall (n,i) \in
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225 \mathds{N}^{\ast} \times \llbracket0;\mathsf{N-1}\rrbracket$:
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227 %\forall n\in \mathds{N}^{\ast }, \forall i\in \llbracket1;\mathsf{N}\rrbracket
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231 x_i^{n-1} & \text{ if }S^n\neq i, \\
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232 \left(f(x^{n-1})\right)_{S^n} & \text{ if }S^n=i.\end{array}\right.\end{equation*}
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236 In other words, at the $n^{th}$ iteration, only the $S^{n}-$th cell is
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237 \textquotedblleft iterated\textquotedblright . Note that in a more general
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238 formulation, $S^n$ can be a subset of components and $f(x^{n-1})_{S^{n}}$ can
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239 be replaced by $f(x^{k})_{S^{n}}$, where $k<n$, describing for example, delays
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240 transmission (see \cite{Bahi2002,bcv06:ij}). For the
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241 general definition of such chaotic iterations, see,
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242 \emph{e.g.},~\cite{Robert1986}.
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246 \subsection{Devaney's chaotic dynamical systems}
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247 \label{sec:Devaney}
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249 Some topological definitions and properties taken from the
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250 mathematical theory of chaos are recalled in this section.
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251 %It is important to understand these notions which will be used in
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252 %the rest of this article especially in the Section~\ref{sec:SCISMM-topological-model}~\vpageref{sec:SCISMM-topological-model} and the Section~\ref{sec:SCISMM-chaos-security}~\vpageref{sec:SCISMM-chaos-security}.
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254 %\subsubsection{Notations}
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257 % \item Let $(\mathcal{X},d)$ be a metric space.
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258 % \item Let $f$ be a continuous function on $(\mathcal{X},d)$.
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261 Let $(\mathcal{X},d)$ be a metric space and $f$ a continuous function on $(\mathcal{X},d)$.
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263 %\subsubsection{Topological definitions used in chaos}
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265 \begin{definition}[Topological transitivity]
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266 $f$ is said to be \emph{topologically transitive} if, for any pair
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267 of open sets $U,V\subset \mathcal{X}$, there exists $k>0$ such that $f^{k}(U)\cap
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268 V\neq\varnothing $.
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269 \label{def:chaos-devaney}
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272 \begin{definition}[Regularity]
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273 $(\mathcal{X},f)$ is said to be \emph{regular} if the set of
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274 periodic points is dense in $\mathcal{X}$.
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277 \begin{definition}[Sensitivity]
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278 $f$ has \emph{sensitive dependence on initial conditions} if there exists $\delta >0$ such that, for any $x\in
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279 \mathcal{X}$ and any neighborhood $V$ of $x$, there exist $y\in V$ and
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280 $n\geqslant 0$ such that $d\left(f^{n}(x),f^{n}(y)\right)>\delta $.
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282 $\delta $ is called the \emph{constant of sensitivity} of $f$.
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286 It is now possible to introduce the well-established mathematical definition of
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287 chaos formulated by Devaney~\cite{Devaney89},
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289 \begin{definition}[Chaotic function]
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290 A function $f:\mathcal{X}\longrightarrow \mathcal{X}$ is said to be
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291 \emph{chaotic} on $\mathcal{X}$ if:
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293 \item $f$ is regular,
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294 \item $f$ is topologically transitive,
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295 \item $f$ has sensitive dependence on initial conditions.
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299 When $f$ is chaotic, then the system $(\mathcal{X}, f)$ is chaotic and quoting
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300 Devaney: ``it is unpredictable because of the sensitive dependence on initial
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301 conditions. It cannot be broken down or simplified into two subsystems which do
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302 not interact because of topological transitivity. And in the midst of this
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303 random behavior, we nevertheless have an element of regularity''. Fundamentally
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304 different behaviors are consequently possible and occur in an unpredictable
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307 Let us finally remark that,
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310 \begin{theorem}[Banks~\cite{Banks92}]\label{theo:banks}
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311 If a function is regular and topologically transitive on a metric space, then the
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312 function is sensitive on initial conditions.
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315 %The main interest of the Banks theorem is to reduce the properties to check when
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316 %we want to prove that a function on a metric space is chaotic.
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317 %This theorm will be used in the
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318 %Section~\ref{sec:SCISMM-chaos-security}~\vpageref{sec:SCISMM-chaos-security}.
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320 %This theorem reduces the number of proofs when establishing the chaotic behavior of an iterate function on a metric space.
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321 %We will use it, for instance, in Section~\ref{sec:SCISMM-chaos-security}.
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323 Let us now study the original relation between chaotic iterations and Devaney's chaos. This relation will be used in the context of information hiding security.
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325 \subsection{Dynamics of Chaotic Iterations}\label{sec:topological}
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328 \noindent In this section, we give the outline proofs we have previously obtained, establishing the topological properties of chaotic iterations \cite{gfb10:ip,guyeux10ter,bg10:ij}. As our scheme is inspired by this work, the proofs detailed in this document will follow a same canvas.
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330 Let us firstly introduce some notations and terminologies in a view to describe chaotic iterations as an iteration function on a metric space. First of all, we show that CIs consist in the operation of a specific function $F_f$ on some components of a system. This can be described by using a discrete Boolean metric, as follows.
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334 \begin{definition}[Discrete boolean metric]
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335 \label{def:discret-boolean-metric}
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336 The \emph{discrete boolean metric} is the
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337 application $\delta: \mathds{B} \longrightarrow \mathds{B}$ defined by
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338 $\delta(x,y)=0\Leftrightarrow x=y.$
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341 We can now define $F_f$, whose goal is to change the $k-$th component of the state $E$ by using $f$:
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343 \begin{definition}[Function $F_{f}$]
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344 Given a function $f:\mathds{B}^{\mathsf{N}} \rightarrow \mathds{B}^{\mathsf{N}}$, the function \emph{$F_{f}$} is defined by:
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347 F_{f}: & \llbracket 0;\mathsf{N-1}\rrbracket \times
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348 \mathds{B}^{\mathsf{N}} \longrightarrow \mathds{B}^{\mathsf{N}} \\
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349 & (k,E) \longmapsto \left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta
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350 (k,j)}\right)_{j\in \llbracket 0;\mathsf{N-1}\rrbracket} \\
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355 As recalled before, the components to update are given by a ``strategy'', \emph{i.e.}, a sequence of integers bounded by the size of the system. This can be formulated as follows,
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357 \begin{definition}[Strategy]
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358 \label{def:strategy-adapter}
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359 Let $\mathsf{k} \in \mathds{N}^\ast$.
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360 A \emph{strategy} is a sequence which elements belong into $\llbracket 0, \mathsf{k-1} \rrbracket$.
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361 The set of all strategies with terms in $\llbracket 0, \mathsf{k-1} \rrbracket$
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362 is denoted by $\mathbb{S}_\mathsf{k}$.
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365 We thus iterate on the Cartesian product $\mathcal{X}_1$ constituted by the set of states and the set of strategies:
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367 \begin{definition}[Phase space]
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368 The phase space used for chaotic iterations is denoted by $\mathcal{X}_1$ and
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369 defined by $\mathcal{X}_1=\mathbb{S}_\mathsf{N} \times
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370 \mathds{B}^{\mathsf{N}}.$
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373 We can remark that~\cite{bg10:ij},
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375 \begin{proposition}%[Cardinality of $\mathcal{X}_1$]
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376 \label{prop:cardinality-X1}
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377 The phase space $\mathcal{X}_1$ has, at least, the cardinality of the
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383 It still remains to describe CIs as a discrete dynamical system $x^{n+1} = G_f(x^n)$ on $\mathcal{X}_1$ for a function $G_f$ to define.
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384 To do so, we will use $F_f$ and two other functions, namely the \emph{initial} and the \emph{shift} functions.
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385 The initial function returns the first term of a given strategy-adapter whereas the shift function removes it:
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387 \begin{definition}[Initial function]
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388 Let $\mathsf{k} \in \mathds{N}^\ast$.
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389 The \emph{initial function} is the map $i_\mathsf{k}$ defined by:
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391 \begin{array}{cccc}
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392 i_\mathsf{k}: & \mathbb{S}_{\mathsf{k}} & \longrightarrow & \llbracket 0, \mathsf{k-1} \rrbracket \\
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393 & (S^{n})_{n\in \mathds{N}} & \longmapsto & S^{0} \\
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401 \begin{definition}[Shift function]
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402 Let $\mathsf{k} \in \mathds{N}^\ast$.
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403 The \emph{shift function} is the map $\sigma_\mathsf{k}$ defined by:
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405 \begin{array}{cccc}
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406 \sigma_\mathsf{k} : & \mathbb{S}_{\mathsf{k}} & \longrightarrow & \mathbb{S}_\mathsf{k}\\
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407 & (S^{n})_{n\in \mathds{N}} & \longmapsto & (S^{n+1})_{n\in \mathds{N}}
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412 We are now able to introduce the function $G_f$:
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414 \begin{definition}[Map $G_{f}$]
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415 Given a function $f:\mathds{B}^{\mathsf{N}} \rightarrow \mathds{B}^{\mathsf{N}}$, the map \emph{$G_{f}$} is defined by:
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417 \begin{array}{cccc}
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418 G_{f}: & \mathcal{X}_1 & \longrightarrow & \mathcal{X}_1 \\
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419 & \left( S,E\right) & \longmapsto &\left( \sigma_\mathsf{N} (S),F_{f}(i_\mathsf{N}(S),E)\right)
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429 In other words, one iteration consists in a shift of strategy $S$ and a replacement of the $i_\mathsf{N}(S)$ cell with $f(E)_{i_\mathsf{N}(S)}$.
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432 With these definitions, CIs can be described by the following iterations of the discrete dynamical system:
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437 X^{0}\in \mathcal{X}_1\\
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438 \forall k \in \mathds{N}^\ast, X^{k+1}=G_{f}(X^{k})
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444 We need a metric space in order to study the topological behavior of the CIs. This is why a new distance $d_1$ between two points has been defined in~\cite{bg10:ij} by:
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447 \begin{definition}[Distance $d_1$ on $\mathcal{X}_1$]\label{def:distance-on-X1}
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448 $\forall (S,E),(\check{S},\check{E} )\in \mathcal{X}_1,$
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449 $d_1((S,E);(\check{S},\check{E}))=d_{\mathds{B}^{\mathsf{N}}}(E,\check{E})+d_{\mathbb{S}_\mathsf{N}}(S,\check{S}),$
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453 $\displaystyle{d_{\mathds{B}^{\mathsf{N}}}(E,\check{E})}=\displaystyle{\sum_{k=0}^{\mathsf{N-1}}\delta
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454 (E_{k},\check{E}_{k})} \in \llbracket 0 ; \mathsf{N} \rrbracket$
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456 $\displaystyle{d_{\mathbb{S}_\mathsf{N}}(S,\check{S})}=\displaystyle{\dfrac{9}{\mathsf{N}}\sum_{k=0}^{\infty
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457 }\dfrac{|S^{k}-\check{S}^{k}|}{10^{k+1}}} \in [0 ; 1[.$
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459 are respectively two distances on $\mathds{B}^{\mathsf{N}}$ and $\mathbb{S}_\mathsf{N}$
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460 ($\forall \mathsf{N} \in \mathds{N}^\ast$).
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464 This new distance has been introduced in \cite{bg10:ij}
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465 to satisfy the following
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466 requirements. When the number of different cells between two systems is
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467 increasing, then their distance should increase too. In addition, if two systems
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468 present the same cells and their respective strategies start with the same
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469 terms, then the distance between these two points must be small, because the
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470 evolution of the two systems will be the same for a while. The distance presented
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471 above follows these recommendations.
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472 Indeed, if the floor value $\lfloor
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473 d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$ differ in $n$
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474 cells. In addition, $d(X,Y) - \lfloor d(X,Y) \rfloor $ is a measure of the
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475 differences between strategies $S$ and $\check{S}$. More precisely, this floating
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476 part is less than $10^{-k}$ if and only if the first $k$ terms of the two
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477 strategies are equal. Moreover, if the $k^{th}$ digit is nonzero, then the
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478 $k^{th}$ terms of the two strategies are different.
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483 We have proven that,
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485 \begin{proposition}\label{Prop:continuite}
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486 $G_f$ is a continuous function on $(\mathcal{X}_1,d_1)$, for all $f:\mathds{B}^\mathsf{N} \rightarrow \mathds{B}^\mathsf{N}$.
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489 Let us finally recall the iteration function used in \cite{guyeux10ter}.
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492 \begin{definition}%[Vectorial negation]
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493 The \emph{vectorial negation} is the function defined by:
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495 \begin{array}{cccc}
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496 f_{0} : & \mathbb{B}^\mathsf{N} & \longrightarrow & \mathbb{B}^\mathsf{N}\\
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497 & (b_0,\cdots,b_\mathsf{N-1}) & \longmapsto &
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498 (\overline{b_0},\cdots,\overline{b_\mathsf{N-1}})\\
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502 %\begin{definition}[Vectorial Negation]\label{Def:vectorial-negation}
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504 %\begin{array}{cccc}
\r
505 %f_0\ :\ & \mathbb{B}^N & \longrightarrow & \mathbb{B}^N \\
\r
506 %& (b_1,\cdots,b_\mathsf{N}) & \longmapsto &
\r
507 %(\overline{b_1},\cdots,\overline{b_\mathsf{N}})\\
\r
511 %It is then checked in \cite{bg10:ij} that i
\r
512 In the metric space $(\mathcal{X}_1,d_1)$, $G_{f_0}$ satisfies the three
\r
513 conditions for Devaney's chaos: regularity, transitivity, and
\r
514 sensibility to the initial condition. %~\cite{bg10:ij}.
\r
517 %the following result.
\r
518 \begin{theorem}%[$G_{f_0}$ is a chaotic map]
\r
519 $G_{f_0}$ is a chaotic map on $(\mathcal{X}_1,d_1)$ according to the Devaney's formulation.
\r
523 \section{A First CIs Watermarking Process called $CI_1$}
\r
524 \label{section:IH based on CIs}
\r
528 In this section is explained how CIs can be used as an information hiding scheme.
\r
530 \subsection{Most and Least Significant Coefficients}\label{sec:msc-lsc}
\r
534 We first take into account the fact that, in a watermarking process, terms of the original content $x$ that may be replaced by terms issued
\r
535 from the watermark $y$ are less important than other: they could be changed
\r
536 without be perceived as such. More generally, a
\r
537 \emph{signification function}
\r
538 attaches a weight to each term defining a digital media,
\r
539 depending on its position $t$ \cite{todo}.
\r
541 \begin{definition}[Signification function]
\r
542 A \emph{signification function} is a real sequence
\r
543 $(u^k)^{k \in \mathds{N}}$. % with a limit equal to 0.
\r
547 \begin{example}\label{Exemple LSC}
\r
548 Let us consider a set of
\r
549 grayscale images stored into portable graymap format (P3-PGM):
\r
550 each pixel ranges between 256 gray levels, \textit{i.e.},
\r
551 is memorized with eight bits.
\r
552 In that context, we consider
\r
553 $u^k = 8 - (k \mod 8)$ to be the $k$-th term of a signification function
\r
554 $(u^k)^{k \in \mathds{N}}$.
\r
555 Intuitively, in each group of eight bits (\textit{i.e.}, for each pixel)
\r
556 the first bit has an importance equal to 8, whereas the last bit has an
\r
557 importance equal to 1. This is compliant with the idea that
\r
558 changing the first bit affects more the image than changing the last one.
\r
561 \begin{definition}[Significance of coefficients]
\r
562 \label{def:msc,lsc}
\r
563 Let $(u^k)^{k \in \mathds{N}}$ be a signification function,
\r
564 $m$ and $M$ be two reals s.t. $m < M$.
\r
566 \item The \emph{most significant coefficients (MSCs)} of $x$ is the finite
\r
567 vector $$u_M = \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k
\r
568 \geqslant M \textrm{ and } k \le \mid x \mid \right);$$
\r
569 \item The \emph{least significant coefficients (LSCs)} of $x$ is the
\r
571 $$u_m = \left( k ~ \big|~ k \in \mathds{N} \textrm{ and } u^k
\r
572 \le m \textrm{ and } k \le \mid x \mid \right);$$
\r
573 \item The \emph{passive coefficients} of $x$ is the finite vector
\r
574 $$u_p = \left( k ~ \big|~ k \in \mathds{N} \textrm{ and }
\r
575 u^k \in ]m;M[ \textrm{ and } k \le \mid x \mid \right).$$
\r
579 For a given host content $x$,
\r
580 MSCs are then ranks of $x$ that describe the relevant part
\r
581 of the image, whereas LSCs translate its less significant parts.
\r
582 These two definitions are illustrated on Figure~\ref{fig:MSCLSC}, where the significance function $(u^k)$ is defined as in Example \ref{Exemple LSC}, $M=5$, and $m=6$.
\r
584 \begin{figure*}[htb]
\r
587 \begin{minipage}[b]{.98\linewidth}
\r
589 \centerline{\includegraphics[width=6.cm]{img/lena512}}
\r
590 %\centerline{\epsfig{figure=img/lena512.pdf,width=4cm}}
\r
591 \centerline{(a) Original Lena.}
\r
593 \begin{minipage}[b]{.49\linewidth}
\r
595 \centerline{\includegraphics[width=6.cm]{img/lena_msb_678}}
\r
596 %\centerline{\epsfig{figure=img/lena_msb_678.pdf,width=4cm}}
\r
597 \centerline{(b) MSCs of Lena.}
\r
600 \begin{minipage}[b]{0.49\linewidth}
\r
602 \centerline{\includegraphics[width=6.cm]{img/lena_lsb_1234_facteur17}}
\r
604 %\centerline{\epsfig{figure=img/lena_lsb_1234_facteur17.pdf,width=4cm}}
\r
605 \centerline{(c) LSCs of Lena ($\times 17$).}
\r
608 \caption{Most and least significant coefficients of Lena.}
\r
623 \subsection{Presentation of the Scheme}
\r
625 We have proposed in \cite{guyeux10ter} to use chaotic iterations as an information hiding scheme, as follows.
\r
628 \item $(K,N) \in [0;1]\times \mathds{N}$ be an embedding key,
\r
629 \item $X \in \mathbb{B}^\mathsf{N}$ be the $\mathsf{N}$ LSCs of a cover $C$,% $X$ be the initial state $X_0$,
\r
630 \item $(S^n)_{n \in \mathds{N}} \in \llbracket 0, \mathsf{N-1}
\r
631 \rrbracket^{\mathds{N}}$ be a strategy, which depends on the message to hide $M \in [0;1]$ and $K$,
\r
632 \item $f_0 : \mathbb{B}^\mathsf{N} \rightarrow \mathbb{B}^\mathsf{N}$ be the vectorial logical negation.
\r
636 So the watermarked media is $C$ whose LSCs are replaced by $Y_K=X^{N}$, where:
\r
642 \forall n < N, X^{n+1} = G_{f_0}\left(X^n\right).\\
\r
643 \end{array} \right.
\r
647 Two ways to generate $(S^n)_{n \in \mathds{N}}$ are given in \cite{trx}, namely
\r
648 Chaotic Iterations with Independent Strategy~(CIIS) and Chaotic Iterations with Dependent
\r
650 In CIIS, the strategy is independent from the cover media $C$, whereas in CIDS the strategy will be dependent on $C$.
\r
654 Revoir ce qui suit.
\r
656 As we will use the CIIS strategy in this document, we recall it below.
\r
657 Finally, MSCs are not used here, as we do not consider the case of authenticated watermarking.
\r
659 \subsection{CIIS Strategy}
\r
661 Let us firstly give the definition of the Piecewise Linear Chaotic Map~(PLCM, see~\cite{Shujun1}):
\r
663 %\begin{definition}%[PLCM]
\r
664 %The \emph{Piecewise Linear Chaotic Map} is defined by
\r
668 x/p & \text{if} & x \in [0;p], \\
\r
669 (x-p)/(\frac{1}{2} - p) & \text{if} & x \in \left[ p; \frac{1}{2} \right],
\r
671 F(1-x,p) & \text{else,} & \\
\r
672 \end{array} \right.
\r
677 \noindent where $p \in \left] 0; \frac{1}{2} \right[$ is a ``control parameter''.
\r
679 The general term of the strategy $(S^n)_n$ in CIIS setup is defined by
\r
680 the following expression: $S^n = \left \lfloor \mathsf{N} \times K^n \right \rfloor +
\r
686 p \in \left[ 0 ; \frac{1}{2} \right] \\
\r
687 K^0 = M \otimes K\\
\r
688 K^{n+1} = F(K^n,p), \forall n \leq N_0, \end{array} \right.
\r
693 \noindent in which $\otimes$ denotes the bitwise exclusive or (XOR) between two floating part numbers (\emph{i.e.}, between their binary digits representations).
\r
696 \subsection{CIDS Strategy}
\r
698 The same notations as above are used.
\r
699 We define CIDS strategy as follows: $\forall k \leqslant N$,
\r
701 \item if $k \leqslant \mathsf{N}$ and $X^k = 1$, then $S^k=k$,
\r
702 \item else $S^k=1$.
\r
704 In this situation, if $N \geqslant \mathsf{N}$, then only two watermarked contents are possible with the information hiding scheme proposed previously, namely: $Y_K=(0,0,\cdots,0)$ and $Y_K=(1,0,\cdots,0)$.
\r
707 \subsection{PSNR evaluation}\label{section:psnr-ci-1}
\r
709 To realize the evaluation of the PSNR of $CI_1$, we have used the same
\r
710 architecture as described in
\r
711 Section~\ref{section:architecture-presentation}~\vpageref{section:architecture-presentation}.
\r
713 It has to be noted that in the $CI_1$ watermarking process, the embedding key
\r
714 correspond to the strategy associated to the number of iterations.
\r
716 Executing the $CI_1$ process on 1000 images, on the one hand with a constant
\r
717 embedding key and one the other hand with a key generated randomly we have
\r
718 obtain the results described in the
\r
719 table~\ref{table:psnr-ci1}~\vpageref{table:psnr-ci1}, an average of and a standard deviation of has been obtained for the PSNR.\newline
\r
726 \begin{tabular}{|c||c|c|c|}
\r
728 \textbf{Strategy} & \textbf{PSNR average} & \textbf{PSNR standard
\r
732 \textbf{Constant} & $21.6653$ & $0.03604$ \\
\r
734 \textbf{Generated randomly} & $13.3909$ & $10.5690$ \\
\r
739 \caption{Evaluation of the PSNR of the watermarking process $CI_1$.(Tests
\r
740 realized on 1000 images)}
\r
742 \label{table:psnr-ci1}
\r
745 Let us now focus on some security aspects of information hiding schemes. In particular, we aim to explain why chaos is relevant in this field.
\r
747 \section{Information hiding security: $CI_1$}
\r
748 \label{section:IH-security}
\r
750 \subsection{Classification of Attacks}\label{sec:attack-classes}
\r
752 In the steganography framework, attacks have been classified
\r
753 in~\cite{Cayre2008} as follows.
\r
757 Watermark-Only Attack (WOA) occurs when an attacker has only access to several watermarked contents.
\r
761 Known-Message Attack (KMA) occurs when an attacker has access to several pairs of watermarked contents and
\r
762 corresponding hidden messages.
\r
766 Known-Original Attack (KOA) is when an attacker has access to several pairs of watermarked contents and
\r
767 their corresponding original versions.
\r
771 Constant-Message Attack (CMA) occurs when the attacker observes several watermarked contents and only knows that the
\r
772 unknown hidden message is the same in all contents.
\r
775 The synthesis of this classification is illustrated in the
\r
776 Table~\ref{table:attack-classification}~\vpageref{table:attack-classification}.
\r
782 \begin{tabular}{|c||c|c|c|}
\r
784 \textbf{Attack} & \textbf{Original content} & \textbf{Stego
\r
785 content} & \textbf{Hidden message}\\
\r
788 \textbf{Watermark-Only Attack (WOA)} & & $\times$ & \\
\r
790 \textbf{Known-Message Attack (KMA)} & & $\times$ & $\times$ \\
\r
792 \textbf{ Kown-Original Attack (KOA)} & $\times$ & $\times$ & \\
\r
794 \textbf{Constant-Message Attack (CMA)} & & & $\times$ \\
\r
797 \caption{Watermarking attacks classification in context of~\cite{Kalker2001}}
\r
799 \label{table:attack-classification}
\r
803 \subsection{Stego-Security of $CI_1$}\label{sec:stego-security}
\r
805 In the prisoner problem of Simmons~\cite{Simmons83,DBLP:conf/ih/BergmairK06}
\r
807 Figure~\ref{fig:simmons-prisonner-problem}~\vpageref{fig:simmons-prisonner-problem}, Alice and Bob are in jail, and they want to, possibly, devise an escape plan by exchanging hidden messages in innocent-looking cover contents. These messages are to be conveyed to one another by a common warden, Eve, who over-drops all contents and can choose to interrupt the communication if they appear to be stego-contents.
\r
811 %\includegraphics[width=10.5cm]{./img/prisoner-problem.png}
\r
812 \includegraphics[width=15cm]{./img/prisoner-problem}
\r
814 \caption{Simmons' prisoner problem~\cite{Simmons83}}
\r
815 \label{fig:simmons-prisonner-problem}
\r
818 The stego-security, defined in this
\r
819 framework, is the highest security level in WOA setup~\cite{Cayre2008}.
\r
820 To recall it, we need the following notations:
\r
822 \item $\mathds{K}$ is the set of embedding keys,
\r
823 \item $p(X)$ is the probabilistic model of $N_0$ initial host contents,
\r
824 \item $p(Y|K_1)$ is the probabilistic model of $N_0$ watermarked contents.
\r
827 Furthermore, it is supposed in this context that each host content has been watermarked with the same secret key $K_1$ and the same embedding function $e$.
\r
830 It is now possible to define the notion of stego-security:
\r
832 \begin{definition}[Stego-Security]
\r
833 \label{Def:Stego-security}
\r
834 The embedding function $e$ is \emph{stego-secure} if and only if:
\r
835 $$\forall K_1 \in \mathds{K}, p(Y|K_1)=p(X).$$
\r
838 To the best of our knowledge, until now, only two schemes have been proven to be stego-secure.
\r
839 On the one hand, the authors of \cite{Cayre2008} have established that the spread spectrum technique called Natural Watermarking is stego-secure when its distortion parameter $\eta$ is equal to $1$.
\r
840 On the other hand, it has been proven in~\cite{gfb10:ip} that:
\r
842 \begin{proposition}
\r
843 Chaotic Iterations with Independent Strategy (CIIS) are stego-secure.
\r
844 \label{prop:CIIS-stego-security}
\r
850 Let us suppose that $X \sim
\r
851 \mathbf{U}\left(\mathbb{B}^N\right)$ in a CIIS setup.
\r
852 We will prove by a mathematical induction that $\forall n \in \mathds{N}, X^n \sim
\r
853 \mathbf{U}\left(\mathbb{B}^N\right)$. The base case is immediate, as $X^0 = X \sim
\r
854 \mathbf{U}\left(\mathbb{B}^N\right)$. Let us now suppose that the statement
\r
856 \mathbf{U}\left(\mathbb{B}^N\right)$ holds for some $n$.
\r
857 Let $e \in \mathbb{B}^N$ and $\mathbf{B}_k=(0,\cdots,0,1,0,\cdots,0) \in \mathbb{B}^N$ (the digit $1$ is in position $k$).
\r
859 So $P\left(X^{n+1}=e\right)=\sum_{k=1}^N
\r
860 P\left(X^n=e+\mathbf{B}_k,S^n=k\right).$
\r
862 These two events are independent in CIIS setup, thus:
\r
864 $P\left(X^{n+1}=e\right)=\sum_{k=1}^N
\r
865 P\left(X^n=e+\mathbf{B}_k\right) \times P\left(S^n=k\right)$.
\r
867 %According to the recurrence hypothesis ($X^n \sim
\r
868 %\mathbf{U}\left(\mathbb{B}^N\right)$):
\r
869 According to the inductive hypothesis:
\r
872 %P\left(X^{n+1}=e\right) & =\sum_{k=1}^N
\r
873 %\frac{1}{2^N} \times P\left(S^n=k\right) \\
\r
874 %& =\frac{1}{2^N} \sum_{k=1}^N
\r
875 % P\left(S^n=k\right)\\
\r
879 $P\left(X^{n+1}=e\right)=\frac{1}{2^N} \sum_{k=1}^N
\r
880 P\left(S^n=k\right)$.
\r
882 The set of events $\left \{ S^n=k \right \}$ for $k \in \llbracket 1;N
\r
883 \rrbracket$ is a partition of the universe of possible, so
\r
884 $\sum_{k=1}^N P\left(S^n=k\right)=1$.
\r
886 %Evaluate now $ P\left(S^n=k\right)$.
\r
888 %We have: $K^0 = M \otimes K \sim
\r
889 %\mathbf{U}\left([0;1]\right)$, so $K^\prime=F\left( K^0,p \right) \sim
\r
890 %\mathbf{U}\left([0;1]\right)$, because for PLCMs, ``Uniform input generate
\r
891 %uniform output''~\cite{Lasota}.
\r
892 %Well with an immediate proof by recurrence we have:
\r
893 %$K^n \sim \mathbf{U}\left([0;1]\right)$. As $S^n = \left \lfloor N \times X^n
\r
894 %\right \rfloor + 1$ we can deduce that: $S^n \sim
\r
895 %\mathbf{U}\left([1;N]\right)$. Therefore: $ P\left(S^n=k\right)=\frac{1}{N}$.
\r
898 %\begin{array}{llr}
\r
899 %P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N
\r
900 % P\left(S^n=k\right) & \\
\r
901 % & =\frac{1}{2^N} \sum_{k=1}^N \frac{1}{N} & \\
\r
902 % & =\frac{1}{2^N} & \\
\r
906 %$\begin{array}{llr}
\r
907 %P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N
\r
908 % P\left(S^n=k\right) & \\
\r
909 %P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N \frac{1}{N} &
\r
910 % =\frac{1}{2^N} \\
\r
912 $P\left(X^{n+1}=e\right)=\frac{1}{2^N}$, which leads to $X^{n+1} \sim
\r
913 \mathbf{U}\left(\mathbb{B}^N\right)$.
\r
914 This result is true $\forall n \in \mathds{N}$, we thus have proven that, $$\forall K \in [0;1], Y_K=X^{N_0} \sim
\r
915 \mathbf{U}\left(\mathbb{B}^N\right) \text{ when } X \sim
\r
916 \mathbf{U}\left(\mathbb{B}^N\right)$$
\r
920 Section~\ref{sec:data-hiding-algo-chaotic iterations} are stego-secure.
\r
923 We will now prove that,
\r
925 \begin{proposition}
\r
927 Chaotic Iterations with Dependent Strategy (CIDS) are not stego-secure.
\r
931 Due to the definition of CIDS, we have $P(Y_K=(1,1,\cdots,1))=0$. So there is
\r
932 no uniform repartition for the stego-contents $Y_K$.
\r
941 \subsection{Chaos-Security of the $CI_1$ scheme}\label{sec:chaos-security}
\r
943 To check whether an information hiding scheme $S$ is chaos-secure or not, $S$
\r
944 must be written as an iterate process $x^{n+1}=f(x^n)$ on a metric space
\r
945 $(\mathcal{X},d)$. This formulation is always possible~\cite{ih10}. So,
\r
948 \begin{definition}[Chaos-Security]
\r
949 \label{Def:chaos-security-definition}
\r
950 An information hiding scheme $S$ is said to be chaos-secure on
\r
951 $(\mathcal{X},d)$ if its iterative process has a chaotic
\r
952 behavior according to Devaney.
\r
955 In the approach presented by Guyeux \emph{et al.}, a data hiding scheme is secure if it is unpredictable.
\r
956 Its iterative process must satisfy the Devaney's chaos property and its level of chaos-security increases with the number of chaotic properties satisfied by it.
\r
958 This new concept of security for data hiding schemes has been proposed in~\cite{ih10} as a complementary approach to the existing framework.
\r
959 It contributes to the reinforcement of confidence into existing secure data hiding schemes.
\r
960 Additionally, the study of security in KMA, KOA, and CMA setups is realizable in this context.
\r
961 Finally, this framework can replace stego-security in situations that are not encompassed by it.
\r
962 In particular, this framework is more relevant to give evaluation of data hiding schemes claimed as chaotic.
\r
967 We can establish that,
\r
969 \begin{proposition}
\r
971 CIIS and CIDS are chaos-secure.
\r
975 It is due to the fact that chaotic iterations have a chaotic behavior, as defined by Devaney.
\r
980 In the two following sections, we will study the qualitative and quantitative
\r
981 properties of chaos-security for chaotic iterations. These properties can measure the
\r
982 disorder generated by our scheme, giving by doing so some important informations about the
\r
983 unpredictability level of such a process.
\r
985 \subsection{Quantitative property of
\r
986 chaotic iterations}\label{sec:chaos-security-quantitative}
\r
988 \begin{definition}[Expansivity]
\r
989 A function $f$ is said to be \emph{expansive} if $
\r
990 \exists \varepsilon >0,\forall x\neq y,\exists n\in \mathds{N}%
\r
991 ,d(f^{n}(x),f^{n}(y))\geqslant \varepsilon .$
\r
994 \begin{proposition}
\r
995 $G_{f_{0}}$ is an expansive chaotic dynamical system on $\mathcal{X}$ with a constant of expansivity is equal to 1.
\r
998 If $(S,E)\neq (\check{S};\check{E})$, then either $E\neq \check{E}$, so at least
\r
999 one cell is not in the same state in $E$ and $\check{E}$. Consequently the distance between $(S,E)$ and $(%
\r
1000 \check{S};\check{E})$ is greater or equal to 1. Or $E=\check{E}$. So the
\r
1001 strategies $S$ and $\check{S}$ are not equal. Let $n_{0}$ be the first index in which the terms $S$ and $\check{S}$
\r
1004 k<n_{0},\tilde{G}_{f_{0}}^{k}(S,E)=\tilde{G}_{f_{0}}^{k}(\check{S},\check{E})$,
\r
1005 and $\tilde{G}_{f_{0}}^{n_{0}}(S,E)\neq \tilde{G}_{f_{0}}^{n_{0}}(\check{S},\check{E})$. As $E=\check{E},$ the cell which has changed in $E$ at the $n_{0}$-th
\r
1006 iterate is not the same as the cell which has changed in $\check{E}$, so
\r
1007 the distance between $\tilde{G}_{f_{0}}^{n_{0}}(S,E)$ and $\tilde{G}_{f_{0}}^{n_{0}}(\check{S%
\r
1008 },\check{E})$ is greater or equal to 2.
\r
1011 \subsection{Qualitative property of
\r
1012 chaotic iterations}\label{sec:chaos-security-qualitative}
\r
1014 \begin{definition}[Topological mixing]
\r
1015 A discrete dynamical system is said to be topologically mixing
\r
1016 if and only if, for any couple of disjoint open set $U, V \neq \varnothing$,
\r
1017 $n_0 \in \mathds{N}$ can be found so that $\forall n \geqslant n_0, f^n(U) \cap
\r
1018 V \neq \varnothing$.
\r
1020 \begin{proposition}
\r
1021 $\tilde{G}_{f_0}$ is topologically mixing on $(\mathcal{X}', d')$.
\r
1024 This result is an immediate consequence of the lemma below.
\r
1027 For any open ball $B$ of $\mathcal{X}'$, an index $n$ can be found such that $\tilde{G}_{f_0}^n(B) = \mathcal{X}'$.
\r
1031 Let $B=B((E,S),\varepsilon)$ be an open ball, which the radius can be considered as strictly less than 1.
\r
1032 All the elements of $B$ have the same state $E$ and are such that an integer $k \left(=-\log_{10}(\varepsilon)\right)$ satisfies:
\r
1034 \item all the strategies of $B$ have the same $k$ first terms,
\r
1035 \item after the index $k$, all values are possible.
\r
1038 Then, after $k$ iterations, the new state of the system is $\tilde{G}_{f_0}^k(E,S)_1$ and all the strategies are possible (all the points $(\tilde{G}_{f_0}^k(E,S)_1,\textrm{\^{S}})$, with any $\textrm{\^{S}} \in \mathbb{S}$, are reachable from $B$).
\r
1040 We will prove that all points of $\mathcal{X}'$ are reachable from $B$. Let
\r
1041 $(E',S') \in \mathcal{X}'$ and $s_i$ be the list of the different cells between
\r
1042 $\tilde{G}_{f_0}^k(E,S)_1$ and $E'$. We denote by $|s|$ the size of the sequence
\r
1043 $s_i$. So the point $(\check{E},\check{S})$ of $B$ defined by: $\check{E} = E$,
\r
1044 $\check{S}^i = S^i, \forall i \leqslant k$, $\check{S}^{k+i} = s_i, \forall i
\r
1045 \leqslant |s|$, and $\forall i \in \mathds{N}, S^{k + |s| + i} = S'^i$
\r
1046 is such that $\tilde{G}_{f_0}^{k+|s|}(\check{E},\check{S}) = (E',S')$. This concludes the proofs of the lemma and of the proposition.
\r
1051 \subsection{Comparison between spread-spectrum and chaotic iterations}\label{sec:comparison-application-context}
\r
1053 The consequences of topological mixing for data hiding are multiple. Firstly,
\r
1054 security can be largely improved by considering
\r
1055 the number of iterations as a secret key. An attacker
\r
1056 will reach all of the possible media when iterating without this key.
\r
1057 Additionally, he cannot benefit from a KOA setup, by studying media in the
\r
1058 neighborhood of the original cover. Moreover, as in a topological mixing
\r
1059 situation, it is possible that any hidden message (the initial condition), is
\r
1060 sent to the same fixed watermarked content (with different numbers of
\r
1061 iterations), the interest to be in a KMA setup is drastically reduced. Lastly, as
\r
1062 all of the watermarked contents are possible for a given hidden message, depending
\r
1063 on the number of iterations, CMA attacks will fail.
\r
1066 %Already corrected
\r
1067 The property of expansivity reinforces drastically the sensitivity in the aims of
\r
1068 reducing the benefits that Eve can obtain from an attack in KMA or KOA setup. For
\r
1069 example, it is impossible to have an estimation of the watermark by moving the
\r
1070 message (or the cover) as a cursor in situation of expansivity: this cursor will
\r
1071 be too much sensitive and the changes will be too important to be useful. On the
\r
1072 contrary, a very large constant of expansivity $\varepsilon$ is unsuitable: the
\r
1073 cover media will be strongly altered whereas the watermark would be undetectable.
\r
1076 % %Indeed, let us consider the same cover $E$ twice with two different watermarks
\r
1077 % $S,S'$. Thus $d(X,Y) < 1$, where $X=(E,S), Y=(E,S')$ and $d$ is for the distance
\r
1078 % previously defined. However, due to expansivity, $\exists n \in \mathds{N},
\r
1079 % d\left(G^n (X) ; G^n (Y) \right) \geqslant \varepsilon.$ Thus, $d_\infty
\r
1080 % \left(G^n (X)_1 ; G^n (Y)_1 \right) \geqslant \varepsilon - 1$, so either
\r
1081 % $d_\infty \left(X_1 ; G^n (X)_1 \right) \geqslant \dfrac{\varepsilon - 1}{2}$, or
\r
1082 % $d_\infty \left(Y_1 ; G^n (Y)_1 \right) \geqslant \dfrac{\varepsilon - 1}{2}$. If
\r
1083 % $\varepsilon$ is large, then at least one of the two watermarked media will be
\r
1084 % very different than its original cover.
\r
1086 Finally, spread-spectrum is relevant when
\r
1087 a discrete and secure data hiding technique is required in WOA setup. However,
\r
1088 this technique should not be used in KOA and KMA setup, due to its lack of
\r
1089 expansivity.% In the next section, we will present a new class of data hiding
\r
1090 schemes, which are expansive.
\r
1093 \section{Study of robustness: $CI_1$}\label{section:robustness-ci-1}
\r
1097 \subsection{Definition of robustness}\label{section:robustness-definition}
\r
1099 \textbf{TODO: Ajouter la definition de robustesse de Kalker}
\r
1101 \textbf{TODO: Ajouter la definition de sécurité de Kalker --> AU BON ENDROIT}
\r
1103 \subsection{Visual scales for robustness
\r
1104 threshold}\label{section:robustness-scale}
\r
1106 \textbf{TODO definition}
\r
1108 Given the robustness definition, it can be said that the notion of robustness is
\r
1109 a subjective notion depending of the context and of the watermarking application
\r
1112 To affirm that a watermarking process is robust or not in such or such other
\r
1113 conditions, it is necessary to define a robustness threshold. If it is
\r
1114 considered an image as watermak, such a threshold can be subjectively defined
\r
1115 looking for a visual difference rate scale in term of bytes differences. It has
\r
1116 been established 3 scales for 3 types of images:
\r
1118 \item Greyscale images,
\r
1119 \item Colour images,
\r
1120 \item And white and black images.
\r
1124 \textbf{Firstly, it can be remark that in the field of watermarking with an image
\r
1125 as watermark, one watermark can be visually considered as identical to an other,
\r
1126 if the second is the same, or correspond to the negative version of the first.}
\r
1128 \subsubsection{\textbf{Scale in greyscale domain}}
\r
1130 The scale in the greyscale domain is presented in
\r
1131 Figure~\ref{fig:visual-scale-greyscale}~\vpageref{fig:visual-scale-greyscale}.
\r
1133 Looking the visual scale, it can be deduced that the in the greyscale domain, a
\r
1134 watermarking process is robust for a difference rate between 0\% and 30\%, and
\r
1135 between 70\% and 100\% for the negative image case.
\r
1137 \textbf{So in this field, the robustness threshold can be fixed to 30\%.}
\r
1139 \begin{figure*}[htb]
\r
1141 \begin{minipage}[b]{.32\linewidth}
\r
1143 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/0_lena}}
\r
1144 \centerline{(0) Difference rate of 0\%}
\r
1147 \begin{minipage}[b]{0.32\linewidth}
\r
1149 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/5_lena}}
\r
1150 \centerline{(1) Difference rate of 5\%}
\r
1153 \begin{minipage}[b]{0.32\linewidth}
\r
1155 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/10_lena}}
\r
1156 \centerline{(2) Difference rate of 10\%}
\r
1158 \begin{minipage}[b]{.32\linewidth}
\r
1160 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/15_lena}}
\r
1161 \centerline{(3) Difference rate of 15\%}
\r
1164 \begin{minipage}[b]{0.32\linewidth}
\r
1166 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/20_lena}}
\r
1167 \centerline{(4) Difference rate of 20\%}
\r
1170 \begin{minipage}[b]{0.32\linewidth}
\r
1172 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/25_lena}}
\r
1173 \centerline{(5) Difference rate of 25\%}
\r
1175 \begin{minipage}[b]{.32\linewidth}
\r
1177 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/30_lena}}
\r
1178 \centerline{(6) Difference rate of 30\%}
\r
1181 \begin{minipage}[b]{0.32\linewidth}
\r
1183 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/35_lena}}
\r
1184 \centerline{(7) Difference rate of 35\%}
\r
1187 \begin{minipage}[b]{0.32\linewidth}
\r
1189 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/40_lena}}
\r
1190 \centerline{(8) Difference rate of 40\%}
\r
1192 \begin{minipage}[b]{.32\linewidth}
\r
1194 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/45_lena}}
\r
1195 \centerline{(9) Difference rate of 45\%}
\r
1198 \begin{minipage}[b]{0.32\linewidth}
\r
1200 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/50_lena}}
\r
1201 \centerline{(10) Difference rate of 50\%}
\r
1204 \begin{minipage}[b]{0.32\linewidth}
\r
1206 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/55_lena}}
\r
1207 \centerline{(11) Difference rate of 55\%}
\r
1209 \begin{minipage}[b]{.32\linewidth}
\r
1211 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/60_lena}}
\r
1212 \centerline{(12) Difference rate of 60\%}
\r
1215 \begin{minipage}[b]{0.32\linewidth}
\r
1217 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/65_lena}}
\r
1218 \centerline{(13) Difference rate of 65\%}
\r
1221 \begin{minipage}[b]{0.32\linewidth}
\r
1223 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/70_lena}}
\r
1224 \centerline{(14) Difference rate of 70\%}
\r
1226 \begin{minipage}[b]{.32\linewidth}
\r
1228 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/75_lena}}
\r
1229 \centerline{(15) Difference rate of 75\%}
\r
1232 \begin{minipage}[b]{0.32\linewidth}
\r
1234 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/80_lena}}
\r
1235 \centerline{(16) Difference rate of 80\%}
\r
1238 \begin{minipage}[b]{0.32\linewidth}
\r
1240 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/85_lena}}
\r
1241 \centerline{(17) Difference rate of 85\%}
\r
1243 \begin{minipage}[b]{.32\linewidth}
\r
1245 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/90_lena}}
\r
1246 \centerline{(18) Difference rate of 90\%}
\r
1249 \begin{minipage}[b]{0.32\linewidth}
\r
1251 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/95_lena}}
\r
1252 \centerline{(19) Difference rate of 95\%}
\r
1255 \begin{minipage}[b]{0.32\linewidth}
\r
1257 \centerline{\includegraphics[width=2.9cm]{img/greyscale-scale/100_lena}}
\r
1258 \centerline{(20) Difference rate of 100\%}
\r
1260 \caption{Visual scale in greyscale domain for robustness threshold evaluation.}
\r
1261 \label{fig:visual-scale-greyscale}
\r
1266 \subsubsection{\textbf{Scale in colour domain}}
\r
1268 The scale in color domain is presented in
\r
1269 Figure~\ref{fig:visual-scale-colour}~\vpageref{fig:visual-scale-colour}.
\r
1271 Looking the visual scale, it can be deduced that the in the color domain, a
\r
1272 watermarking process is robust for a difference rate between 0\% and 40\%, and
\r
1273 between 60\% and 100\% for the negative image case.
\r
1275 \textbf{So in this field, the robustness threshold can be fixed to 40\%.}
\r
1277 \begin{figure*}[htb]
\r
1279 \begin{minipage}[b]{.32\linewidth}
\r
1281 \centerline{\includegraphics[width=2.9cm]{img/color-scale/0_singe}}
\r
1282 \centerline{(0) Difference rate of 0\%}
\r
1285 \begin{minipage}[b]{0.32\linewidth}
\r
1287 \centerline{\includegraphics[width=2.9cm]{img/color-scale/5_singe}}
\r
1288 \centerline{(1) Difference rate of 5\%}
\r
1291 \begin{minipage}[b]{0.32\linewidth}
\r
1293 \centerline{\includegraphics[width=2.9cm]{img/color-scale/10_singe}}
\r
1294 \centerline{(2) Difference rate of 10\%}
\r
1296 \begin{minipage}[b]{.32\linewidth}
\r
1298 \centerline{\includegraphics[width=2.9cm]{img/color-scale/15_singe}}
\r
1299 \centerline{(3) Difference rate of 15\%}
\r
1302 \begin{minipage}[b]{0.32\linewidth}
\r
1304 \centerline{\includegraphics[width=2.9cm]{img/color-scale/20_singe}}
\r
1305 \centerline{(4) Difference rate of 20\%}
\r
1308 \begin{minipage}[b]{0.32\linewidth}
\r
1310 \centerline{\includegraphics[width=2.9cm]{img/color-scale/25_singe}}
\r
1311 \centerline{(5) Difference rate of 25\%}
\r
1313 \begin{minipage}[b]{.32\linewidth}
\r
1315 \centerline{\includegraphics[width=2.9cm]{img/color-scale/30_singe}}
\r
1316 \centerline{(6) Difference rate of 30\%}
\r
1319 \begin{minipage}[b]{0.32\linewidth}
\r
1321 \centerline{\includegraphics[width=2.9cm]{img/color-scale/35_singe}}
\r
1322 \centerline{(7) Difference rate of 35\%}
\r
1325 \begin{minipage}[b]{0.32\linewidth}
\r
1327 \centerline{\includegraphics[width=2.9cm]{img/color-scale/40_singe}}
\r
1328 \centerline{(8) Difference rate of 40\%}
\r
1330 \begin{minipage}[b]{.32\linewidth}
\r
1332 \centerline{\includegraphics[width=2.9cm]{img/color-scale/45_singe}}
\r
1333 \centerline{(9) Difference rate of 45\%}
\r
1336 \begin{minipage}[b]{0.32\linewidth}
\r
1338 \centerline{\includegraphics[width=2.9cm]{img/color-scale/50_singe}}
\r
1339 \centerline{(10) Difference rate of 50\%}
\r
1342 \begin{minipage}[b]{0.32\linewidth}
\r
1344 \centerline{\includegraphics[width=2.9cm]{img/color-scale/55_singe}}
\r
1345 \centerline{(11) Difference rate of 55\%}
\r
1347 \begin{minipage}[b]{.32\linewidth}
\r
1349 \centerline{\includegraphics[width=2.9cm]{img/color-scale/60_singe}}
\r
1350 \centerline{(12) Difference rate of 60\%}
\r
1353 \begin{minipage}[b]{0.32\linewidth}
\r
1355 \centerline{\includegraphics[width=2.9cm]{img/color-scale/65_singe}}
\r
1356 \centerline{(13) Difference rate of 65\%}
\r
1359 \begin{minipage}[b]{0.32\linewidth}
\r
1361 \centerline{\includegraphics[width=2.9cm]{img/color-scale/70_singe}}
\r
1362 \centerline{(14) Difference rate of 70\%}
\r
1364 \begin{minipage}[b]{.32\linewidth}
\r
1366 \centerline{\includegraphics[width=2.9cm]{img/color-scale/75_singe}}
\r
1367 \centerline{(15) Difference rate of 75\%}
\r
1370 \begin{minipage}[b]{0.32\linewidth}
\r
1372 \centerline{\includegraphics[width=2.9cm]{img/color-scale/80_singe}}
\r
1373 \centerline{(16) Difference rate of 80\%}
\r
1376 \begin{minipage}[b]{0.32\linewidth}
\r
1378 \centerline{\includegraphics[width=2.9cm]{img/color-scale/85_singe}}
\r
1379 \centerline{(17) Difference rate of 85\%}
\r
1381 \begin{minipage}[b]{.32\linewidth}
\r
1383 \centerline{\includegraphics[width=2.9cm]{img/color-scale/90_singe}}
\r
1384 \centerline{(18) Difference rate of 90\%}
\r
1387 \begin{minipage}[b]{0.32\linewidth}
\r
1389 \centerline{\includegraphics[width=2.9cm]{img/color-scale/95_singe}}
\r
1390 \centerline{(19) Difference rate of 95\%}
\r
1393 \begin{minipage}[b]{0.32\linewidth}
\r
1395 \centerline{\includegraphics[width=2.9cm]{img/color-scale/100_singe}}
\r
1396 \centerline{(20) Difference rate of 100\%}
\r
1398 \caption{Visual scale in colour domain for robustness threshold evaluation.}
\r
1399 \label{fig:visual-scale-colour}
\r
1403 \subsubsection{\textbf{Scale in white and black domain}}
\r
1405 The scale in white and black domain is presented in
\r
1406 Figure~\ref{fig:visual-scale-white-black}~\vpageref{fig:visual-scale-white-black}.
\r
1408 Looking the visual scale, it can be deduced that the in the white and black domain, a
\r
1409 watermarking process is robust for a difference rate between 0\% and 20\%, and
\r
1410 between 80\% and 100\% for the negative image case.
\r
1412 \textbf{So in this field, the robustness threshold can be fixed to 20\%.}
\r
1414 \begin{figure*}[htb]
\r
1416 \begin{minipage}[b]{.32\linewidth}
\r
1418 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/0_invader}}
\r
1419 \centerline{(0) Difference rate of 0\%}
\r
1422 \begin{minipage}[b]{0.32\linewidth}
\r
1424 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/5_invader}}
\r
1425 \centerline{(1) Difference rate of 5\%}
\r
1428 \begin{minipage}[b]{0.32\linewidth}
\r
1430 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/10_invader}}
\r
1431 \centerline{(2) Difference rate of 10\%}
\r
1433 \begin{minipage}[b]{.32\linewidth}
\r
1435 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/15_invader}}
\r
1436 \centerline{(3) Difference rate of 15\%}
\r
1439 \begin{minipage}[b]{0.32\linewidth}
\r
1441 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/20_invader}}
\r
1442 \centerline{(4) Difference rate of 20\%}
\r
1445 \begin{minipage}[b]{0.32\linewidth}
\r
1447 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/25_invader}}
\r
1448 \centerline{(5) Difference rate of 25\%}
\r
1450 \begin{minipage}[b]{.32\linewidth}
\r
1452 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/30_invader}}
\r
1453 \centerline{(6) Difference rate of 30\%}
\r
1456 \begin{minipage}[b]{0.32\linewidth}
\r
1458 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/35_invader}}
\r
1459 \centerline{(7) Difference rate of 35\%}
\r
1462 \begin{minipage}[b]{0.32\linewidth}
\r
1464 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/40_invader}}
\r
1465 \centerline{(8) Difference rate of 40\%}
\r
1467 \begin{minipage}[b]{.32\linewidth}
\r
1469 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/45_invader}}
\r
1470 \centerline{(9) Difference rate of 45\%}
\r
1473 \begin{minipage}[b]{0.32\linewidth}
\r
1475 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/50_invader}}
\r
1476 \centerline{(10) Difference rate of 50\%}
\r
1479 \begin{minipage}[b]{0.32\linewidth}
\r
1481 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/55_invader}}
\r
1482 \centerline{(11) Difference rate of 55\%}
\r
1484 \begin{minipage}[b]{.32\linewidth}
\r
1486 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/60_invader}}
\r
1487 \centerline{(12) Difference rate of 60\%}
\r
1490 \begin{minipage}[b]{0.32\linewidth}
\r
1492 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/65_invader}}
\r
1493 \centerline{(13) Difference rate of 65\%}
\r
1496 \begin{minipage}[b]{0.32\linewidth}
\r
1498 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/70_invader}}
\r
1499 \centerline{(14) Difference rate of 70\%}
\r
1501 \begin{minipage}[b]{.32\linewidth}
\r
1503 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/75_invader}}
\r
1504 \centerline{(15) Difference rate of 75\%}
\r
1507 \begin{minipage}[b]{0.32\linewidth}
\r
1509 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/80_invader}}
\r
1510 \centerline{(16) Difference rate of 80\%}
\r
1513 \begin{minipage}[b]{0.32\linewidth}
\r
1515 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/85_invader}}
\r
1516 \centerline{(17) Difference rate of 85\%}
\r
1518 \begin{minipage}[b]{.32\linewidth}
\r
1520 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/90_invader}}
\r
1521 \centerline{(18) Difference rate of 90\%}
\r
1524 \begin{minipage}[b]{0.32\linewidth}
\r
1526 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/95_invader}}
\r
1527 \centerline{(19) Difference rate of 95\%}
\r
1530 \begin{minipage}[b]{0.32\linewidth}
\r
1532 \centerline{\includegraphics[width=2.9cm]{img/W_B-scale/100_invader}}
\r
1533 \centerline{(20) Difference rate of 100\%}
\r
1535 \caption{Visual scale in white and black domain for robustness threshold
\r
1537 \label{fig:visual-scale-white-black}
\r
1541 \subsubsection{\textbf{Conclusion concerning the robustness threshold}}
\r
1543 \textbf{From the results of the previous study, it can be concluded that the
\r
1544 robustness threshold depends of the nature of the watermark.}\newline
\r
1546 \textbf{So for a picture as watermark, the highest robustness threshold is
\r
1547 obtain with color images.}
\r
1549 \subsection{Presentation of the
\r
1550 architecture}\label{section:architecture-presentation}
\r
1552 Computations have been
\r
1553 performed on the supercomputer facilities of the Mésocentre de calcul de Franche-Comt\'{e}.
\r
1555 In order to take benefits of parallelism, we have used Jace~\cite{bhm09:ip}, a
\r
1556 grid-enabled programming and execution environment allowing a simple and efficient implementation
\r
1557 of parallel and distributed applications. Roughly speaking, Jace builds a virtual parallel machine by
\r
1558 connecting a set of heterogeneous and distant computers.
\r
1559 It schedules tasks, executes them, and returns results to the user. It
\r
1560 also proposes a simple programming interface for the implementation of applications
\r
1561 using a message passing model.
\r
1563 In our case, tasks are independent Python programs with different parameters. Jace takes care to execute them in parallel ensuring optimal load balancing and fault tolerance.
\r
1565 For these experiments we have used HPC resources
\r
1566 from Mesocentre of Franche-Comt\'{e}. This platform
\r
1567 is currently composed of 720 cores with 11 TeraFLOPS of power.
\r
1569 The different tests have been realized on several images from the Wikimedia
\r
1570 Commons repository~\cite{wiki:wikimedia-commons}.
\r
1573 \textbf{FAIRE AUSSI DES TESTS AVEC LA BASE DE BOSS}
\r
1574 The conditions of all tests are described for each one in the caption of the graph result.
\r
1576 %je ne sais pas comment expliquer vos configs !
\r
1577 %In the following experiments, we have chosen different configuration files,
\r
1578 %about 1000 images from wikimedia.....
\r
1582 \subsection{Tests realized}\label{sec:tests-realized-ci-1}
\r
1584 In this section, it is studied the robustness of the watermarking process
\r
1585 $CI_1$. To do this, it has been executed two test batteries in order to evaluate
\r
1586 the evolution of the bytes difference rate in function of the geometrical
\r
1587 attacks parameters, between the watermark inserted and the watermark extracted
\r
1590 In the first battery of tests we have used the same embedding key (a constant
\r
1591 strategy,the same strategy has been used to watermark all the
\r
1592 images in this test) to embed the watermark. In the second battery of tests,
\r
1593 in order to see the influence of the embedding key on the robustness, we have embedded all
\r
1594 the watermark with strategies generated randomly(a
\r
1595 different strategy has been used to watermark each image).
\r
1597 To obtain the following results, we have realized a great number of attacks on
\r
1598 600 images, and we have traced for each attack, two graphics (average and
\r
1599 standard deviation), in order to determine the acceptability
\r
1600 threshold for our watermarking process $CI_1$.
\r
1602 \subsection{Tests results}\label{sec:tests-results-ci-1}
\r
1604 The exhaustive list of the geometrical attacks tested and the
\r
1605 corresponding graphs results is following:
\r
1608 \item \textbf{Robustness facing a resizing attack.}\\
\r
1609 See Figure~\ref{fig:resizing-attack-constant-strategy}~\vpageref{fig:resizing-attack-constant-strategy}.
\r
1610 \item \textbf{Robustness facing a JPEG attack.}\\See
\r
1611 Figure~\ref{fig:jpeg-attack-constant-strategy}~\vpageref{fig:jpeg-attack-constant-strategy}.
\r
1612 \item \textbf{Robustness facing a Gaussian blur
\r
1614 Figure~\ref{fig:gaussian-blur-attack-constant-strategy}~\vpageref{fig:gaussian-blur-attack-constant-strategy}.
\r
1615 \item \textbf{Robustness facing a rotation
\r
1617 Figure~\ref{fig:rotation-attack-constant-strategy}~\vpageref{fig:rotation-attack-constant-strategy}.
\r
1618 \item \textbf{Robustness facing a blur
\r
1620 Figure~\ref{fig:blur-attack-constant-strategy}~\vpageref{fig:blur-attack-constant-strategy}.
\r
1621 \item \textbf{Robustness facing a contrast
\r
1623 Figure~\ref{fig:contrast-attack-constant-strategy}~\vpageref{fig:contrast-attack-constant-strategy}.
\r
1624 \item \textbf{Robustness facing a cropping
\r
1626 Figure~\ref{fig:cropping-attack-constant-strategy}~\vpageref{fig:cropping-attack-constant-strategy}.
\r
1632 % %\includegraphics[width=10.5cm]{./img/prisoner-problem.png}
\r
1633 % \includegraphics[width=15cm]{./img/prisoner-problem}
\r
1636 % \caption{Simmons' prisoner problem~\cite{Simmons83}}
\r
1640 \begin{figure*}[htb]
\r
1641 \begin{minipage}[b]{.45\linewidth}
\r
1643 \centerline{\includegraphics[width=9cm]{graphs/graph_1_attack=redimensionnement-average}}
\r
1644 \centerline{(a) Average}
\r
1647 \begin{minipage}[b]{0.45\linewidth}
\r
1649 \centerline{\includegraphics[width=9cm]{graphs/graph_1_attack=redimensionnement-sd}}
\r
1650 \centerline{(b) Standard deviation}
\r
1652 \caption{Robustness of $CI_1$ facing a resizing attack.(600 images and constant strategy)}
\r
1653 \label{fig:resizing-attack-constant-strategy}
\r
1656 \begin{figure*}[htb]
\r
1657 \begin{minipage}[b]{.45\linewidth}
\r
1659 \centerline{\includegraphics[width=9cm]{graphs/graph_2_attack=jpeg-average}}
\r
1660 \centerline{(a) Average}
\r
1663 \begin{minipage}[b]{0.45\linewidth}
\r
1665 \centerline{\includegraphics[width=9cm]{graphs/graph_2_attack=jpeg-sd}}
\r
1666 \centerline{(b) Standard deviation}
\r
1668 \caption{Robustness of $CI_1$ facing a JPEG compression attack.(600 images and
\r
1669 constant strategy)}
\r
1670 \label{fig:jpeg-attack-constant-strategy}
\r
1674 \begin{figure*}[htb]
\r
1675 \begin{minipage}[b]{.45\linewidth}
\r
1677 \centerline{\includegraphics[width=9cm]{graphs/graph_3_attack=gaussien-average}}
\r
1678 \centerline{(a) Average}
\r
1681 \begin{minipage}[b]{0.45\linewidth}
\r
1683 \centerline{\includegraphics[width=9cm]{graphs/graph_3_attack=gaussien-sd}}
\r
1684 \centerline{(b) Standard deviation}
\r
1686 \caption{Robustness of $CI_1$ facing a Gaussian blur attack.(600 images and
\r
1687 constant strategy)}
\r
1688 \label{fig:gaussian-blur-attack-constant-strategy}
\r
1691 \begin{figure*}[htb]
\r
1692 \begin{minipage}[b]{.45\linewidth}
\r
1694 \centerline{\includegraphics[width=9cm]{graphs/graph_4_attack=rotation-average}}
\r
1695 \centerline{(a) Average}
\r
1698 \begin{minipage}[b]{0.45\linewidth}
\r
1700 \centerline{\includegraphics[width=9cm]{graphs/graph_4_attack=rotation-sd}}
\r
1701 \centerline{(b) Standard deviation}
\r
1703 \caption{Robustness of $CI_1$ facing a rotation attack.(600 images and constant
\r
1705 \label{fig:rotation-attack-constant-strategy}
\r
1709 \begin{figure*}[htb]
\r
1710 \begin{minipage}[b]{.45\linewidth}
\r
1712 \centerline{\includegraphics[width=9cm]{graphs/graph_5_attack=flou-average}}
\r
1713 \centerline{(a) Average}
\r
1716 \begin{minipage}[b]{0.45\linewidth}
\r
1718 \centerline{\includegraphics[width=9cm]{graphs/graph_5_attack=flou-sd}}
\r
1719 \centerline{(b) Standard deviation}
\r
1721 \caption{Robustness of $CI_1$ facing a blur attack.(600 images and constant
\r
1723 \label{fig:blur-attack-constant-strategy}
\r
1727 \begin{figure*}[htb]
\r
1728 \begin{minipage}[b]{.45\linewidth}
\r
1730 \centerline{\includegraphics[width=9cm]{graphs/graph_6_attack=contraste-average}}
\r
1731 \centerline{(a) Average}
\r
1734 \begin{minipage}[b]{0.45\linewidth}
\r
1736 \centerline{\includegraphics[width=9cm]{graphs/graph_6_attack=contraste-sd}}
\r
1737 \centerline{(b) Standard deviation}
\r
1739 \caption{Robustness of $CI_1$ facing a contrast attack.(600 images and constant
\r
1741 \label{fig:contrast-attack-constant-strategy}
\r
1745 \begin{figure*}[htb]
\r
1746 \begin{minipage}[b]{.45\linewidth}
\r
1748 \centerline{\includegraphics[width=9cm]{graphs/graph_7_attack=decoupage-average}}
\r
1749 \centerline{(a) Average}
\r
1752 \begin{minipage}[b]{0.45\linewidth}
\r
1754 \centerline{\includegraphics[width=9cm]{graphs/graph_7_attack=decoupage-sd}}
\r
1755 \centerline{(b) Standard deviation}
\r
1757 \caption{Robustness of $CI_1$ facing a cropping attack.(600 images and constant
\r
1759 \label{fig:cropping-attack-constant-strategy}
\r
1762 \subsection{Robustness evaluation}\label{sec:tests-results-ci-1}
\r
1764 From the results presented in the previous section, the robustness of the
\r
1765 process $CI_1$ can be evaluated.
\r
1768 As a first conclusion, it can be note that in the $CI_1$ watermarking process, the strategy has no
\r
1769 influence on the robustness. Indeed, analyzing all the curves, a strong
\r
1770 similarity between the curves obtained with a constant strategy and the curves obtained with a strategy generated randomly can
\r
1773 As a second conclusion, it can be note that the $CI_1$ watermarking process is
\r
1774 not robust in spatial domain facing attacks of resizing, of JPEG compression
\r
1775 with a compression rate lesser than 97\%, of Gaussian blur with a blur rate
\r
1776 greater than 42\%, of classic blur, of contrast, and of cropping with a number
\r
1777 of pixels greater than 180.
\r
1779 Inversely, $CI_1$ is robust in spatial domain facing geometrical attacks of JPEG
\r
1780 compression with a compression rate greater than 97\%, of Gaussian blur with a blur rate
\r
1781 lesser than 42\%, of rotation with angles between 0° and 30°, and of cropping
\r
1782 with a number of pixels lesser than 180.
\r
1786 \item \textbf{Robustness facing a resizing attack.}\\
\r
1787 See Figure~\ref{fig:resizing-attack-constant-strategy}~\vpageref{fig:resizing-attack-constant-strategy}.
\r
1788 \item \textbf{Robustness facing a JPEG compression
\r
1790 Figure~\ref{fig:jpeg-attack-constant-strategy}~\vpageref{fig:jpeg-attack-constant-strategy}.
\r
1791 \item \textbf{Robustness facing a Gaussian blur
\r
1793 Figure~\ref{fig:gaussian-blur-attack-constant-strategy}~\vpageref{fig:gaussian-blur-attack-constant-strategy}.
\r
1794 \item \textbf{Robustness facing a rotation
\r
1796 Figure~\ref{fig:rotation-attack-constant-strategy}~\vpageref{fig:rotation-attack-constant-strategy}.
\r
1797 \item \textbf{Robustness facing a blur
\r
1799 Figure~\ref{fig:blur-attack-constant-strategy}~\vpageref{fig:blur-attack-constant-strategy}.
\r
1800 \item \textbf{Robustness facing a contrast
\r
1802 Figure~\ref{fig:contrast-attack-constant-strategy}~\vpageref{fig:contrast-attack-constant-strategy}.
\r
1803 \item \textbf{Robustness facing a cropping
\r
1805 Figure~\ref{fig:cropping-attack-constant-strategy}~\vpageref{fig:cropping-attack-constant-strategy}.
\r
1808 ensuite il faut reprendre chaque courbe et énoncer une phrase du genre :
\r
1809 " on voit qu'on est loin d'une génération aléatoire des bits qui serait environ de 50%"
\r
1810 et donc on peut affirmer avec certitude que telle ou telle image est tatouée ou non.
\r
1815 To conclude, at this point, only two information hiding schemes are both stego-secure and chaos-secure \cite{gfb10:ip}.
\r
1816 The first one is based on a spread spectrum technique called Natural Watermarking.
\r
1817 It is stego-secure when its parameter $\eta$ is equal to $1$ \cite{Cayre2008}.
\r
1818 Unfortunately, this scheme is neither robust, nor able to face an attacker in KOA and KMA setups, due to its lack of a topological property called expansivity \cite{gfb10:ip}.
\r
1819 The second scheme both chaos-secure and stego-secure is based on chaotic iterations \cite{guyeux10ter}.
\r
1820 However, its first versions called $CI_1$ allows to embed securely only one bit per embedding parameters.
\r
1821 The objective of the next sections is to improve the $CI_1$ scheme presented in \cite{guyeux10ter}, in such a way that more than one bit can be embedded.
\r
1824 \section{Improved process for steganography: $CI_2$}
\r
1825 \label{section:process-ci2}
\r
1827 \noindent In this section is given a new algorithm that generalize the
\r
1828 $CI_1$ scheme presented above.
\r
1830 Let us firstly introduce the following notations:
\r
1832 \subsection{Notations}
\r
1834 %\item Let $\mathbb{S}_k=\llbracket 0, \mathsf{k-1} \rrbracket^{\mathds{N}}$ be
\r
1835 %the set of all strategies with values in $\llbracket 0, \mathsf{k-1}
\r
1837 \item $x^0 \in \mathbb{B}^\mathsf{N}$ is the $\mathsf{N}$ least
\r
1838 significant coefficients of a given cover media $C$.% $X$ be the initial state $X_0$,
\r
1839 %\item The vectorial negation $f_0$ (defined in~\ref{Def:vectorial-negation})
\r
1840 %as iteration function.
\r
1841 \item $m^0 \in \mathbb{B}^\mathsf{P}$ is the watermark to embed into $x^0$.
\r
1842 \item $S_p \in \mathbb{S}_N$ is a strategy called \textbf{place strategy}.
\r
1843 \item $S_c \in \mathbb{S}_P$ is a strategy called \textbf{choice strategy}.
\r
1844 \item Lastly, $S_m \in \mathbb{S}_P$ is a strategy called \textbf{mixing strategy}.
\r
1847 %\subsection{Algorithm SCISMM in details}\label{sec:algo-SCISMM}
\r
1849 \subsection{Presentation of the scheme}
\r
1851 Our information hiding scheme called Steganography by Chaotic Iterations and
\r
1852 Substitution with Mixing Message (SCISMM) is defined by
\r
1853 %\begin{definition}[SCISMM]\label{def:algo}
\r
1854 $\forall (n,i,j) \in
\r
1855 \mathds{N}^{\ast} \times \llbracket 0;\mathsf{N-1}\rrbracket \times \llbracket
\r
1856 0;\mathsf{P-1}\rrbracket$:
\r
1863 x_i^{n-1} & \text{ if }S_p^n\neq i \\
\r
1864 m_{S_c^n} & \text{ if }S_p^n=i.
\r
1871 m_j^{n-1} & \text{ if }S_m^n\neq j \\
\r
1873 \overline{m_j^{n-1}} & \text{ if }S_m^n=j.
\r
1880 where $\overline{m_j^{n-1}}$ is the boolean negation of $m_j^{n-1}$.
\r
1881 %\subsection{Stego-content by SCISMM}\label{sec:stego-content}
\r
1883 The stego-content is the boolean vector $y = x^P \in
\r
1884 \mathbb{B}^\mathsf{N}$.
\r
1887 \section{Study of security of $CI_2$}
\r
1889 \subsection{Study of stego-security}\label{sec:stego-security-ci-2}
\r
1892 \noindent Let us prove that,
\r
1894 \begin{proposition}
\r
1895 SCISMM is stego-secure.
\r
1899 %We suppose that $K,M \sim
\r
1900 %\mathbf{U}\left([0;1]\right)$, and $X \sim
\r
1901 %We consider a strategy in the CIIS scheme, and we
\r
1902 Let us suppose that $x^0 \sim
\r
1903 \mathbf{U}\left(\mathbb{B}^N\right)$ and $m^0 \sim
\r
1904 \mathbf{U}\left(\mathbb{B}^P\right)$ in a SCISMM setup.
\r
1905 We will prove by a mathematical induction that $\forall n \in \mathds{N}, x^n
\r
1906 \sim \mathbf{U}\left(\mathbb{B}^N\right)$. The base case is obvious according
\r
1907 to the uniform repartition hypothesis.
\r
1909 Let us now suppose that the statement $x^n \sim
\r
1910 \mathbf{U}\left(\mathbb{B}^N\right)$ holds for some $n$.
\r
1911 For a given $k \in \mathbb{B}^N$,
\r
1912 we denote by $\tilde{k_i} \in \mathbb{B}^N$ the vector defined by: $\forall i
\r
1913 \in \llbracket 0;\mathsf{N-1}\rrbracket,$
\r
1915 $k=\left(k_0,k_1,\hdots,k_i,\hdots,k_{\mathsf{N}-2},k_{\mathsf{N}-1}\right)$,\newline
\r
1916 then $\tilde{k}_i=\left( k_0,k_1,\hdots,\overline{k_i},\hdots,k_{\mathsf{N}-2},k_{\mathsf{N}-1} \right).$
\r
1919 %\begin{equation*}
\r
1921 $E_{i,j}$ be the following events: $$\forall (i,j) \in \llbracket
\r
1922 0;\mathsf{N-1}\rrbracket \times \llbracket 0;\mathsf{P-1}\rrbracket, E_{i,j}=$$
\r
1923 $$S_p^{n+1}=i \wedge S_c^{n+1}=j \wedge m_j^{n+1}=k_i \wedge \left(x^n=k \vee
\r
1924 x^n=\tilde{k}_i\right),$$ and
\r
1925 $p=P\left(x^{n+1}=k\right).$
\r
1928 p=P\left(\bigvee_{i \in \llbracket
\r
1929 0;\mathsf{N-1}\rrbracket, j \in \llbracket 0;\mathsf{P-1}\rrbracket}
\r
1932 We now introduce the following notation: $P_1(i)=P\left(S_p^{n+1}=i\right),$
\r
1933 $P_2(j)=P\left(S_c^{n+1}=j\right),$
\r
1934 $P_3(i,j)=P\left(m_j^{n+1}=k_i\right),$
\r
1935 and $P_4(i)=P\left(x^n=k \vee x^n=\tilde{k}_i\right).$
\r
1938 These four events are independent in SCISMM setup, thus:
\r
1940 p=\sum_{i \in \llbracket
\r
1941 0;\mathsf{N-1}\rrbracket, j \in \llbracket 0;\mathsf{P-1}\rrbracket}
\r
1942 P_1(i)P_2(i)P_3(i,j)P_4(i).
\r
1945 Proposition~\ref{prop:CIIS-stego-security}, $P\left(m_j^{n+1}=k_i\right)=\frac{1}{2}$.
\r
1946 As the two events are incompatible:
\r
1947 $$P\left(x^n=k \vee x^n=\tilde{k}_i\right)=P\left(x^n=k\right) + P\left(x^n=\tilde{k}_i\right).$$
\r
1950 Then, by using the inductive hypothesis:
\r
1951 $P\left(x^n=k\right)=\frac{1}{2^N},$
\r
1953 $P\left(x^n=\tilde{k}_i\right)=\frac{1}{2^N}.$
\r
1955 Let $S$ be defined by $$S=\sum_{i
\r
1956 \in \llbracket 0;\mathsf{N-1}\rrbracket, j \in \llbracket 0;\mathsf{P-1}\rrbracket}
\r
1958 Then $p = 2 \times \frac{1}{2} \times \frac{1}{2^N} \times S = \frac{1}{2^N} \times S$.
\r
1960 $S$ can now be evaluated:
\r
1962 \begin{array}{lll}
\r
1964 \in \llbracket 0;\mathsf{N-1}\rrbracket, j \in \llbracket 0;\mathsf{P-1}\rrbracket}
\r
1966 & = & \sum_{i \in \llbracket 0;\mathsf{N-1}\rrbracket} P_1(i) \times \sum_{j \in \llbracket 0;\mathsf{P-1}\rrbracket}
\r
1971 The set of events $\left \{ S_p^{n+1}=i \right \}$ for $i \in \llbracket 0;N-1
\r
1972 \rrbracket$ and the set of events $\left \{ S_c^{n+1}=j \right \}$ for $j \in
\r
1973 \llbracket 0;P-1 \rrbracket$ are both a partition of the universe of possible,
\r
1976 %Evaluate now $ P\left(S^n=k\right)$.
\r
1978 %We have: $K^0 = M \otimes K \sim
\r
1979 %\mathbf{U}\left([0;1]\right)$, so $K^\prime=F\left( K^0,p \right) \sim
\r
1980 %\mathbf{U}\left([0;1]\right)$, because for PLCMs, ``Uniform input generate
\r
1981 %uniform output''~\cite{Lasota}.
\r
1982 %Well with an immediate proof by recurrence we have:
\r
1983 %$K^n \sim \mathbf{U}\left([0;1]\right)$. As $S^n = \left \lfloor N \times X^n
\r
1984 %\right \rfloor + 1$ we can deduce that: $S^n \sim
\r
1985 %\mathbf{U}\left([1;N]\right)$. Therefore: $ P\left(S^n=k\right)=\frac{1}{N}$.
\r
1987 %\begin{equation*}
\r
1988 %\begin{array}{llr}
\r
1989 %P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N
\r
1990 % P\left(S^n=k\right) & \\
\r
1991 % & =\frac{1}{2^N} \sum_{k=1}^N \frac{1}{N} & \\
\r
1992 % & =\frac{1}{2^N} & \\
\r
1995 %\begin{equation*}
\r
1996 %$\begin{array}{llr}
\r
1997 %P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N
\r
1998 % P\left(S^n=k\right) & \\
\r
1999 %P\left(X^{n+1}=e\right) & =\frac{1}{2^N} \sum_{k=1}^N \frac{1}{N} &
\r
2000 % =\frac{1}{2^N} \\
\r
2002 $P\left(x^{n+1}=k\right)=\frac{1}{2^N}$, which leads to $x^{n+1} \sim
\r
2003 \mathbf{U}\left(\mathbb{B}^N\right)$.
\r
2004 This result is true $\forall n \in \mathds{N}$, we thus have proven that the
\r
2005 stego-content $y$ is uniform in the set of possible stego-content, so
\r
2006 $y \sim \mathbf{U}\left(\mathbb{B}^N\right) \text{
\r
2007 when } x \sim \mathbf{U}\left(\mathbb{B}^N\right).$
\r
2011 \subsection{Study of chaos security of $CI_2$}\label{sec:chaos-security-ci2}
\r
2012 \subsubsection{\textbf{Topological model}}
\r
2013 \label{sec:SCISMM-topological-model}
\r
2015 %\section{THE NEW TOPOLOGICAL SPACE}
\r
2016 \noindent In this section, we prove that SCISMM can be modeled as a
\r
2017 discret dynamical system in a topological space.
\r
2018 We will show in the next section that SCISMM is a case of topological chaos in the sense of Devaney.
\r
2021 \paragraph{\textbf{Iteration Function and Phase Space}}\label{Defining}
\r
2024 %\begin{definition}[Function $F$]
\r
2028 F: & \llbracket0;\mathsf{N-1}\rrbracket \times
\r
2029 \mathds{B}^{\mathsf{N}}\times \llbracket0;\mathsf{P-1}\rrbracket \times
\r
2030 \mathds{B}^{\mathsf{P}}
\r
2031 \longrightarrow \mathds{B}^{\mathsf{N}} \\
\r
2032 & (k,x,\lambda,m) \longmapsto \left( \delta
\r
2033 (k,j).x_j+\overline{\delta (k,j)}.m_{\lambda}\right) _{j\in
\r
2034 \llbracket0;\mathsf{N-1}\rrbracket}\\
\r
2039 \noindent where + and . are the boolean addition and product operations.
\r
2041 Consider the phase space $\mathcal{X}_2$ defined as follow:
\r
2043 %\begin{definition}[Phase space $\mathcal{X}_2$]
\r
2046 \mathcal{X}_2 = \mathbb{S}_N \times \mathds{B}^\mathsf{N} \times \mathbb{S}_P
\r
2047 \times \mathds{B}^\mathsf{P} \times \mathbb{S}_P,
\r
2049 where $\mathbb{S}_N$ and $\mathbb{S}_P$ are the sets introduced in Section \ref{section:Algorithm}.
\r
2052 We define the map $\mathcal{G}_{f_0}:\mathcal{X}_2 \longrightarrow \mathcal{X}_2$ by:
\r
2055 \mathcal{G}_{f_0}\left(S_p,x,S_c,m,S_m\right) =
\r
2058 \left(\sigma_N(S_p),
\r
2059 F(i_N(S_p),x,i_P(S_c),m),\sigma_P(S_c),G_{f_0}(m,S_m),\sigma_P(S_m)\right)
\r
2062 \noindent Then SCISMM can be described by the
\r
2063 iterations of the following discret dynamical system:
\r
2065 %\begin{definition}[Discret Dynamical System for SCISMM]
\r
2069 X^0 \in \mathcal{X}_2 \\
\r
2070 X^{k+1}=\mathcal{G}_{f_0}(X^k).%
\r
2074 %\end{definition}%
\r
2077 \paragraph{\textbf{Cardinality of $\mathcal{X}_2$}}
\r
2079 By comparing $\mathcal{X}_2$ and $\mathcal{X}_1$, we have the following result.
\r
2081 \begin{proposition}
\r
2082 The phase space $\mathcal{X}_2$ has, at least, the cardinality of the continuum.
\r
2086 Let $\varphi$ be the map defined as follow:
\r
2089 \begin{array}{lcll}
\r
2090 \varphi: & \mathcal{X}_1 & \longrightarrow & \mathcal{X}_2 \\
\r
2091 & (S,x) & \longmapsto &(S,x,0,0,0)\\
\r
2095 \noindent $\varphi$ is injective.
\r
2096 So the cardinality of $\mathcal{X}_2$ is greater than or equal to the
\r
2097 cardinality of $\mathcal{X}_1$. And consequently
\r
2098 $\mathcal{X}_2$ has at least the cardinality of the continuum.
\r
2103 This result is independent on the number of cells of the system.
\r
2107 \paragraph{\textbf{A New Distance on $\mathcal{X}_2$}}
\r
2109 We define a new distance on $\mathcal{X}_2$ as follow:
\r
2110 %\begin{definition}[Distance $d_2$ on $\mathcal{X}_2$]
\r
2111 $\forall X,\check{X} \in \mathcal{X}_2,$ if $X =(S_p,x,S_c,m,S_m)$ and
\r
2112 $\check{X} = (\check{S_p},\check{x},\check{S_c},\check{m}, \check{S_m}),$ then:
\r
2114 \begin{array}{lll}
\r
2115 d_2(X,\check{X}) & = &
\r
2116 d_{\mathds{B}^{\mathsf{N}}}(x,\check{x})+d_{\mathds{B}^{\mathsf{P}}}(m,\check{m})\\
\r
2118 d_{\mathbb{S}_N}(S_p,\check{S_p})+d_{\mathbb{S}_P}(S_c,\check{S_c})+d_{\mathbb{S}_P}(S_m,\check{S_m}),\\
\r
2121 where $d_{\mathds{B}^{\mathsf{N}}}$, $d_{\mathds{B}^{\mathsf{P}}}$,
\r
2122 $d_{\mathbb{S}_N}$, and $d_{\mathbb{S}_P}$ are the same distances than in
\r
2123 Definition~\ref{def:distance-on-X1}.
\r
2131 \subsubsection{\textbf{Continuity of $CI_2$}}
\r
2133 To prove that SCISMM is another example of topological chaos in the
\r
2134 sense of Devaney, $\mathcal{G}_{f_0}$ must be continuous on the metric
\r
2135 space $(\mathcal{X}_2,d_2)$.
\r
2137 \begin{proposition}
\r
2138 $\mathcal{G}_{f_0}$ is a continuous function on $(\mathcal{X}_2,d_2)$.
\r
2142 We use the sequential continuity.
\r
2144 Let $((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)_{n\in \mathds{N}}$ be a sequence of the
\r
2145 phase space $\mathcal{X}_2$, which converges to $(S_p,x,S_c,m,S_m)$. We will
\r
2146 prove that $\left( \mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)\right)
\r
2147 _{n\in \mathds{N}}$ converges to $\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)$. Let us
\r
2148 recall that for all $n$, $(S_p)^n$, $(S_c)^n$ and $(S_m)^n$ are strategies, thus
\r
2149 we consider a sequence of strategies (\emph{i.e.}, a sequence of sequences).
\r
2152 As $d_2(((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n),(S_p,x,S_c,m,S_m))$ converges to 0,
\r
2154 $d_{\mathds{B}^{\mathsf{N}}}(x^n,x)$, $d_{\mathds{B}^{\mathsf{P}}}(m^n,m)$,
\r
2155 $d_{\mathbb{S}_N}((S_p)^n,S_p)$, $d_{\mathbb{S}_P}((S_c)^n,S_c)$, and
\r
2156 $d_{\mathbb{S}_P}((S_m)^n,S_m)$ converges to 0. But
\r
2157 $d_{\mathds{B}^{\mathsf{N}}}(x^n,x)$ and $d_{\mathds{B}^{\mathsf{P}}}(m^n,m)$
\r
2158 are integers, so $\exists n_{0}\in \mathds{N}, \forall
\r
2160 d_{\mathds{B}^{\mathsf{N}}}(x^n,x)=0$ and $\exists n_{1}\in \mathds{N},
\r
2161 \forall n \geqslant n_1, d_{\mathds{B}^{\mathsf{P}}}(m^n,m)=0$.
\r
2163 Let $n_3=Max(n_0,n_1)$.
\r
2164 In other words, there exists a threshold $n_{3}\in \mathds{N}$ after which no
\r
2165 cell will change its state:
\r
2166 $\exists n_{3}\in \mathds{N},n\geqslant n_{3}\Longrightarrow (x^n = x) \wedge
\r
2169 In addition, $d_{\mathbb{S}_N}((S_p)^n,S_p)\longrightarrow 0$,
\r
2170 $d_{\mathbb{S}_P}((S_c)^n,S_c)\longrightarrow 0$, and
\r
2171 $d_{\mathbb{S}_P}((S_m)^n,S_m)\longrightarrow 0$, so $\exists n_4,n_5,n_6\in
\r
2174 \item $\forall n \geqslant n_4, d_{\mathbb{S}_N}((S_p)^n,S_p)<10^{-1}$,
\r
2175 \item $\forall n \geqslant n_5, d_{\mathbb{S}_P}((S_c)^n,S_c)<10^{-1}$,
\r
2176 \item $\forall n \geqslant n_6, d_{\mathbb{S}_P}((S_m)^n,S_m)<10^{-1}$.
\r
2179 Let $n_7=Max(n_4,n_5,n_6)$. For
\r
2180 $n\geqslant n_7$, all the strategies $(S_p)^n$, $(S_c)^n$, and $(S_m)^n$ have
\r
2181 the same first term, which are respectively $(S_p)_0$,$(S_c)_0$ and $(S_m)_0$
\r
2182 :$\forall n\geqslant n_7,$ $$((S_p)_0^n={(S_p)}_0)\wedge
\r
2183 ((S_c)_0^n={(S_c)}_0)\wedge ((S_m)_0^n={(S_m)}_0).
\r
2186 Let $n_8=Max(n_3,n_7)$.
\r
2187 After the $n_8-$th term, states of $x^n$ and $x$ on the one hand, and $m^n$ and $m$ on the other hand, are
\r
2188 identical. Additionally, strategies $(S_p)^n$ and $S_p$, $(S_c)^n$ and $S_c$,
\r
2189 and $(S_m)^n$ and $S_m$ start with the same first term.
\r
2191 Consequently, states of
\r
2192 $\mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)$ and
\r
2193 $\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)$ are equal, so, after the $(n_8)^{th}$ term,
\r
2194 the distance $d_2$ between these two points is strictly smaller than $3.10^{-1}$, so strictly smaller than 1.
\r
2196 We now prove that the distance between $\left(
\r
2197 \mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)\right) $ and $\left(
\r
2198 \mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)\right) $ is convergent to 0.
\r
2199 Let $\varepsilon >0$.
\r
2202 \item If $\varepsilon \geqslant 1$, we have seen that distance
\r
2203 between $\mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)$ and
\r
2204 $\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)$ is
\r
2205 strictly less than 1 after the $(n_8)^{th}$ term (same state).
\r
2208 \item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
\r
2209 \frac{\varepsilon}{3} \geqslant 10^{-(k+1)}$. As
\r
2210 $d_{\mathbb{S}_N}((S_p)^n,S_p)$, $d_{\mathbb{S}_P}((S_c)^n,S_c)$ and
\r
2211 $d_{\mathbb{S}_P}((S_m)^n,S_m)$ converges to 0, we have:
\r
2213 \item $\exists n_9\in \mathds{N},\forall n\geqslant
\r
2214 n_9,d_{\mathbb{S}_N}((S_p)^n,S_p)<10^{-(k+2)}$,
\r
2215 \item $\exists n_{10}\in \mathds{N},\forall n\geqslant
\r
2216 n_{10},d_{\mathbb{S}_P}((S_c)^n,S_c)<10^{-(k+2)}$,
\r
2217 \item $\exists n_{11}\in \mathds{N},\forall n\geqslant
\r
2218 n_{11},d_{\mathbb{S}_P}((S_m)^n,S_m)<10^{-(k+2)}$.
\r
2221 Let $n_{12}=Max(n_9,n_{10},n_{11})$
\r
2222 thus after $n_{12}$, the $k+2$ first terms of $(S_p)^n$ and $S_p$, $(S_c)^n$ and
\r
2223 $S_c$, and $(S_m)^n$ and $S_m$, are equal.
\r
2226 \noindent As a consequence, the $k+1$ first entries of the strategies of
\r
2227 $\mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n)$ and $
\r
2228 \mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)$ are the same
\r
2229 (due to the shift of strategies) and following the definition of
\r
2230 $d_{\mathbb{S}_N}$ and $d_{\mathbb{S}_P}$:
\r
2231 $$d_2\left(\mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n);\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)\right)$$
\r
2233 $$d_{\mathbb{S}_N}((S_p)^n,S_p) + d_{\mathbb{S}_P}((S_c)^n,S_c) +
\r
2234 d_{\mathbb{S}_P}((S_m)^n,S_m)$$
\r
2235 which is smaller than $3.10^{-(k+1)} \leqslant 3.\frac{\varepsilon}{3} =\varepsilon$.
\r
2237 Let $N_{0}=max(n_{8},n_{12})$. We can claim that
\r
2239 \forall \varepsilon >0,\exists N_{0}\in \mathds{N},\forall n\geqslant N_{0},
\r
2242 d_2\left(\mathcal{G}_{f_0}((S_p)^n,x^n,(S_c)^n,m^n,(S_m)^n);\mathcal{G}_{f_0}(S_p,x,S_c,m,S_m)\right)
\r
2243 \leqslant \varepsilon .
\r
2245 $\mathcal{G}_{f_0}$ is consequently continuous on $(\mathcal{X}_2,d_2)$.
\r
2250 % >>>>>>>>>>>>>>>>>>>>>> Discrete chaotic iterations as topological chaos <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<
\r
2251 %\subsection{Study of chaos-security}\label{section:chaos-security-ci-2}
\r
2253 \noindent To prove that we are in the framework of Devaney's topological chaos,
\r
2254 we have to check the regularity, transitivity, and sensitivity conditions.
\r
2255 %It will be proven that $\mathcal{G}_{f_0}$ function as introduce in
\r
2256 %definition~\ref{} satisfies these three hypotheses.
\r
2260 \subsubsection{\textbf{Regularity}}
\r
2262 \begin{proposition}%[Regularity of $\mathcal{G}_{f_0}$]
\r
2263 \label{prop:regularite}
\r
2264 Periodic points of $\mathcal{G}_{f_0}$ are dense in $\mathcal{X}_2$.
\r
2268 Let $(\check{S_p},\check{x},\check{S_c},\check{m}, \check{S_m})\in
\r
2269 \mathcal{X}_2$ and $\varepsilon >0$. We are looking for a periodic point
\r
2270 $(\widetilde{S_p},\widetilde{x},\widetilde{S_c},\widetilde{m}, \widetilde{S_m})$
\r
2271 satisfying $d_2((\check{S_p},\check{x},\check{S_c},\check{m},
\r
2272 \check{S_m});(\widetilde{S_p},\widetilde{x},\widetilde{S_c},\widetilde{m},
\r
2273 \widetilde{S_m}))<\varepsilon$.
\r
2275 As $\varepsilon$ can be strictly lesser than 1, we must choose
\r
2276 $\widetilde{x} = \check{x}$ and
\r
2277 $\widetilde{m} = \check{m}$.
\r
2278 Let us define $k_0(\varepsilon) =\lfloor
\r
2279 - log_{10}(\frac{\varepsilon}{3} )\rfloor +1$ and consider the set:
\r
2280 $\mathcal{S}_{\check{S_p},\check{S_c},\check{S_m}, k_0(\varepsilon)} =$
\r
2282 \in \mathbb{S}_N \times \mathbb{S}_P \times \mathbb{S}_P/ ((S_p)^k =
\r
2283 \check{S_p}^k) \wedge ((S_c)^k =\check{S_c}^k))\right.$
\r
2284 $\left.\wedge ((S_m)^k=\check{S_m}^k)), \forall k \leqslant k_0(\varepsilon)
\r
2289 $\forall (S_p,S_c,S_m) \in
\r
2290 \mathcal{S}_{\check{S_p},\check{S_c},\check{S_m},k_0(\varepsilon)},$
\r
2291 $d_2((S_p,\check{x},S_c,\check{m},S_m);(\check{S_p},\check{x},\check{S_c},\check{m},
\r
2292 \check{S_m})) < 3.\frac{\varepsilon}{3}=\varepsilon.$
\r
2293 It remains to choose $(\widetilde{S_p},\widetilde{S_c},\widetilde{S_m}) \in
\r
2294 \mathcal{S}_{\check{S_p},\check{S_c},\check{S_m}, k_0(\varepsilon)}$ such that
\r
2295 $(\widetilde{S_p},\widetilde{x},\widetilde{S_c},\widetilde{m}, \widetilde{S_m})
\r
2296 = (\widetilde{S_p},\check{x},\widetilde{S_c},\check{m}, \widetilde{S_m})$ is
\r
2297 a periodic point for $\mathcal{G}_{f_0}$.
\r
2299 Let $\mathcal{J} =$
\r
2300 $\left\{ i \in \llbracket0;\mathsf{N-1}\rrbracket / x_i \neq
\r
2301 \check{x_i}, \text{ where }\right.$
\r
2302 $\left.(S_p,x,S_c,m,S_m) = \mathcal{G}_{f_0}^{k_0}
\r
2303 (\check{S_p},\check{x},\check{S_c},\check{m},\check{S_m}) \right\},$
\r
2304 $\lambda = card(\mathcal{J})$, and
\r
2305 $j_0 <j_1 < ... < j_{\lambda-1}$ the elements of $ \mathcal{J}$.
\r
2308 \item Let us firstly build three strategies: $S_p^\ast$, $S_c^\ast$, and
\r
2309 $S_m^\ast$, as follows.
\r
2312 $(S_p^\ast)^k=\check{S_p}^k$, $(S_c^\ast)^k=\check{S_c}^k$, and
\r
2313 $(S_m^\ast)^k=\check{S_m}^k$, if $k \leqslant k_0(\varepsilon).$
\r
2314 \item Let us now explain how to replace $\check{x}_{j_q}$, $\forall q \in
\r
2315 \llbracket0;\mathsf{\lambda-1}\rrbracket$:\newline
\r
2316 First of all, we must replace $\check{x}_{j_0}$:
\r
2318 \item If $\exists \lambda_0 \in \llbracket0;\mathsf{P-1}\rrbracket /
\r
2319 \check{x}_{j_0}=m_{\lambda_0}$, then we can choose
\r
2320 $(S_p^\ast)^{k_0+1}=j_0$, $(S_c^\ast)^{k_0+1}=\lambda_0$,
\r
2321 $(S_m^\ast)^{k_0+1}=\lambda_0$, and so $I_{j_0}$ will be equal to $1$.
\r
2322 \item If such a $\lambda_0$ does not exist, we choose:\newline
\r
2323 $(S_p^\ast)^{k_0+1}=j_0$, $(S_c^\ast)^{k_0+1}=0$, $(S_m^\ast)^{k_0+1}=0$,\newline $(S_p^\ast)^{k_0+2}=j_0$,
\r
2324 $(S_c^\ast)^{k_0+2}=0$, $(S_m^\ast)^{k_0+2}=0$, \newline and
\r
2325 $I_{j_0}=2$.\newline
\r
2327 All of the $\check{x}_{j_q}$ are replaced similarly.
\r
2328 The other terms of $S_p^\ast$, $S_c^\ast$, and
\r
2329 $S_m^\ast$ are constructed identically, and the values of $I_{j_q}$ are defined in
\r
2330 the same way.\newline
\r
2331 Let $\gamma = \sum_{q=0}^{\lambda-1}I_{j_q}$.
\r
2334 \item Finally, let $(S_p^\ast)^{k}=(S_p^\ast)^{j}$,
\r
2335 $(S_c^\ast)^{k}=(S_c^\ast)^{j}$, and $(S_m^\ast)^{k}=(S_m^\ast)^{j}$, where
\r
2336 $j\leqslant k_{0}(\varepsilon )+\gamma$ is satisfying $j\equiv k~[\text{mod } (k_{0}(\varepsilon )+\gamma)]$, if
\r
2337 $k>k_{0}(\varepsilon )+\gamma$.
\r
2340 $\mathcal{G}_{f_0}^{k_{0}(\varepsilon
\r
2341 )+\gamma}(S_p^\ast,\check{x},S_c^\ast,\check{m},S_m^\ast)=(S_p^\ast,\check{x},S_c^\ast,m,S_m^\ast).$
\r
2342 Let $\mathcal{K} =\left\{ i \in \llbracket0;\mathsf{P-1}\rrbracket / m_i \neq
\r
2343 \check{m_i}, \text{ where }\right.$\newline
\r
2344 $\left.\mathcal{G}_{f_0}^{k_{0}(\varepsilon
\r
2345 )+\gamma}(S_p^\ast,\check{x},S_c^\ast,\check{m},S_m^\ast)=(S_p^\ast,\check{x},S_c^\ast,m,S_m^\ast)
\r
2346 \right\},$\newline
\r
2347 $\mu = card(\mathcal{K})$, and
\r
2348 $r_0 <r_1 < ... < r_{\mu-1}$ the elements of $ \mathcal{K}$.
\r
2352 \item Let us now build the strategies $\widetilde{S_p}$,
\r
2353 $\widetilde{S_c}$, $\widetilde{S_m}$.
\r
2355 \item Firstly, let $\widetilde{S_p}^k=(S_p^\ast)^k$,
\r
2356 $\widetilde{S_c}^k=(S_c^\ast)^k$, and $\widetilde{S_m}^k=(S_m^\ast)^k$, if $k
\r
2357 \leqslant k_0(\varepsilon) + \gamma.$
\r
2358 \item How to replace $\check{m}_{r_q}, \forall q \in
\r
2359 \llbracket0;\mathsf{\mu-1}\rrbracket$:\newline
\r
2361 First of all, let us explain how to replace $\check{m}_{r_0}$:
\r
2363 \item If $\exists \mu_0 \in \llbracket0;\mathsf{N-1}\rrbracket /
\r
2364 \check{x}_{\mu_0}=m_{r_0}$, then
\r
2365 we can choose $\widetilde{S_p}^{k_0+\gamma +1}=\mu_0$,
\r
2366 $\widetilde{S_c}^{k_0+\gamma +1}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2367 +1}=r_0$.\newline In that situation, we define $J_{r_0}=1$.
\r
2368 \item If such a $\mu_0$ does not exist, then we can choose:\newline
\r
2369 $\widetilde{S_p}^{k_0+\gamma +1}=0$, $\widetilde{S_c}^{k_0+\gamma
\r
2370 +1}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2371 +1}=r_0$,\newline
\r
2372 $\widetilde{S_p}^{k_0+\gamma +2}=0$, $\widetilde{S_c}^{k_0+\gamma
\r
2373 +2}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2375 $\widetilde{S_p}^{k_0+\gamma +3}=0$, $\widetilde{S_c}^{k_0+\gamma
\r
2376 +3}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2380 Then the other $\check{m}_{r_q}$ are replaced as previously,
\r
2381 the other terms of $\widetilde{S_p}$,
\r
2382 $\widetilde{S_c}$, and $\widetilde{S_m}$ are constructed in the same way, and
\r
2383 the values of $J_{r_q}$ are defined similarly.\newline
\r
2384 Let $\alpha = \sum_{q=0}^{\mu-1}J_{r_q}$.
\r
2386 \item Finally, let $\widetilde{S_p}^{k}=\widetilde{S_p}^{j}$,
\r
2387 $\widetilde{S_c}^{k}=\widetilde{S_c}^{j}$, and
\r
2388 $\widetilde{S_m}^{k}=\widetilde{S_m}^{j}$ where $j\leqslant k_{0}(\varepsilon
\r
2389 )+\gamma +\alpha$ is satisfying $j\equiv k~[\text{mod } (k_{0}(\varepsilon
\r
2390 )+\gamma + \alpha)]$, if $k>k_{0}(\varepsilon )+\gamma + \alpha$.
\r
2395 So, $\mathcal{G}_{f_0}^{k_{0}(\varepsilon
\r
2396 )+\gamma+\alpha}(\widetilde{S_p},\check{x},\widetilde{S_c},\check{m},\widetilde{S_m})=(\widetilde{S_p},\check{x},\widetilde{S_c},\check{m},\widetilde{S_m})$
\r
2401 Then, $(\widetilde{S_p},\widetilde{S_c},\widetilde{S_m}) \in
\r
2402 \mathcal{S}_{\check{S_p},\check{S_c},\check{S_m},k_0(\varepsilon)}$ defined as
\r
2403 previous is such that $(\widetilde{S_m},\check{x},\widetilde{S_m},\check{m},\widetilde{S_m})$
\r
2404 is a periodic point, of period $k_{0}(\varepsilon
\r
2405 )+\gamma+\alpha$, which is $\varepsilon -$close to
\r
2406 $(\check{S_p},\check{x},\check{S_c},\check{m}, \check{S_m})$.
\r
2408 As a conclusion, $(\mathcal{X}_2,\mathcal{G}_{f_0})$ is regular.
\r
2411 \subsubsection{\textbf{Transitivity}}\label{sec:transitivite}
\r
2413 \begin{proposition}%[Transitivity of $ \mathcal{G}_{f_0}$]
\r
2414 \label{prop:transitivite}
\r
2415 $(\mathcal{X}_2,\mathcal{G}_{f_0})$ is topologically transitive.
\r
2419 Let us define $\mathcal{X}:\mathcal{X}_2\rightarrow \mathbb{B}^{\mathsf{N}},$
\r
2420 such that $\mathcal{X}(S_p,x,S_c,m,S_m)=x$ and
\r
2421 $\mathcal{M}:\mathcal{X}_2\rightarrow \mathbb{B}^{\mathsf{P}},$ such that
\r
2422 $\mathcal{M}(S_p,x,S_c,m,S_m)=m$. Let \linebreak
\r
2423 $\mathcal{B}_{A}=\mathcal{B}(X_{A},r_{A}) $ and
\r
2424 $\mathcal{B}_{B}=\mathcal{B}(X_{B},r_{B})$ be two open balls of $\mathcal{X}_2$,
\r
2425 with\linebreak $X_{A}=((S_p)_{A},x_{A},(S_c)_{A},m_{A},(S_m)_{A})$ and
\r
2426 $X_{B}=((S_p)_{B},x_{B},(S_c)_{B},m_{B},(S_m)_{B})$. We are looking for
\r
2427 $\widetilde{X}=(\widetilde{S_p},\widetilde{x},\widetilde{S_c},\widetilde{m},\widetilde{S_m})$
\r
2428 in $\mathcal{B}_{A} $ such that $\exists n_{0}\in
\r
2429 \mathbb{N},\mathcal{G}_{f_0}^{n_{0}}(\widetilde{X})\in \mathcal{B}_{B}$.\newline
\r
2430 $\widetilde{X}$ must be in $\mathcal{B}_{A}$ and $r_{A}$ can be strictly
\r
2431 lesser than 1, so $\widetilde{x}=x_{A}$ and $\widetilde{m}=m_{A}$. Let
\r
2432 $k_{0}=\lfloor - \log _{10}(\frac{r_{A}}{3})+1\rfloor $.
\r
2433 Let us notice $\mathcal{S}_{X_{A}, k_0} =\left\{ (S_p,S_c,S_m)
\r
2434 \in \mathbb{S}_\mathsf{N}\times (\mathbb{S}_\mathsf{P})^2/ \forall k\leqslant k_{0},\right.$
\r
2435 $\left.(S_p^{k}=(S_p)_{A}^{k})\wedge
\r
2436 (S_c^{k}=(S_c)_{A}^{k})\wedge (S_m^{k}=(S_m)_{A}^{k}))\right\}$.
\r
2438 Then $\forall (S_p,S_c,S_m) \in \mathcal{S}_{X_{A}, k_0},
\r
2439 (S_p,\widetilde{x},S_c,\widetilde{m},S_m)\in \mathcal{B}_{A}.$
\r
2442 Let $\mathcal{J} =\left\{i\in
\r
2443 \llbracket0,\mathsf{N-1}\rrbracket/\check{x}_{i}\neq
\r
2444 \mathcal{X}(X_{B})_{i}, \text{ where }\right.$\newline
\r
2445 $\left.(\check{S_p},\check{x},\check{S_c},\check{m},\check{S_m})=\mathcal{G}_{f_0}^{k_{0}}(X_{A})\right\},$
\r
2446 $\lambda = card(\mathcal{J})$,\newline and
\r
2447 $j_0 <j_1 < ... < j_{\lambda-1}$ the elements of $ \mathcal{J}$.
\r
2450 \item Let us firstly build three strategies: $S_p^\ast$, $S_c^\ast$, and
\r
2451 $S_m^\ast$ as follows.
\r
2454 $(S_p^\ast)^k=(S_p)_A^k$, $(S_c^\ast)^k=(S_c)_A^k$, and
\r
2455 $(S_m^\ast)^k=(S_m)_A^k$, if $k \leqslant k_0.$
\r
2456 \item Let us now explain how to replace $\mathcal{X}(X_{B})_{j_q}$, $\forall q \in
\r
2457 \llbracket0;\mathsf{\lambda-1}\rrbracket$:\newline
\r
2458 First of all, we must replace $\mathcal{X}(X_{B})_{j_0}$:
\r
2460 \item If $\exists \lambda_0 \in \llbracket0;\mathsf{P-1}\rrbracket /
\r
2461 \mathcal{X}(X_{B})_{j_0}=\check{m}_{\lambda_0}$, then we can choose
\r
2462 $(S_p^\ast)^{k_0+1}=j_0$, $(S_c^\ast)^{k_0+1}=\lambda_0$,
\r
2463 $(S_m^\ast)^{k_0+1}=\lambda_0$, and so $I_{j_0}$ will be equal to $1$.
\r
2464 \item If such a $\lambda_0$ does not exist, we choose:\newline
\r
2465 $(S_p^\ast)^{k_0+1}=j_0$, $(S_c^\ast)^{k_0+1}=0$,
\r
2466 $(S_m^\ast)^{k_0+1}=0$,\newline $(S_p^\ast)^{k_0+2}=j_0$,
\r
2467 $(S_c^\ast)^{k_0+2}=0$, $(S_m^\ast)^{k_0+2}=0$ \newline and so let us notice
\r
2468 $I_{j_0}=2$.\newline
\r
2470 All of the $\mathcal{X}(X_{B})_{j_q}$ are replaced similarly.
\r
2471 The other terms of $S_p^\ast$, $S_c^\ast$, and $S_m^\ast$ are constructed
\r
2472 identically, and the values of $I_{j_q}$ are defined on the same way.\newline Let $\gamma = \sum_{q=0}^{\lambda-1}I_{j_q}$.
\r
2475 \item $(S_p^\ast)^{k}=(S_p^\ast)^{j}$, $(S_c^\ast)^{k}=(S_c^\ast)^{j}$
\r
2476 and $(S_m^\ast)^{k}=(S_m^\ast)^{j}$ where $j\leqslant k_{0}+\gamma$
\r
2477 is satisfying $j\equiv k~[\text{mod } (k_{0}+\gamma)]$, if
\r
2480 So,$\mathcal{G}_{f_0}^{k_{0}+\gamma}((S_p^\ast,x_A,S_c^\ast,m_A,S_m^\ast))=(S_p^\ast,x_B,S_c^\ast,m,S_m^\ast)$
\r
2482 Let $\mathcal{K} =\left\{ i \in \llbracket0;\mathsf{P-1}\rrbracket / m_i \neq
\r
2483 \mathcal{M}(X_{B})_{i}, \text{ where }\right.$\newline
\r
2484 $\left.(S_p^\ast,x_B,S_c^\ast,m,S_m^\ast)=\mathcal{G}_{f_0}^{k_{0}+\gamma}((S_p^\ast,x_A,S_c^\ast,m_A,S_m^\ast))\right\},$\newline
\r
2485 $\mu = card(\mathcal{K})$ and $r_0 <r_1 < ... < r_{\mu-1}$ the elements of $
\r
2489 \item Let us secondly build three other strategies: $\widetilde{S_p}$,
\r
2490 $\widetilde{S_c}$, $\widetilde{S_m}$ as follows.
\r
2492 \item $\widetilde{S_p}^k=(S_p^\ast)^k$, $\widetilde{S_c}^k=(S_c^\ast)^k$, and
\r
2493 $\widetilde{S_m}^k=(S_m^\ast)^k$, if $k \leqslant k_0 + \gamma.$
\r
2494 \item Let us now explain how to replace $\mathcal{M}(X_{B})_{r_q}, \forall q
\r
2495 \in \llbracket0;\mathsf{\mu-1}\rrbracket$:
\r
2497 First of all, we must replace $\mathcal{M}(X_{B})_{r_0}$:
\r
2499 \item If $\exists \mu_0 \in \llbracket0;\mathsf{N-1}\rrbracket /
\r
2500 \mathcal{M}(X_{B})_{r_0}=(x_B)_{\mu_0}$, then we can choose
\r
2501 $\widetilde{S_p}^{k_0+\gamma +1}=\mu_0$, $\widetilde{S_c}^{k_0+\gamma
\r
2502 +1}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2503 +1}=r_0$, and $J_{r_0}$ will be equal to $1$.
\r
2504 \item If such a $\mu_0$ does not exist, we choose:
\r
2505 $\widetilde{S_p}^{k_0+\gamma +1}=0$, $\widetilde{S_c}^{k_0+\gamma
\r
2506 +1}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2507 +1}=r_0$,\newline
\r
2508 $\widetilde{S_p}^{k_0+\gamma +2}=0$, $\widetilde{S_c}^{k_0+\gamma
\r
2509 +2}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2511 $\widetilde{S_p}^{k_0+\gamma +3}=0$, $\widetilde{S_c}^{k_0+\gamma
\r
2512 +3}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2514 and so let us notice $J_{r_0}=3$.\newline
\r
2516 All the $\mathcal{M}(X_{B})_{r_q}$ are replaced similarly.
\r
2517 The other terms of $\widetilde{S_p}$,
\r
2518 $\widetilde{S_c}$, and $\widetilde{S_m}$ are constructed identically, and the
\r
2519 values of $J_{r_q}$ are defined on the same way.\newline
\r
2520 Let $\alpha = \sum_{q=0}^{\mu-1}J_{r_q}$.
\r
2522 \item $\forall k \in \mathbb{N}^\ast, \widetilde{S_p}^{k_0+ \gamma + \alpha
\r
2523 + k}=(S_p)_B^{k}$, $\widetilde{S_c}^{k_0+ \gamma + \alpha + k}=(S_c)_B^{k}$, and
\r
2524 $\widetilde{S_m}^{k_0+ \gamma + \alpha + k}=(S_m)_B^{k}$.
\r
2529 $\mathcal{G}_{f_0}^{k_{0}+\gamma+\alpha}(\widetilde{S_p},x_A,\widetilde{S_c},m_A,\widetilde{S_m})=X_B$,
\r
2530 with $(\widetilde{S_p},\widetilde{S_c},\widetilde{S_m}) \in \mathcal{S}_{X_{A},
\r
2532 Then $\widetilde{X} = (\widetilde{S_p},x_A,\widetilde{S_c},m_A,\widetilde{S_m})
\r
2533 \in \mathcal{X}_2$ is such that $\widetilde{X} \in \mathcal{B}_A$ and
\r
2534 $\mathcal{G}_{f_0}^{k_0+\gamma+\alpha}( \widetilde{X}) \in \mathcal{B}_B$.
\r
2535 Finally we have proven the result.
\r
2538 \subsubsection{\textbf{Sensibility to the Initial
\r
2539 Condition}}\label{sec:sensibilite}
\r
2541 \begin{proposition}%[Sensitivity of $,\mathcal{G}_{f_0}$]
\r
2542 \label{prop:sensibilite}
\r
2543 $(\mathcal{X}_2,\mathcal{G}_{f_0})$ has sensitive dependence on initial conditions.
\r
2547 $\mathcal{G}_{f_0}$ is regular and transitive. Due to Theorem~\ref{theo:banks}, $\mathcal{G}_{f_0}$ is sensitive.
\r
2550 %In a following section, we will show that this result can be stated independently of regularity,
\r
2551 %which must be redefined in the context of the finite set of
\r
2552 %machine numbers (see section \ref{Concerning}).
\r
2554 %In addition, the constant of sensitivity will be computed.
\r
2557 %This sensitive dependence could be stated as a consequence of regularity and
\r
2558 %transitivity (by using the theorem of Banks~\cite{Banks92}). However, we
\r
2559 %prefer to prove this result independently of regularity, because the
\r
2560 %notion of regularity must be redefined in the context of the finite set of
\r
2561 %machine numbers (see section \ref{Concerning}).
\r
2563 %In addition, the constant of sensitivity has been computed.
\r
2565 \subsubsection{\textbf{Devaney's chaos}}
\r
2569 In conclusion, $(\mathcal{X}_2,\mathcal{G}_{f_0})$ is topologically transitive, regular,
\r
2570 and has sensitive dependence on initial conditions. Then according to
\r
2572 defintition~\ref{def:chaos-devaney}~\vpageref{def:chaos-devaney}, we have the result.
\r
2574 \begin{theorem}%[$\mathcal{G}_{f_0}$ is a chaotic map]
\r
2575 \label{theo:chaotic}
\r
2576 $\mathcal{G}_{f_0}$ is a chaotic map on $(\mathcal{X}_2,d_2)$ in the sense of Devaney.
\r
2579 So we can claim that:
\r
2581 \begin{theorem}%[Chaos-security of SCISMM]
\r
2582 \label{theo:SCISMM-chaosecurity}
\r
2583 SCISMM is chaos-secure.
\r
2587 \subsection{Quantitative property of
\r
2588 $CI_2$}\label{sec:quantitative-properties-ci2}
\r
2590 In this section, it is studied one of the quantitative properties of the
\r
2591 chaos-security of the steganographic process $CI_2$: the constant of sensitivity
\r
2594 \begin{proposition}%[Constant of sensitivity of $ \mathcal{G}_{f_0}$]
\r
2595 \label{prop:sensitivity-constant-evaluation}
\r
2597 $(\mathcal{X}_2,\mathcal{G}_{f_0})$ has sensitive dependence on initial
\r
2598 conditions, and it constant of sensitivity is equal to $N-1$.
\r
2602 Let us define $\mathcal{X}:\mathcal{X}_2\rightarrow \mathbb{B}^{\mathsf{N}},$
\r
2603 such that $\mathcal{X}(S_p,x,S_c,m,S_m)=x$ and
\r
2604 $\mathcal{M}:\mathcal{X}_2\rightarrow \mathbb{B}^{\mathsf{P}},$ such that
\r
2605 $\mathcal{M}(S_p,x,S_c,m,S_m)=m$.
\r
2607 Let $\check{X}=(\check{S_p},\check{x},\check{S_c},\check{m}, \check{S_m})\in
\r
2608 \mathcal{X}_2$. We are looking for
\r
2609 $\widetilde{X}=(\widetilde{S_p},\widetilde{x},\widetilde{S_c},\widetilde{m},\widetilde{S_m})$
\r
2610 in $\mathcal{X}_2$ such that $d_2\left(\check{X},\widetilde{X} \right) \leq
\r
2611 \delta$ and $\exists n_0 \in \mathds{N},
\r
2612 d_2\left(\mathcal{G}_{f_0}^{n_0}(\check{X}),\mathcal{G}_{f_0}^{n_0}(\widetilde{X})
\r
2613 \right) \geq N-1$.
\r
2614 Let $k_{0}=\lfloor - \log _{10}(\frac{\delta}{3}\rfloor +1$,
\r
2615 and consider the set:
\r
2616 $\mathcal{S}_{\check{S_p},\check{S_c},\check{S_m}, k_0} =$
\r
2618 \in \mathbb{S}_N \times \mathbb{S}_P \times \mathbb{S}_P/ ((S_p)^k =
\r
2619 \check{S_p}^k) \wedge ((S_c)^k =\check{S_c}^k))\right.$
\r
2620 $\left.\wedge ((S_m)^k=\check{S_m}^k)), \forall k \leqslant k_0(\varepsilon)
\r
2623 $\forall (S_p,S_c,S_m) \in
\r
2624 \mathcal{S}_{\check{S_p},\check{S_c},\check{S_m},k_0},$
\r
2625 $d_2((S_p,\check{x},S_c,\check{m},S_m);(\check{S_p},\check{x},\check{S_c},\check{m},
\r
2626 \check{S_m})) < 3.\frac{\delta}{3}=\delta.$
\r
2628 Let $\mathcal{J} =\left\{i\in
\r
2629 \llbracket0,\mathsf{N-1}\rrbracket/
\r
2630 \mathcal{X}\left(\mathcal{G}_{f_0}^{k_{0}}(\check{S})\right)_{i} \neq
\r
2631 \mathcal{X}\left(\mathcal{G}_{f_0}^{k_{0}+N}(\check{S})\right)_{i}\right\}$ the
\r
2632 set of indexes of cells of the second component different between the two states
\r
2633 $\mathcal{G}_{f_0}^{k_{0}}(\check{X})$ and
\r
2634 $\mathcal{G}_{f_0}^{k_{0}+N}(\check{X})$.
\r
2636 Let $\lambda = card(\mathcal{J})$ the number of differences.
\r
2639 \item If $\lambda = N$, then the point
\r
2640 $(\widetilde{S_p},\widetilde{x},\widetilde{S_c},\widetilde{m},\widetilde{S_m})$
\r
2641 defined by: \begin{enumerate}
\r
2642 \item $\widetilde{x}$ = $\check{x}$ and $\widetilde{m}$ = $\check{m}$ in
\r
2643 order to have the same cells that $\check{X}$, and thus be at a distance less than 1 of $\check{X}$.
\r
2644 \item $\forall k \leqslant k_0, \widetilde{S_p}^k= \check{S_p}^k$,
\r
2645 $\widetilde{S_c}^k= \check{S_c}^k$, and $\widetilde{S_m}^k= \check{S_m}^k$.
\r
2646 \item Let us now explain how to replace $\mathcal{X}(X_{B})_{j_q}$, $\forall q \in
\r
2647 \llbracket0;\mathsf{\lambda-1}\rrbracket$:\newline
\r
2648 First of all, we must replace $\mathcal{X}(X_{B})_{j_0}$:
\r
2650 \item If $\exists \lambda_0 \in \llbracket0;\mathsf{P-1}\rrbracket /
\r
2651 \mathcal{X}(X_{B})_{j_0}=\check{m}_{\lambda_0}$, then we can choose
\r
2652 $(S_p^\ast)^{k_0+1}=j_0$, $(S_c^\ast)^{k_0+1}=\lambda_0$,
\r
2653 $(S_m^\ast)^{k_0+1}=\lambda_0$, and so $I_{j_0}$ will be equal to $1$.
\r
2654 \item If such a $\lambda_0$ does not exist, we choose:\newline
\r
2655 $(S_p^\ast)^{k_0+1}=j_0$, $(S_c^\ast)^{k_0+1}=0$,
\r
2656 $(S_m^\ast)^{k_0+1}=0$,\newline $(S_p^\ast)^{k_0+2}=j_0$,
\r
2657 $(S_c^\ast)^{k_0+2}=0$, $(S_m^\ast)^{k_0+2}=0$ \newline and so let us notice
\r
2658 $I_{j_0}=2$.\newline
\r
2660 All of the $\mathcal{X}(X_{B})_{j_q}$ are replaced similarly.
\r
2661 The other terms of $S_p^\ast$, $S_c^\ast$, and $S_m^\ast$ are constructed
\r
2662 identically, and the values of $I_{j_q}$ are defined on the same way.\newline Let $\gamma = \sum_{q=0}^{\lambda-1}I_{j_q}$.
\r
2665 \item $(S_p^\ast)^{k}=(S_p^\ast)^{j}$, $(S_c^\ast)^{k}=(S_c^\ast)^{j}$
\r
2666 and $(S_m^\ast)^{k}=(S_m^\ast)^{j}$ where $j\leqslant k_{0}+\gamma$
\r
2667 is satisfying $j\equiv k~[\text{mod } (k_{0}+\gamma)]$, if
\r
2670 So,$\mathcal{G}_{f_0}^{k_{0}+\gamma}((S_p^\ast,x_A,S_c^\ast,m_A,S_m^\ast))=(S_p^\ast,x_B,S_c^\ast,m,S_m^\ast)$
\r
2672 Let $\mathcal{K} =\left\{ i \in \llbracket0;\mathsf{P-1}\rrbracket / m_i \neq
\r
2673 \mathcal{M}(X_{B})_{i}, \text{ where }\right.$\newline
\r
2674 $\left.(S_p^\ast,x_B,S_c^\ast,m,S_m^\ast)=\mathcal{G}_{f_0}^{k_{0}+\gamma}((S_p^\ast,x_A,S_c^\ast,m_A,S_m^\ast))\right\},$\newline
\r
2675 $\mu = card(\mathcal{K})$ and $r_0 <r_1 < ... < r_{\mu-1}$ the elements of $
\r
2679 \item Let us secondly build three other strategies: $\widetilde{S_p}$,
\r
2680 $\widetilde{S_c}$, $\widetilde{S_m}$ as follows.
\r
2682 \item $\widetilde{S_p}^k=(S_p^\ast)^k$, $\widetilde{S_c}^k=(S_c^\ast)^k$, and
\r
2683 $\widetilde{S_m}^k=(S_m^\ast)^k$, if $k \leqslant k_0 + \gamma.$
\r
2684 \item Let us now explain how to replace $\mathcal{M}(X_{B})_{r_q}, \forall q
\r
2685 \in \llbracket0;\mathsf{\mu-1}\rrbracket$:
\r
2687 First of all, we must replace $\mathcal{M}(X_{B})_{r_0}$:
\r
2689 \item If $\exists \mu_0 \in \llbracket0;\mathsf{N-1}\rrbracket /
\r
2690 \mathcal{M}(X_{B})_{r_0}=(x_B)_{\mu_0}$, then we can choose
\r
2691 $\widetilde{S_p}^{k_0+\gamma +1}=\mu_0$, $\widetilde{S_c}^{k_0+\gamma
\r
2692 +1}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2693 +1}=r_0$, and $J_{r_0}$ will be equal to $1$.
\r
2694 \item If such a $\mu_0$ does not exist, we choose:
\r
2695 $\widetilde{S_p}^{k_0+\gamma +1}=0$, $\widetilde{S_c}^{k_0+\gamma
\r
2696 +1}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2697 +1}=r_0$,\newline
\r
2698 $\widetilde{S_p}^{k_0+\gamma +2}=0$, $\widetilde{S_c}^{k_0+\gamma
\r
2699 +2}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2701 $\widetilde{S_p}^{k_0+\gamma +3}=0$, $\widetilde{S_c}^{k_0+\gamma
\r
2702 +3}=r_0$, $\widetilde{S_m}^{k_0+\gamma
\r
2704 and so let us notice $J_{r_0}=3$.\newline
\r
2706 All the $\mathcal{M}(X_{B})_{r_q}$ are replaced similarly.
\r
2707 The other terms of $\widetilde{S_p}$,
\r
2708 $\widetilde{S_c}$, and $\widetilde{S_m}$ are constructed identically, and the
\r
2709 values of $J_{r_q}$ are defined on the same way.\newline
\r
2710 Let $\alpha = \sum_{q=0}^{\mu-1}J_{r_q}$.
\r
2712 \item $\forall k \in \mathbb{N}^\ast, \widetilde{S_p}^{k_0+ \gamma + \alpha
\r
2713 + k}=(S_p)_B^{k}$, $\widetilde{S_c}^{k_0+ \gamma + \alpha + k}=(S_c)_B^{k}$, and
\r
2714 $\widetilde{S_m}^{k_0+ \gamma + \alpha + k}=(S_m)_B^{k}$.
\r
2719 $\mathcal{G}_{f_0}^{k_{0}+\gamma+\alpha}(\widetilde{S_p},x_A,\widetilde{S_c},m_A,\widetilde{S_m})=X_B$,
\r
2720 with $(\widetilde{S_p},\widetilde{S_c},\widetilde{S_m}) \in \mathcal{S}_{X_{A},
\r
2722 Then $\widetilde{X} = (\widetilde{S_p},x_A,\widetilde{S_c},m_A,\widetilde{S_m})
\r
2723 \in \mathcal{X}_2$ is such that $\widetilde{X} \in \mathcal{B}_A$ and
\r
2724 $\mathcal{G}_{f_0}^{k_0+\gamma+\alpha}( \widetilde{X}) \in \mathcal{B}_B$.
\r
2725 Finally we have proven the result.
\r
2729 \subsection{Qualitative property of
\r
2730 $CI_2$}\label{sec:qualitative-properties-ci2}
\r
2732 In this section, it is studied one of the qualitative properties of the
\r
2733 chaos-security of the steganographic process $CI_2$: the strong transitivity.
\r
2739 \subsection{Consequences in attacks
\r
2740 frameworks}\label{sec:qualitative-properties-ci2}
\r
2742 \textbf{TODO} In this section, it is given the implication of the qualitative
\r
2743 and quantitative properties of $CI_2$ in KOA, KMA,CMA and WOA setups.
\r
2747 \section{Conclusion and future work}\label{sec:conclusion}
\r
2749 In this research work, the two information hiding schemes based on chaotic
\r
2750 iterations proposed in \cite{gfb10:ip} and \cite{fgb11:ip}
\r
2751 have been recalled.
\r
2752 Additionally, all the previous contributions in the field of chaos-based
\r
2753 security for information hiding have been summed up. Furthermore, the robustness
\r
2754 study of the first scheme initiated in \cite{guyeux10ter} has been widely
\r
2755 improved, by using a large number of images, parameters, and attacks. Finally,
\r
2756 the security study of the new version of our information hiding scheme has been
\r
2757 deepened, by giving qualitative and quantitative measures of its level of
\r
2758 chaos-security. In particular, the constant of sensibility and the qualitative
\r
2759 properties of strong transitivity and indecomposability have been
\r
2760 established.\newline
\r
2762 The comparison between the currently only stego and chaos secure schemes is
\r
2764 Table~\ref{table:security-processes-comparison}~\vpageref{table:security-processes-comparison}.\newline
\r
2766 In future work, we intend to compare the robustness and security of these
\r
2767 schemes with other existing information hiding algorithms. Among other things, a
\r
2768 complete comparison between spread-spectrum and chaotic iterations will be
\r
2769 realized, and the security of other existing schemes will be studied in the
\r
2770 framework of chaos-security. Concerning the theoretical aspects of security, we
\r
2771 intend to understand links and implications between the two existing security
\r
2772 frameworks for information hiding: the stego-security and the chaos-security.
\r
2773 Finally, their relation with intended requirements in information hiding will be
\r
2777 \newcolumntype{V}{>{\centering\arraybackslash} m{1cm} }
\r
2778 %\resizebox{8cm}{!} {
\r
2781 %\begin{tabularx}{8cm}{|c|V|V|V|V|}
\r
2782 \begin{tabular}{|c|c|c|c|c|c|c|}\cline{2-3}
\r
2784 & \multirow{2}{*}{\textbf{KOA}} & \multirow{2}{*}{\textbf{KMA}} &
\r
2785 \multirow{2}{*}{\textbf{WOA}} & \multirow{2}{*}{\textbf{CMA}} &
\r
2786 \textbf{Bits}\\ & & & & & \textbf{embeded}\\
\r
2791 \textbf{$\mathcal{NW}$ \tiny($\eta=1$)}
\r
2793 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-normal-jaune}}
\r
2795 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-normal-jaune}}
\r
2797 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
\r
2799 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/interrogation}}
\r
2805 \textbf{$\mathcal{CIW}_1$} &
\r
2806 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
\r
2808 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
\r
2810 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
\r
2812 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
\r
2818 \textbf{$\mathcal{CIS}_2$} &
\r
2819 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
\r
2821 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
\r
2823 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
\r
2825 \parbox[c]{0.4cm}{\includegraphics[width=0.5cm]{img/smiley-sourire-vert}}
\r
2832 \caption{$\mathcal{NW}$, $\mathcal{CIW}_1$, and $\mathcal{CIS}_2$ security.
\r
2833 Embedding capacity.}
\r
2834 \label{table:security-processes-comparison}
\r
2839 \textcolor{red}{\textbf{ENVISAGER DE PARLER DE $\mathcal{DI}_1$ POUR LE
\r
2840 COMPARER AUX AUTRES}}
\r
2845 % To cite in reference all the Bibtex entries.
\r
2848 %\bibliographystyle{compj}
\r
2849 %\bibliography{ModellingBidders}
\r
2851 \bibliographystyle{compj}
\r
2852 \bibliography{jabref}
\r
