-Both these experimentations allows us to conclude that it is possible to
-generate a huge number of pseudorandom numbers with the xor-like version and
-about tens times less with the BBS based version. The former version has only
-chaotic properties whereas the latter also has cryptographically properties.
-
-
-%% \section{Cryptanalysis of the Proposed PRNG}
-
-
-%% Mettre ici la preuve de PCH
-
-%\section{The relativity of disorder}
-%\label{sec:de la relativité du désordre}
-
-%In the next two sections, we investigate the impact of the choices that have
-%lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}.
-
-%\subsection{Impact of the topology's finenesse}
-
-%Let us firstly introduce the following notations.
-
-%\begin{notation}
-%$\mathcal{X}_\tau$ will denote the topological space
-%$\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set
-%of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply
-%$\mathcal{V} (x)$, if there is no ambiguity).
-%\end{notation}
-
-
-
-%\begin{theorem}
-%\label{Th:chaos et finesse}
-%Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t.
-%$\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous
-%both for $\tau$ and $\tau'$.
-
-%If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then
-%$(\mathcal{X}_\tau,f)$ is chaotic too.
-%\end{theorem}
-
-%\begin{proof}
-%Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$.
-
-%Let $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in
-%\tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we
-%can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) =
-%\varnothing$. Consequently, $f$ is $\tau-$transitive.
-
-%Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for
-%all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a
-%periodic point for $f$ into $V$.
-
-%Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood
-%of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.
-
-%But $\tau \subset \tau'$, so $\omega \in \tau'$, and then $V \in
-%\mathcal{V}_{\tau'} (x)$. As $(\mathcal{X}_{\tau'},f)$ is regular, there is a
-%periodic point for $f$ into $V$, and the regularity of $(\mathcal{X}_\tau,f)$ is
-%proven.
-%\end{proof}
-
-%\subsection{A given system can always be claimed as chaotic}
-
-%Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point.
-%Then this function is chaotic (in a certain way):
-
-%\begin{theorem}
-%Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having
-%at least a fixed point.
-%Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete)
-%topology on $\X$.
-%\end{theorem}
-
-
-%\begin{proof}
-%$f$ is transitive when $\forall \omega, \omega' \in \tau_0 \setminus
-%\{\varnothing\}, \exists n \in \mathds{N}, f^{(n)}(\omega) \cap \omega' \neq
-%\varnothing$.
-%As $\tau_0 = \left\{ \varnothing, \X \right\}$, this is equivalent to look for
-%an integer $n$ s.t. $f^{(n)}\left( \X \right) \cap \X \neq \varnothing$. For
-%instance, $n=0$ is appropriate.
-
-%Let us now consider $x \in \X$ and $V \in \mathcal{V}_{\tau_0} (x)$. Then $V =
-%\mathcal{X}$, so $V$ has at least a fixed point for $f$. Consequently $f$ is
-%regular, and the result is established.
-%\end{proof}
-
-
-
-
-%\subsection{A given system can always be claimed as non-chaotic}
-
-%\begin{theorem}
-%Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$.
-%If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic
-%(for the Devaney's formulation), where $\tau_\infty$ is the discrete topology.
-%\end{theorem}
-
-%\begin{proof}
-%Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty},
-%f\right)$ is both transitive and regular.
-
-%Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must
-%contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty},
-%f\right)$ is regular. Then $x$ must be a periodic point of $f$.
-
-%Let $I_x = \left\{ f^{(n)}(x), n \in \mathds{N}\right\}$. This set is finite
-%because $x$ is periodic, and $\mathcal{X}$ is infinite, then $\exists y \in
-%\mathcal{X}, y \notin I_x$.
-
-%As $\left(\X_{\tau_\infty}, f\right)$ must be transitive, for all open nonempty
-%sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq
-%\varnothing$. However $\{x\}$ and $\{y\}$ are open sets and $y \notin I_x
-%\Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$.
-%\end{proof}
-
-
-
-
-
-
-%\section{Chaos on the order topology}
-%\label{sec: chaos order topology}
-%\subsection{The phase space is an interval of the real line}
-
-%\subsubsection{Toward a topological semiconjugacy}
-
-%In what follows, our intention is to establish, by using a topological
-%semiconjugacy, that chaotic iterations over $\mathcal{X}$ can be described as
-%iterations on a real interval. To do so, we must firstly introduce some
-%notations and terminologies.
-
-%Let $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket
-%1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N}
-%\times \B^\mathsf{N}$.
-
-
-%\begin{definition}
-%The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[
-%0, 2^{10} \big[$ is defined by:
-%\begin{equation}
-% \begin{array}{cccl}
-%\varphi: & \mathcal{X}_{10} = \mathcal{S}_{10} \times\mathds{B}^{10}&
-%\longrightarrow & \big[ 0, 2^{10} \big[ \\
-% & (S,E) = \left((S^0, S^1, \hdots ); (E_0, \hdots, E_9)\right) & \longmapsto &
-%\varphi \left((S,E)\right)
-%\end{array}
-%\end{equation}
-%where $\varphi\left((S,E)\right)$ is the real number:
-%\begin{itemize}
-%\item whose integral part $e$ is $\displaystyle{\sum_{k=0}^9 2^{9-k} E_k}$, that
-%is, the binary digits of $e$ are $E_0 ~ E_1 ~ \hdots ~ E_9$.
-%\item whose decimal part $s$ is equal to $s = 0,S^0~ S^1~ S^2~ \hdots =
-%\sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$
-%\end{itemize}
-%\end{definition}
-
-
-
-%$\varphi$ realizes the association between a point of $\mathcal{X}_{10}$ and a
-%real number into $\big[ 0, 2^{10} \big[$. We must now translate the chaotic
-%iterations $\Go$ on this real interval. To do so, two intermediate functions
-%over $\big[ 0, 2^{10} \big[$ must be introduced:
-
-
-%\begin{definition}
-%\label{def:e et s}
-%Let $x \in \big[ 0, 2^{10} \big[$ and:
-%\begin{itemize}
-%\item $e_0, \hdots, e_9$ the binary digits of the integral part of $x$:
-%$\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{9} 2^{9-k} e_k}$.
-%\item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, where the chosen decimal
-%decomposition of $x$ is the one that does not have an infinite number of 9:
-%$\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k 10^{-k-1}}$.
-%\end{itemize}
-%$e$ and $s$ are thus defined as follows:
-%\begin{equation}
-%\begin{array}{cccl}
-%e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\
-% & x & \longmapsto & (e_0, \hdots, e_9)
-%\end{array}
-%\end{equation}
-%and
-%\begin{equation}
-% \begin{array}{cccc}
-%s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9
-%\rrbracket^{\mathds{N}} \\
-% & x & \longmapsto & (s^k)_{k \in \mathds{N}}
-%\end{array}
-%\end{equation}
-%\end{definition}
-
-%We are now able to define the function $g$, whose goal is to translate the
-%chaotic iterations $\Go$ on an interval of $\mathds{R}$.
-
-%\begin{definition}
-%$g:\big[ 0, 2^{10} \big[ \longrightarrow \big[ 0, 2^{10} \big[$ is defined by:
-%\begin{equation}
-%\begin{array}{cccc}
-%g: & \big[ 0, 2^{10} \big[ & \longrightarrow & \big[ 0, 2^{10} \big[ \\
-% & x & \longmapsto & g(x)
-%\end{array}
-%\end{equation}
-%where g(x) is the real number of $\big[ 0, 2^{10} \big[$ defined bellow:
-%\begin{itemize}
-%\item its integral part has a binary decomposition equal to $e_0', \hdots,
-%e_9'$, with:
-% \begin{equation}
-%e_i' = \left\{
-%\begin{array}{ll}
-%e(x)_i & \textrm{ if } i \neq s^0\\
-%e(x)_i + 1 \textrm{ (mod 2)} & \textrm{ if } i = s^0\\
-%\end{array}
-%\right.
-%\end{equation}
-%\item whose decimal part is $s(x)^1, s(x)^2, \hdots$
-%\end{itemize}
-%\end{definition}
-
-%\bigskip
-
-
-%In other words, if $x = \displaystyle{\sum_{k=0}^{9} 2^{9-k} e_k +
-%\sum_{k=0}^{+\infty} s^{k} ~10^{-k-1}}$, then:
-%\begin{equation}
-%g(x) =
-%\displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) +
-%\sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}.
-%\end{equation}
-
-
-%\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$}
-
-%Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most
-%usual one being the Euclidian distance recalled bellow:
-
-%\begin{notation}
-%\index{distance!euclidienne}
-%$\Delta$ is the Euclidian distance on $\big[ 0, 2^{10} \big[$, that is,
-%$\Delta(x,y) = |y-x|^2$.
-%\end{notation}
-
-%\medskip
-
-%This Euclidian distance does not reproduce exactly the notion of proximity
-%induced by our first distance $d$ on $\X$. Indeed $d$ is finer than $\Delta$.
-%This is the reason why we have to introduce the following metric:
-
-
-
-%\begin{definition}
-%Let $x,y \in \big[ 0, 2^{10} \big[$.
-%$D$ denotes the function from $\big[ 0, 2^{10} \big[^2$ to $\mathds{R}^+$
-%defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s\left(s(x),s(y)\right)$,
-%where:
-%\begin{center}
-%$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k,
-%\check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty
-%\dfrac{|S^k-\check{S}^k|}{10^k}}$.
-%\end{center}
-%\end{definition}
