+\begin{proof}
+We use the sequential continuity.
+Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
+\mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
+G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
+G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy,
+thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of
+sequences).\newline
+As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
+to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
+d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline
+In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
+cell will change its state:
+$\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$
+
+In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
+\mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
+n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
+first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$
+
+Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
+identical and strategies $S^n$ and $S$ start with the same first term.\newline
+Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
+so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline
+\noindent We now prove that the distance between $\left(
+G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to
+0. Let $\varepsilon >0$. \medskip
+\begin{itemize}
+\item If $\varepsilon \geqslant 1$, we see that the distance
+between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
+strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
+\medskip
+\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
+\varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
+\begin{equation*}
+\exists n_{2}\in \mathds{N},\forall n\geqslant
+n_{2},d_{s}(S^n,S)<10^{-(k+2)},
+\end{equation*}%
+thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
+\end{itemize}
+\noindent As a consequence, the $k+1$ first entries of the strategies of $%
+G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
+the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
+10^{-(k+1)}\leqslant \varepsilon $.
+
+In conclusion,
+%%RAPH : ici j'ai rajouté une ligne
+%%TOF : ici j'ai rajouté un commentaire
+%%TOF : ici aussi
+$
+\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
+,$ $\forall n\geqslant N_{0},$
+$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
+\leqslant \varepsilon .
+$
+$G_{f}$ is consequently continuous.
+\end{proof}
+
+
+It is now possible to study the topological behavior of the general chaotic
+iterations. We will prove that,
+
+\begin{theorem}
+\label{t:chaos des general}
+ The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
+the Devaney's property of chaos.
+\end{theorem}
+
+Let us firstly prove the following lemma.
+
+\begin{lemma}[Strong transitivity]
+\label{strongTrans}
+ For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can
+find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
+\end{lemma}
+
+\begin{proof}
+ Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$.
+Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$,
+are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define
+$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
+We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
+that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
+the form $(S',E')$ where $E'=E$ and $S'$ starts with
+$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
+\begin{itemize}
+ \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
+ \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
+\end{itemize}
+Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$,
+where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
+claimed in the lemma.
+\end{proof}
+
+We can now prove the Theorem~\ref{t:chaos des general}.
+
+\begin{proof}[Theorem~\ref{t:chaos des general}]
+Firstly, strong transitivity implies transitivity.
+
+Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
+prove that $G_f$ is regular, it is sufficient to prove that
+there exists a strategy $\tilde S$ such that the distance between
+$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
+$(\tilde S,E)$ is a periodic point.
+
+Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
+configuration that we obtain from $(S,E)$ after $t_1$ iterations of
+$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$
+and $t_2\in\mathds{N}$ such
+that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
+
+Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
+of $S$ and the first $t_2$ terms of $S'$:
+%%RAPH : j'ai coupé la ligne en 2
+$$\tilde
+S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,$$$$\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
+is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
+$t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
+point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
+have $d((S,E),(\tilde S,E))<\epsilon$.
+\end{proof}
+
+
+%\section{Statistical Improvements Using Chaotic Iterations}
+
+%\label{The generation of pseudorandom sequence}
+
+
+%Let us now explain why we have reasonable ground to believe that chaos
+%can improve statistical properties.
+%We will show in this section that chaotic properties as defined in the
+%mathematical theory of chaos are related to some statistical tests that can be found
+%in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with
+%chaotic iterations, the new generator presents better statistical properties
+%(this section summarizes and extends the work of~\cite{bfg12a:ip}).
+
+
+
+%\subsection{Qualitative relations between topological properties and statistical tests}
+
+
+%There are various relations between topological properties that describe an unpredictable behavior for a discrete
+%dynamical system on the one
+%hand, and statistical tests to check the randomness of a numerical sequence
+%on the other hand. These two mathematical disciplines follow a similar
+%objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a
+%recurrent sequence), with two different but complementary approaches.
+%It is true that the following illustrative links give only qualitative arguments,
+%and proofs should be provided later to make such arguments irrefutable. However
+%they give a first understanding of the reason why we think that chaotic properties should tend
+%to improve the statistical quality of PRNGs.
+%%
+%Let us now list some of these relations between topological properties defined in the mathematical
+%theory of chaos and tests embedded into the NIST battery. %Such relations need to be further
+%%investigated, but they presently give a first illustration of a trend to search similar properties in the
+%%two following fields: mathematical chaos and statistics.
+
+
+%\begin{itemize}
+% \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney}, a chaotic dynamical system must
+%have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of
+%a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity
+%is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a
+%knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in
+%the two following NIST tests~\cite{Nist10}:
+% \begin{itemize}
+% \item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern.
+% \item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are close one to another) in the tested sequence that would indicate a deviation from the assumption of randomness.
+% \end{itemize}
+
+%\item \textbf{Transitivity}. This topological property previously introduced states that the dynamical system is intrinsically complicated: it cannot be simplified into
+%two subsystems that do not interact, as we can find in any neighborhood of any point another point whose orbit visits the whole phase space.
+%This focus on the places visited by the orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory
+%of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention
+%is brought on the states visited during a random walk in the two tests below~\cite{Nist10}:
+% \begin{itemize}
+% \item \textbf{Random Excursions Variant Test}. Detect deviations from the expected number of visits to various states in the random walk.
+% \item \textbf{Random Excursions Test}. Determine if the number of visits to a particular state within a cycle deviates from what one would expect for a random sequence.
+% \end{itemize}
+
+%\item \textbf{Chaos according to Li and Yorke}. Two points of the phase space $(x,y)$ define a couple of Li-Yorke when $\limsup_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))>0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillate as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according
+%to Li-Yorke~\cite{Li75,Ruette2001}. A similar property is regarded in the following NIST test~\cite{Nist10}.
+% \begin{itemize}
+% \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow.
+% \end{itemize}
+% \item \textbf{Topological entropy}. The desire to formulate an equivalency of the thermodynamics entropy
+%has emerged both in the topological and statistical fields. Once again, a similar objective has led to two different
+%rewritting of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach,
+%whereas topological entropy is defined as follows:
+%$x,y \in \mathcal{X}$ are $\varepsilon-$\emph{separated in time $n$} if there exists $k \leqslant n$ such that $d\left(f^{(k)}(x),f^{(k)}(y)\right)>\varepsilon$. Then $(n,\varepsilon)-$separated sets are sets of points that are all $\varepsilon-$separated in time $n$, which
+%leads to the definition of $s_n(\varepsilon,Y)$, being the maximal cardinality of all $(n,\varepsilon)-$separated sets. Using these notations,
+%the topological entropy is defined as follows: $$h_{top}(\mathcal{X},f) = \displaystyle{\lim_{\varepsilon \rightarrow 0} \Big[ \limsup_{n \rightarrow +\infty} \dfrac{1}{n} \log s_n(\varepsilon,\mathcal{X})\Big]}.$$
+%This value measures the average exponential growth of the number of distinguishable orbit segments.
+%In this sense, it measures the complexity of the topological dynamical system, whereas
+%the Shannon approach comes to mind when defining the following test~\cite{Nist10}:
+% \begin{itemize}
+%\item \textbf{Approximate Entropy Test}. Compare the frequency of the overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence.
+% \end{itemize}
+
+% \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are
+%not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}.
+% \begin{itemize}
+%\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence.
+%\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random.
+% \end{itemize}
+%\end{itemize}
+
+
+%We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques} are, among other
+%things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke,
+%and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$,
+%where $\mathsf{N}$ is the size of the iterated vector.
+%These topological properties make that we are ground to believe that a generator based on chaotic
+%iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like
+%the NIST one. The following subsections, in which we prove that defective generators have their
+%statistical properties improved by chaotic iterations, show that such an assumption is true.
+
+%\subsection{Details of some Existing Generators}
+
+%The list of defective PRNGs we will use
+%as inputs for the statistical tests to come is introduced here.
+
+%Firstly, the simple linear congruency generators (LCGs) will be used.
+%They are defined by the following recurrence:
+%\begin{equation}
+%x^n = (ax^{n-1} + c)~mod~m,
+%\label{LCG}
+%\end{equation}
+%where $a$, $c$, and $x^0$ must be, among other things, non-negative and inferior to
+%$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer to two (resp. three)
+%combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
+
+%Secondly, the multiple recursive generators (MRGs) which will be used,
+%are based on a linear recurrence of order
+%$k$, modulo $m$~\cite{LEcuyerS07}:
+%\begin{equation}
+%x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m .
+%\label{MRG}
+%\end{equation}
+%The combination of two MRGs (referred as 2MRGs) is also used in these experiments.
+
+%Generators based on linear recurrences with carry will be regarded too.
+%This family of generators includes the add-with-carry (AWC) generator, based on the recurrence:
+%\begin{equation}
+%\label{AWC}
+%\begin{array}{l}
+%x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
+%c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
+%the SWB generator, having the recurrence:
+%\begin{equation}
+%\label{SWB}
+%\begin{array}{l}
+%x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
+%c^n=\left\{
+%\begin{array}{l}
+%1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
+%0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
+%and the SWC generator, which is based on the following recurrence:
+%\begin{equation}
+%\label{SWC}
+%\begin{array}{l}
+%x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
+%c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}
+
+%Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
+%\begin{equation}
+%x^n = x^{n-r} \oplus x^{n-k} .
+%\label{GFSR}
+%\end{equation}
+
+
+%Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:
+
+%\begin{equation}
+%\label{INV}
+%\begin{array}{l}
+%x^n=\left\{
+%\begin{array}{ll}
+%(a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\
+%a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
+
+
+
+%\begin{table}
+%%\renewcommand{\arraystretch}{1}
+%\caption{TestU01 Statistical Test Failures}
+%\label{TestU011}
+%\centering
+% \begin{tabular}{lccccc}
+% \toprule
+%Test name &Tests& Logistic & XORshift & ISAAC\\
+%Rabbit & 38 &21 &14 &0 \\
+%Alphabit & 17 &16 &9 &0 \\
+%Pseudo DieHARD &126 &0 &2 &0 \\
+%FIPS\_140\_2 &16 &0 &0 &0 \\
+%SmallCrush &15 &4 &5 &0 \\
+%Crush &144 &95 &57 &0 \\
+%Big Crush &160 &125 &55 &0 \\ \hline
+%Failures & &261 &146 &0 \\
+%\bottomrule
+% \end{tabular}
+%\end{table}
+
+
+
+%\begin{table}
+%%\renewcommand{\arraystretch}{1}
+%\caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
+%\label{TestU01 for Old CI}
+%\centering
+% \begin{tabular}{lcccc}
+% \toprule
+%\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
+%&Logistic& XORshift& ISAAC&ISAAC \\
+%&+& +& + & + \\
+%&Logistic& XORshift& XORshift&ISAAC \\ \cmidrule(r){2-5}
+%Rabbit &7 &2 &0 &0 \\
+%Alphabit & 3 &0 &0 &0 \\
+%DieHARD &0 &0 &0 &0 \\
+%FIPS\_140\_2 &0 &0 &0 &0 \\
+%SmallCrush &2 &0 &0 &0 \\
+%Crush &47 &4 &0 &0 \\
+%Big Crush &79 &3 &0 &0 \\ \hline
+%Failures &138 &9 &0 &0 \\
+%\bottomrule
+% \end{tabular}
+%\end{table}
+
+
+
+
+
+%\subsection{Statistical tests}
+%\label{Security analysis}
+
+%Three batteries of tests are reputed and regularly used
+%to evaluate the statistical properties of newly designed pseudorandom
+%number generators. These batteries are named DieHard~\cite{Marsaglia1996},
+%the NIST suite~\cite{ANDREW2008}, and the most stringent one called
+%TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
+
+
+
+%\label{Results and discussion}
+%\begin{table*}
+%%\renewcommand{\arraystretch}{1}
+%\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
+%\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
+%\centering
+% \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
+% \hline\hline
+%Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
+%NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15} & 14/15 & 14/15 & 14/15 & 14/15& 14/15& 14/15 \\ \hline
+%DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
+%\end{tabular}
+%\end{table*}
+
+%Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the
+%results on the two first batteries recalled above, indicating that all the PRNGs presented
+%in the previous section
+%cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot
+%fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
+%iterations can solve this issue.
+%%More precisely, to
+%%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
+%%\begin{enumerate}
+%% \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
+%% \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
+%% \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
+%%\begin{equation}
+%%\begin{array}{l}
+%%\left\{
+%%\begin{array}{l}
+%%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
+%%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
+%%\end{equation}
+%%$m$ is called the \emph{functional power}.
+%%\end{enumerate}
+%%
+%The obtained results are reproduced in Table
+%\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
+%The scores written in boldface indicate that all the tests have been passed successfully, whereas an
+%asterisk ``*'' means that the considered passing rate has been improved.
+%The improvements are obvious for both the ``Old CI'' and the ``New CI'' generators.
+%Concerning the ``Xor CI PRNG'', the score is less spectacular. Because of a large speed improvement, the statistics
+% are not as good as for the two other versions of these CIPRNGs.
+%However 8 tests have been improved (with no deflation for the other results).
+
+
+%\begin{table*}
+%%\renewcommand{\arraystretch}{1.3}
+%\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
+%\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
+%\centering
+% \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
+% \hline
+%Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
+%Old CIPRNG\\ \hline \hline
+%NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
+%New CIPRNG\\ \hline \hline
+%NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
+%Xor CIPRNG\\ \hline\hline
+%NIST & 14/15*& \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} \\ \hline
+%DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
+%\end{tabular}
+%\end{table*}
+
+
+%We have then investigated in~\cite{bfg12a:ip} if it were possible to improve
+%the statistical behavior of the Xor CI version by combining more than one
+%$\oplus$ operation. Results are summarized in Table~\ref{threshold}, illustrating
+%the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in.
+%Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained
+%using chaotic iterations on defective generators.
+
+%\begin{table*}
+%%\renewcommand{\arraystretch}{1.3}
+%\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
+%\label{threshold}
+%\centering
+% \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
+% \hline
+%Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3 & MRG2 \\ \hline\hline
+%Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline
+%\end{tabular}
+%\end{table*}
+
+%Finally, the TestU01 battery has been launched on three well-known generators
+%(a logistic map, a simple XORshift, and the cryptographically secure ISAAC,
+%see Table~\ref{TestU011}). These results can be compared with
+%Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
+%Old CI PRNG that has received these generators.
+%The obvious improvement speaks for itself, and together with the other
+%results recalled in this section, it reinforces the opinion that a strong
+%correlation between topological properties and statistical behavior exists.
+
+
+%The next subsection will now give a concrete original implementation of the Xor CI PRNG, the
+%fastest generator in the chaotic iteration based family. In the remainder,
+%this generator will be simply referred to as CIPRNG, or ``the proposed PRNG'', if this statement does not
+%raise ambiguity.
+
+
+\section{Toward Efficiency and Improvement for CI PRNG}
+\label{sec:efficient PRNG}
+
+\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
+%
+%Based on the proof presented in the previous section, it is now possible to
+%improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}.
+%The first idea is to consider
+%that the provided strategy is a pseudorandom Boolean vector obtained by a
+%given PRNG.
+%An iteration of the system is simply the bitwise exclusive or between
+%the last computed state and the current strategy.
+%Topological properties of disorder exhibited by chaotic
+%iterations can be inherited by the inputted generator, we hope by doing so to
+%obtain some statistical improvements while preserving speed.
+%
+%%RAPH : j'ai viré tout ca
+%% Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
+%% are
+%% done.
+%% Suppose that $x$ and the strategy $S^i$ are given as
+%% binary vectors.
+%% Table~\ref{TableExemple} shows the result of $x \oplus S^i$.
+
+%% \begin{table}
+%% \begin{scriptsize}
+%% $$
+%% \begin{array}{|cc|cccccccccccccccc|}
+%% \hline
+%% x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
+%% \hline
+%% S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
+%% \hline
+%% x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
+%% \hline
+
+%% \hline
+%% \end{array}
+%% $$
+%% \end{scriptsize}
+%% \caption{Example of an arbitrary round of the proposed generator}
+%% \label{TableExemple}
+%% \end{table}
+
+
+
+
+\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label={algo:seqCIPRNG}}
+\begin{small}
