+%\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}
