X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/9e057cd5768916849c2767ef4bd0f54dd9adc3b4..ddb01e4b5bfe53afe6dba0b77f3d5322ac38c81f:/prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 90f00f8..3228083 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -418,7 +418,7 @@ the metric space $(\mathcal{X},d)$.
 \end{proposition}
 
 The chaotic property of $G_f$ has been firstly established for the vectorial
-Boolean negation $f(x_1,\hdots, x_\mathsf{N}) =  (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly
+Boolean negation $f_0(x_1,\hdots, x_\mathsf{N}) =  (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly
 introduced the notion of asynchronous iteration graph recalled bellow.
 
 Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The
@@ -475,33 +475,58 @@ Let us finally remark that the vectorial negation satisfies the hypotheses of bo
 
 We have proposed in~\cite{bgw09:ip} a new family of generators that receives 
 two PRNGs as inputs. These two generators are mixed with chaotic iterations, 
-leading thus to a new PRNG that improves the statistical properties of each
-generator taken alone. Furthermore, our generator 
-possesses various chaos properties that none of the generators used as input
+leading thus to a new PRNG that 
+\begin{color}{red}
+should improves the statistical properties of each
+generator taken alone. 
+Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input
 present.
 
 
+
 \begin{algorithm}[h!]
 \begin{small}
 \KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$
 ($n$ bits)}
 \KwOut{a configuration $x$ ($n$ bits)}
 $x\leftarrow x^0$\;
-$k\leftarrow b + \textit{XORshift}(b)$\;
+$k\leftarrow b + PRNG_1(b)$\;
 \For{$i=0,\dots,k$}
 {
-$s\leftarrow{\textit{XORshift}(n)}$\;
+$s\leftarrow{PRNG_2(n)}$\;
 $x\leftarrow{F_f(s,x)}$\;
 }
 return $x$\;
 \end{small}
-\caption{PRNG with chaotic functions}
+\caption{An arbitrary round of $Old~ CI~ PRNG_f(PRNG_1,PRNG_2)$}
 \label{CI Algorithm}
 \end{algorithm}
 
 
 
 
+This generator is synthesized in Algorithm~\ref{CI Algorithm}.
+It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:CaractÃ©risation   des   IC   chaotiques};
+an integer $b$, ensuring that the number of executed iterations
+between two outputs is at least $b$
+and at most $2b+1$; and an initial configuration $x^0$.
+It returns the new generated configuration $x$.  Internally, it embeds two
+inputted generators $PRNG_i(k), i=1,2$,
+ which must return integers
+uniformly distributed
+into $\llbracket 1 ; k \rrbracket$.
+For instance, these PRNGs can be the \textit{XORshift}~\cite{Marsaglia2003},
+being a category of very fast PRNGs designed by George Marsaglia
+that repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number
+with a bit shifted version of it. Such a PRNG, which has a period of
+$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. 
+This XORshift, or any other reasonable PRNG, is used
+in our own generator to compute both the number of iterations between two
+outputs (provided by $PRNG_1$) and the strategy elements ($PRNG_2$).
+
+%This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones.
+
+
 \begin{algorithm}[h!]
 \begin{small}
 \KwIn{the internal configuration $z$ (a 32-bit word)}
@@ -517,31 +542,95 @@ return $y$\;
 \end{algorithm}
 
 
+\subsection{A ``New CI PRNG''}
+
+In order to make the Old CI PRNG usable in practice, we have proposed 
+an adapted version of the chaotic iteration based generator in~\cite{bg10:ip}.
+In this ``New CI PRNG'', we prevent from changing twice a given
+bit between two outputs.
+This new generator is designed by the following process. 
+
+First of all, some chaotic iterations have to be done to generate a sequence 
+$\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$ 
+of Boolean vectors, which are the successive states of the iterated system. 
+Some of these vectors will be randomly extracted and our pseudo-random bit 
+flow will be constituted by their components. Such chaotic iterations are 
+realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean 
+vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in 
+\llbracket 1, 32 \rrbracket^\mathds{N}$ is
+an \emph{irregular decimation} of $PRNG_2$ sequence, as described in 
+Algorithm~\ref{Chaotic iteration1}.
+
+Then, at each iteration, only the $S^n$-th component of state $x^n$ is 
+updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$.
+Such a procedure is equivalent to achieve chaotic iterations with
+the Boolean vectorial negation $f_0$ and some well-chosen strategies.
+Finally, some $x^n$ are selected
+by a sequence $m^n$ as the pseudo-random bit sequence of our generator.
+$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers.
 
+The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}.
+The internal state is $x$, the output state is $r$. $a$ and $b$ are those computed by the two input
+PRNGs. Lastly, the value $g(a)$ is an integer defined as in Eq.~\ref{Formula}.
+This function is required to make the outputs uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$
+(the reader is referred to~\cite{bg10:ip} for more information).
 
+\begin{equation}
+\label{Formula}
+m^n = g(y^n)=
+\left\{
+\begin{array}{l}
+0 \text{ if }0 \leqslant{y^n}<{C^0_{32}},\\
+1 \text{ if }{C^0_{32}} \leqslant{y^n}<\sum_{i=0}^1{C^i_{32}},\\
+2 \text{ if }\sum_{i=0}^1{C^i_{32}} \leqslant{y^n}<\sum_{i=0}^2{C^i_{32}},\\
+\vdots~~~~~ ~~\vdots~~~ ~~~~\\
+N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
+\end{array}
+\right.
+\end{equation}
 
-This generator is synthesized in Algorithm~\ref{CI Algorithm}.
-It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:CaractÃ©risation   des   IC   chaotiques};
-an integer $b$, ensuring that the number of executed iterations is at least $b$
-and at most $2b+1$; and an initial configuration $x^0$.
-It returns the new generated configuration $x$.  Internally, it embeds two
-\textit{XORshift}$(k)$ PRNGs~\cite{Marsaglia2003} that return integers
-uniformly distributed
-into $\llbracket 1 ; k \rrbracket$.
-\textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia,
-which repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number
-with a bit shifted version of it. This PRNG, which has a period of
-$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used
-in our PRNG to compute the strategy length and the strategy elements.
+\begin{algorithm}
+\textbf{Input:} the internal state $x$ (32 bits)\\
+\textbf{Output:} a state $r$ of 32 bits
+\begin{algorithmic}[1]
+\FOR{$i=0,\dots,N$}
+{
+\STATE$d_i\leftarrow{0}$\;
+}
+\ENDFOR
+\STATE$a\leftarrow{PRNG_1()}$\;
+\STATE$m\leftarrow{g(a)}$\;
+\STATE$k\leftarrow{m}$\;
+\WHILE{$i=0,\dots,k$}
+
+\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
+\STATE$S\leftarrow{b}$\;
+    \IF{$d_S=0$}
+    {
+\STATE      $x_S\leftarrow{ \overline{x_S}}$\;
+\STATE      $d_S\leftarrow{1}$\;
 
-This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones.
+    }
+    \ELSIF{$d_S=1$}
+    {
+\STATE      $k\leftarrow{ k+1}$\;
+    }\ENDIF
+\ENDWHILE\\
+\STATE $r\leftarrow{x}$\;
+\STATE return $r$\;
+\medskip
+\caption{An arbitrary round of the new CI generator}
+\label{Chaotic iteration1}
+\end{algorithmic}
+\end{algorithm}
+\end{color}
 
 \subsection{Improving the Speed of the Former Generator}
 
-Instead of updating only one cell at each iteration, we can try to choose a
-subset of components and to update them together. Such an attempt leads
-to a kind of merger of the two sequences used in Algorithm 
-\ref{CI Algorithm}. When the updating function is the vectorial negation,
+Instead of updating only one cell at each iteration,\begin{color}{red} we now propose to choose a
+subset of components and to update them together, for speed improvements. Such a proposition leads\end{color}
+to a kind of merger of the two sequences used in Algorithms 
+\ref{CI Algorithm} and \ref{Chaotic iteration1}. When the updating function is the vectorial negation,
 this algorithm can be rewritten as follows:
 
 \begin{equation}
@@ -551,7 +640,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N
 \forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
 \end{array}
 \right.
-\label{equation Oplus}
+\label{equation Oplus0}
 \end{equation}
 where $\oplus$ is for the bitwise exclusive or between two integers. 
 This rewriting can be understood as follows. The $n-$th term $S^n$ of the
@@ -561,7 +650,7 @@ as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th
 component of this state (a binary digit) changes if and only if the $k-$th 
 digit in the binary decomposition of $S^n$ is 1.
 
-The single basic component presented in Eq.~\ref{equation Oplus} is of 
+The single basic component presented in Eq.~\ref{equation Oplus0} is of 
 ordinary use as a good elementary brick in various PRNGs. It corresponds
 to the following discrete dynamical system in chaotic iterations:
 
@@ -582,8 +671,8 @@ than the ones presented in Definition \ref{Def:chaotic iterations} because, inst
 we select a subset of components to change.
 
 
-Obviously, replacing Algorithm~\ref{CI Algorithm} by 
-Equation~\ref{equation Oplus}, which is possible when the iteration function is
+Obviously, replacing the previous CI PRNG Algorithms by 
+Equation~\ref{equation Oplus0}, which is possible when the iteration function is
 the vectorial negation, leads to a speed improvement. However, proofs
 of chaos obtained in~\cite{bg10:ij} have been established
 only for chaotic iterations of the form presented in Definition 
@@ -838,166 +927,43 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 
 
 \begin{color}{red}
-\section{Improving Statistical Properties Using Chaotic Iterations}
-
-
-\subsection{The CIPRNG family}
-
-Three categories of PRNGs have been derived from chaotic iterations. They are
-recalled in what follows.
-
-\subsubsection{Old CIPRNG}
-
-Let $\mathsf{N} = 4$. Some chaotic iterations are fulfilled to generate a sequence $\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^4\right)^\mathds{N}$ of Boolean vectors: the successive states of the iterated system. Some of these vectors are randomly extracted and their components constitute our pseudorandom bit flow~\cite{bgw09:ip}.
-Chaotic iterations are realized as follows. Initial state $x^0 \in \mathds{B}^4$ is a Boolean vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in \llbracket 1, 4 \rrbracket^\mathds{N}$ is constructed with $PRNG_2$. Lastly, iterate function $f$ is the vectorial Boolean negation.
-At each iteration, only the $S^n$-th component of state $x^n$ is updated. Finally, some $x^n$ are selected by a sequence $m^n$, provided by a second generator $PRNG_1$, as the pseudorandom bit sequence of our generator.
-
-The basic design procedure of the Old CI generator is summed up in Algorithm~\ref{Chaotic iteration}.
-The internal state is $x$, the output array is $r$. $a$ and $b$ are those computed by $PRNG_1$ and $PRNG_2$.
-
-
-\begin{algorithm}
-\textbf{Input:} the internal state $x$ (an array of 4-bit words)\\
-\textbf{Output:} an array $r$ of 4-bit words
-\begin{algorithmic}[1]
-
-\STATE$a\leftarrow{PRNG_1()}$;
-\STATE$m\leftarrow{a~mod~2+13}$;
-\WHILE{$i=0,\dots,m$}
-\STATE$b\leftarrow{PRNG_2()}$;
-\STATE$S\leftarrow{b~mod~4}$;
-\STATE$x_S\leftarrow{ \overline{x_S}}$;
-\ENDWHILE
-\STATE$r\leftarrow{x}$;
-\STATE return $r$;
-\medskip
-\caption{An arbitrary round of the old CI generator}
-\label{Chaotic iteration}
-\end{algorithmic}
-\end{algorithm}
-
-\subsubsection{New CIPRNG}
-
-The New CI generator is designed by the following process~\cite{bg10:ip}. First of all, some chaotic iterations have to be done to generate a sequence $\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$ of Boolean vectors, which are the successive states of the iterated system. Some of these vectors will be randomly extracted and our pseudo-random bit flow will be constituted by their components. Such chaotic iterations are realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in \llbracket 1, 32 \rrbracket^\mathds{N}$ is
-an \emph{irregular decimation} of $PRNG_2$ sequence, as described in Algorithm~\ref{Chaotic iteration1}.
+\section{Statistical Improvements Using Chaotic Iterations}
 
-Another time, at each iteration, only the $S^n$-th component of state $x^n$ is updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$.
-Finally, some $x^n$ are selected
-by a sequence $m^n$ as the pseudo-random bit sequence of our generator.
-$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers.
-
-The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}.
-The internal state is $x$, the output state is $r$. $a$ and $b$ are those computed by the two input
-PRNGs. Lastly, the value $g_1(a)$ is an integer defined as in Eq.~\ref{Formula}.
-
-\begin{equation}
-\label{Formula}
-m^n = g_1(y^n)=
-\left\{
-\begin{array}{l}
-0 \text{ if }0 \leqslant{y^n}<{C^0_{32}},\\
-1 \text{ if }{C^0_{32}} \leqslant{y^n}<\sum_{i=0}^1{C^i_{32}},\\
-2 \text{ if }\sum_{i=0}^1{C^i_{32}} \leqslant{y^n}<\sum_{i=0}^2{C^i_{32}},\\
-\vdots~~~~~ ~~\vdots~~~ ~~~~\\
-N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
-\end{array}
-\right.
-\end{equation}
-
-\begin{algorithm}
-\textbf{Input:} the internal state $x$ (32 bits)\\
-\textbf{Output:} a state $r$ of 32 bits
-\begin{algorithmic}[1]
-\FOR{$i=0,\dots,N$}
-{
-\STATE$d_i\leftarrow{0}$\;
-}
-\ENDFOR
-\STATE$a\leftarrow{PRNG_1()}$\;
-\STATE$m\leftarrow{f(a)}$\;
-\STATE$k\leftarrow{m}$\;
-\WHILE{$i=0,\dots,k$}
-
-\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
-\STATE$S\leftarrow{b}$\;
-    \IF{$d_S=0$}
-    {
-\STATE      $x_S\leftarrow{ \overline{x_S}}$\;
-\STATE      $d_S\leftarrow{1}$\;
-
-    }
-    \ELSIF{$d_S=1$}
-    {
-\STATE      $k\leftarrow{ k+1}$\;
-    }\ENDIF
-\ENDWHILE\\
-\STATE $r\leftarrow{x}$\;
-\STATE return $r$\;
-\medskip
-\caption{An arbitrary round of the new CI generator}
-\label{Chaotic iteration1}
-\end{algorithmic}
-\end{algorithm}
-
-
-\subsubsection{Xor CIPRNG}
-
-Instead of updating only one cell at each iteration as Old CI and New CI, we can try to choose a
-subset of components and to update them together. Such an attempt leads
-to a kind of merger of the two random sequences. When the updating function is the vectorial negation,
-this algorithm can be rewritten as follows~\cite{arxivRCCGPCH}:
-
-\begin{equation}
-\left\{
-\begin{array}{l}
-x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\
-\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
-\end{array}
-\right.
-\label{equation Oplus}
-\end{equation}
-%This rewriting can be understood as follows. The $n-$th term $S^n$ of the
-%sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents
-%the list of cells to update in the state $x^n$ of the system (represented
-%as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th
-%component of this state (a binary digit) changes if and only if the $k-$th
-%digit in the binary decomposition of $S^n$ is 1.
-
-The single basic component presented in Eq.~\ref{equation Oplus} is of
-ordinary use as a good elementary brick in various PRNGs. It corresponds
-to the discrete dynamical system in chaotic iterations.
-
-\subsection{About some Well-known PRNGs}
 \label{The generation of pseudo-random sequence}
 
 
-
-
-Let us now give illustration on the fact that chaos appears to improve statistical properties.
+Let us now explain why we are reasonable grounds to believe that chaos 
+can improve statistical properties.
+We will show in this section that, when mixing defective PRNGs with
+chaotic iterations, the result presents better statistical properties
+(this section summarizes the work of~\cite{bfg12a:ip}).
 
 \subsection{Details of some Existing Generators}
 
-Here are the modules of PRNGs we have chosen to experiment.
+The list of defective PRNGs we will use 
+as inputs for the statistical tests to come is introduced here.
 
-\subsubsection{LCG}
-This PRNG implements either the simple or the combined linear congruency generator (LCGs). The simple LCG is defined by the recurrence:
+Firstly, the simple linear congruency generator (LCGs) will be used. 
+It is 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 less than $m$~\cite{testU01}. In what follows, 2LCGs and 3LCGs refer as two (resp. three) combinations of such LCGs.
-For further details, see~\cite{combined_lcg}.
+where $a$, $c$, and $x^0$ must be, among other things, non-negative and less than 
+$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer as two (resp. three) 
+combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
 
-\subsubsection{MRG}
-This module implements multiple recursive generators (MRGs), based on a linear recurrence of order $k$, modulo $m$~\cite{testU01}:
+Secondly, the multiple recursive generators (MRGs) will be used too, which
+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}
-Combination of two MRGs (referred as 2MRGs) is also be used in this paper.
+Combination of two MRGs (referred as 2MRGs) is also used in these experimentations.
 
-\subsubsection{UCARRY}
-Generators based on linear recurrences with carry are implemented in this module. This includes the add-with-carry (AWC) generator, based on the recurrence:
+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}
@@ -1019,16 +985,14 @@ and the SWC generator designed by R. Couture, which is based on the following re
 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}
 
-\subsubsection{GFSR}
-This module implements the generalized feedback shift register (GFSR) generator, that is:
+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}
 
 
-\subsubsection{INV}
-Finally, this module implements the nonlinear inversive generator, as defined in~\cite{testU01}, which is:
+Finally, the nonlinear inversive generator~\cite{LEcuyerS07} has been regarded too, which is:
 
 \begin{equation}
 \label{INV}
@@ -1045,45 +1009,14 @@ a^1 & \text{if}~  z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
 \subsection{Statistical tests}
 \label{Security analysis}
 
-%A theoretical proof for the randomness of a generator is impossible to give, therefore statistical inference based on observed sample sequences produced by the generator seems to be the best option.
-Considering the properties of binary random sequences, various statistical tests can be designed to evaluate the assertion that the sequence is generated by a perfectly random source. We have performed some statistical tests for the CIPRNGs proposed here. These tests include NIST suite~\cite{ANDREW2008} and DieHARD battery of tests~\cite{DieHARD}. For completeness and for reference, we give in the following subsection a brief description of each of the aforementioned tests.
-
-
-
-\subsubsection{NIST statistical tests suite}
-
-Among the numerous standard tests for pseudo-randomness, a convincing way to show the randomness of the produced sequences is to confront them to the NIST (National Institute of  Standards and Technology) statistical tests, being an up-to-date tests suite proposed by the Information Technology Laboratory (ITL). A new version of the Statistical tests suite has been released in August 11, 2010.
+Three batteries of tests are reputed and usually 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.
 
-The NIST tests suite SP 800-22 is a statistical package consisting of 15 tests. They were developed to test the randomness of binary sequences produced by hardware or software based cryptographic pseudorandom number generators. These tests focus on a variety of different types of non-randomness that could exist in a sequence.
 
-For each statistical test, a set of $P-values$ (corresponding to the set of sequences) is produced.
-The interpretation of empirical results can be conducted in various ways.
-In this paper, the examination of the distribution of P-values to check for uniformity ($ P-value_{T}$) is used.
-The distribution of $P-values$ is examined to ensure uniformity.
-If $P-value_{T} \geqslant 0.0001$, then the sequences can be considered to be uniformly distributed.
 
-In our experiments, 100 sequences (s = 100), each with 1,000,000-bit long, are generated and tested. If the $P-value_{T}$ of any test is smaller than 0.0001, the sequences are considered to be not good enough and the generating algorithm is not suitable for usage.
-
-
-
-
-
-\subsubsection{DieHARD battery of tests}
-The DieHARD battery of tests has been the most sophisticated standard for over a decade. Because of the stringent requirements in the DieHARD tests suite, a generator passing this battery of
-tests can be considered good as a rule of thumb.
-
-The DieHARD battery of tests consists of 18 different independent statistical tests. This collection
- of tests is based on assessing the randomness of bits comprising 32-bit integers obtained from
-a random number generator. Each test requires $2^{23}$ 32-bit integers in order to run the full set
-of tests. Most of the tests in DieHARD return a $P-value$, which should be uniform on $[0,1)$ if the input file
-contains truly independent random bits.  These $P-values$ are obtained by
-$P=F(X)$, where $F$ is the assumed distribution of the sample random variable $X$ (often normal).
-But that assumed $F$ is just an asymptotic approximation, for which the fit will be worst
-in the tails. Thus occasional $P-values$ near 0 or 1, such as 0.0012 or 0.9983, can occur.
-An individual test is considered to be failed if the $P-value$ approaches 1 closely, for example $P>0.9999$.
-
-
-\subsection{Results and discussion}
 \label{Results and discussion}
 \begin{table*}
 \renewcommand{\arraystretch}{1.3}
@@ -1099,16 +1032,21 @@ DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18
 \end{tabular}
 \end{table*}
 
-Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the results on the batteries recalled above, indicating that almost all the PRNGs cannot pass all their tests. In other words, the statistical quality of these PRNGs cannot fulfill the up-to-date standards presented previously. We will show that the CIPRNG can solve this issue.
-
-To illustrate the effects of this CIPRNG in detail, experiments will be divided in three parts:
+Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the 
+results on the two firsts 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 $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket,$
+  \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}
-x_i^n=\left\{
+\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}
