\usepackage[ruled,vlined]{algorithm2e}
\usepackage{listings}
\usepackage[standard]{ntheorem}
+\usepackage{algorithmic}
+\usepackage{slashbox}
% Pour mathds : les ensembles IR, IN, etc.
\usepackage{dsfont}
generator is cryptographically secure, then it is the case too for the
generator provided by the post-treatment.
Such a proof leads to the proposition of a cryptographically secure and
-chaotic generator on GPU based on the famous Blum Blum Shum
+chaotic generator on GPU based on the famous Blum Blum Shub
in Section~\ref{sec:CSGPU}, and to an improvement of the
Blum-Goldwasser protocol in Sect.~\ref{Blum-Goldwasser}.
This research work ends by a conclusion section, in which the contribution is
are continuous. For further explanations, see, e.g., \cite{guyeux10}.
Let $\delta $ be the \emph{discrete Boolean metric}, $\delta
-(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function:
-%%RAPH : ici j'ai coupé la dernière ligne en 2, c'est moche mais bon
-\begin{equation}
+(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function
+$F_{f}: \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}}
+\longrightarrow \mathds{B}^{\mathsf{N}}$
+\begin{equation*}
\begin{array}{lrll}
-F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} &
-\longrightarrow & \mathds{B}^{\mathsf{N}} \\
-& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+ \right.\\
-& & & \left. f(E)_{k}.\overline{\delta
-(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+ f(E)_{k}.\overline{\delta
+(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%
\end{array}%
-\end{equation}%
+\end{equation*}%
\noindent where + and . are the Boolean addition and product operations.
Consider the phase space:
\begin{equation}
\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
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:Caractérisation 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)}
\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:Caractérisation 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.
-
-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}
+\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 $d_S\leftarrow{1}$\;
+
+ }
+ \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}
\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
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:
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
-the vectorial negation, leads to a speed improvement. However, proofs
+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
+(the resulting generator will be referred as ``Xor CI PRNG''
+in what follows).
+However, proofs
of chaos obtained in~\cite{bg10:ij} have been established
only for chaotic iterations of the form presented in Definition
\ref{Def:chaotic iterations}. The question is now to determine whether the
\subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
\label{deuxième def}
Let us consider the discrete dynamical systems in chaotic iterations having
-the general form:
+the general form: $\forall n\in \mathds{N}^{\ast }$, $ \forall i\in
+\llbracket1;\mathsf{N}\rrbracket $,
\begin{equation}
-\forall n\in \mathds{N}^{\ast }, \forall i\in
-\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+ x_i^n=\left\{
\begin{array}{ll}
x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\
\left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.
Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
-%%RAPH : j'ai coupé la dernière ligne en 2, c'est moche
-\begin{equation}
-\begin{array}{lrll}
-F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} &
-\longrightarrow & \mathds{B}^{\mathsf{N}} \\
-& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+\right.\\
-& & &\left.f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+$F_{f}: \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}}
+\longrightarrow \mathds{B}^{\mathsf{N}}$
+\begin{equation*}
+\begin{array}{rll}
+ (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%
\end{array}%
-\end{equation}%
+\end{equation*}%
where + and . are the Boolean addition and product operations, and $\overline{x}$
is the negation of the Boolean $x$.
Consider the phase space:
\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 $.\bigskip \newline
+10^{-(k+1)}\leqslant \varepsilon $.
+
In conclusion,
%%RAPH : ici j'ai rajouté une ligne
-\begin{flushleft}$$
-\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}%
-,\forall n\geqslant N_{0},$$
-$$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
+$
+\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
+,$ $\forall n\geqslant N_{0},$
+$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
\leqslant \varepsilon .
-$$
-\end{flushleft}
+$
$G_{f}$ is consequently continuous.
\end{proof}
claimed in the lemma.
\end{proof}
-<<<<<<< HEAD
We can now prove the Theorem~\ref{t:chaos des general}.
-=======
-We can now prove Theorem~\ref{t:chaos des general}...
->>>>>>> e55d237aba022a66cc2d7650d295b29169878f45
\begin{proof}[Theorem~\ref{t:chaos des general}]
Firstly, strong transitivity implies transitivity.
\end{proof}
+\begin{color}{red}
+\section{Statistical Improvements Using Chaotic Iterations}
+
+\label{The generation of pseudo-random sequence}
+
+
+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}
+
+The list of defective PRNGs we will use
+as inputs for the statistical tests to come is introduced here.
+
+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{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}.
+
+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 used in these experimentations.
-\section{Efficient PRNG based on Chaotic Iterations}
+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 designed by R. Couture, 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 generator~\cite{LEcuyerS07} has been regarded too, 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}
+
+
+
+
+
+\subsection{Statistical tests}
+\label{Security analysis}
+
+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.
+
+
+
+\label{Results and discussion}
+\begin{table*}
+\renewcommand{\arraystretch}{1.3}
+\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 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 $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
+%\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 ``New CI'' generators.
+Concerning the ``Xor CI PRNG'', the speed improvement makes that statistical
+results are not as good as for the two other versions of these CIPRNGs.
+
+
+\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 investigate in~\cite{bfg12a:ip} if it is possible to improve
+the statistical behavior of the Xor CI version by combining more than one
+$\oplus$ operation. Results are summarized in~\ref{threshold}, showing
+that 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*}
+
+Next subsection gives a concrete implementation of this Xor CI PRNG, which will
+new be simply called CIPRNG, or ``the proposed PRNG'', if this statement does not
+raise ambiguity.
+\end{color}
+
+\subsection{Efficient PRNG based on Chaotic Iterations}
\label{sec:efficient PRNG}
Based on the proof presented in the previous section, it is now possible to
algorithm (Algorithm~\ref{algo:gpu_kernel2}). Due to Proposition~\ref{cryptopreuve},
it simply consists in replacing
the {\it xor-like} PRNG by a cryptographically secure one.
-We have chosen the Blum Blum Shum generator~\cite{BBS} (usually denoted by BBS) having the form:
+We have chosen the Blum Blum Shub generator~\cite{BBS} (usually denoted by BBS) having the form:
$$x_{n+1}=x_n^2~ mod~ M$$ where $M$ is the product of two prime numbers (these
prime numbers need to be congruent to 3 modulus 4). BBS is known to be
very slow and only usable for cryptographic applications.
+\begin{color}{red}
+\subsection{Practical Security Evaluation}
+
+Suppose now that the PRNG will work during
+$M=100$ time units, and that during this period,
+an attacker can realize $10^{12}$ clock cycles.
+We thus wonder whether, during the PRNG's
+lifetime, the attacker can distinguish this
+sequence from truly random one, with a probability
+greater than $\varepsilon = 0.2$.
+We consider that $N$ has 900 bits.
+
+The random process is the BBS generator, which
+is cryptographically secure. More precisely, it
+is $(T,\varepsilon)-$secure: no
+$(T,\varepsilon)-$distinguishing attack can be
+successfully realized on this PRNG, if~\cite{Fischlin}
+$$
+T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
+$$
+where $M$ is the length of the output ($M=100$ in
+our example), and $L(N)$ is equal to
+$$
+2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right)
+$$
+is the number of clock cycles to factor a $N-$bit
+integer.
+
+A direct numerical application shows that this attacker
+cannot achieve its $(10^{12},0.2)$ distinguishing
+attack in that context.
+
+\end{color}
+
\subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
\label{Blum-Goldwasser}
We finish this research work by giving some thoughts about the use of
Furthermore, we have shown that when the inputted generator is cryptographically
secure, then it is the case too for the PRNG we propose, thus leading to
the possibility to develop fast and secure PRNGs using the GPU architecture.
-Thoughts about an improvement of the Blum-Goldwasser cryptosystem, using the
-proposed method, has been finally proposed.
+\begin{color}{red} An improvement of the Blum-Goldwasser cryptosystem, making it
+behaves chaotically, has finally been proposed. \end{color}
-In future work we plan to extend these researches, building a parallel PRNG for clusters or
+In future work we plan to extend this research, building a parallel PRNG for clusters or
grid computing. Topological properties of the various proposed generators will be investigated,
and the use of other categories of PRNGs as input will be studied too. The improvement
of Blum-Goldwasser will be deepened. Finally, we
