supplementary

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index bf745396e5de1554ae158241292cc6c1b56293eb..0ab28a170efca640d062d65a664bcf8fc7507550 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -27,7 +27,6 @@
 % Pour faire des sous-figures dans les figures
 \usepackage{subfigure}
 
 % Pour faire des sous-figures dans les figures
 \usepackage{subfigure}
 

-\usepackage{color}

 
 \newtheorem{notation}{Notation}
 
 
 \newtheorem{notation}{Notation}
 


@@ -40,6 +39,8 @@
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 

+
+

 \title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
 \begin{document}
 
 \title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
 \begin{document}
 


@@ -90,7 +91,13 @@ On the other side, speed is not the main requirement in cryptography: the great
 need is to define \emph{secure} generators able to withstand malicious
 attacks. Roughly speaking, an attacker should not be able in practice to make 
 the distinction between numbers obtained with the secure generator and a true random
 need is to define \emph{secure} generators able to withstand malicious
 attacks. Roughly speaking, an attacker should not be able in practice to make 
 the distinction between numbers obtained with the secure generator and a true random

-sequence. 
+sequence.  However, in an equivalent formulation, he or she should not be
+able (in practice) to predict the next bit of the generator, having the knowledge of all the 
+binary digits that have been already released. ``Being able in practice'' refers here
+to the possibility to achieve this attack in polynomial time, and to the exponential growth
+of the difficulty of this challenge when the size of the parameters of the PRNG increases.
+
+

 Finally, a small part of the community working in this domain focuses on a
 third requirement, that is to define chaotic generators.
 The main idea is to take benefits from a chaotic dynamical system to obtain a
 Finally, a small part of the community working in this domain focuses on a
 third requirement, that is to define chaotic generators.
 The main idea is to take benefits from a chaotic dynamical system to obtain a


@@ -124,18 +131,16 @@ statistical perfection refers to the ability to pass the whole
 {\it BigCrush} battery of tests, which is widely considered as the most
 stringent statistical evaluation of a sequence claimed as random.
 This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}.
 {\it BigCrush} battery of tests, which is widely considered as the most
 stringent statistical evaluation of a sequence claimed as random.
 This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}.

-Chaos, for its part, refers to the well-established definition of a
-chaotic dynamical system proposed by Devaney~\cite{Devaney}.
-\begin{color}{red}

 More precisely, each time we performed a test on a PRNG, we ran it
 More precisely, each time we performed a test on a PRNG, we ran it

-twice in order to observe if all p-values are inside [0.01, 0.99]. In
-fact, we observed that few p-values (less than ten) are sometimes
+twice in order to observe if all $p-$values are inside [0.01, 0.99]. In
+fact, we observed that few $p-$values (less than ten) are sometimes

 outside this interval but inside [0.001, 0.999], so that is why a
 second run allows us to confirm that the values outside are not for
 the same test. With this approach all our PRNGs pass the {\it
 outside this interval but inside [0.001, 0.999], so that is why a
 second run allows us to confirm that the values outside are not for
 the same test. With this approach all our PRNGs pass the {\it

-  BigCrush} successfully and all p-values are at least once inside
+  BigCrush} successfully and all $p-$values are at least once inside

 [0.01, 0.99].
 [0.01, 0.99].

-\end{color}
+Chaos, for its part, refers to the well-established definition of a
+chaotic dynamical system proposed by Devaney~\cite{Devaney}.

 
 In a previous work~\cite{bgw09:ip,guyeux10} we have proposed a post-treatment on PRNGs making them behave
 as a chaotic dynamical system. Such a post-treatment leads to a new category of
 
 In a previous work~\cite{bgw09:ip,guyeux10} we have proposed a post-treatment on PRNGs making them behave
 as a chaotic dynamical system. Such a post-treatment leads to a new category of


@@ -160,23 +165,47 @@ property.
 Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric
 key encryption protocol by using the proposed method.
 
 Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric
 key encryption protocol by using the proposed method.
 

+
+{\bf Main contributions.} In this paper a new PRNG using chaotic iteration
+is defined. From a theoretical point of view, it is proven that it has fine
+topological chaotic properties and that it is cryptographically secured (when
+the initial PRNG is also cryptographically secured). From a practical point of
+view, experiments point out a very good statistical behavior. An optimized
+original implementation of this PRNG is also proposed and experimented.
+Pseudorandom numbers are generated at a rate of 20GSamples/s, which is faster
+than in~\cite{conf/fpga/ThomasHL09,Marsaglia2003} (and with a better
+statistical behavior). Experiments are also provided using BBS as the initial
+random generator. The generation speed is significantly weaker.
+Note also that an original qualitative comparison between topological chaotic
+properties and statistical test is also proposed.
+
+
+
+

 The remainder of this paper  is organized as follows. In Section~\ref{section:related
   works} we  review some GPU implementations  of PRNGs.  Section~\ref{section:BASIC
   RECALLS} gives some basic recalls  on the well-known Devaney's formulation of chaos, 
   and on an iteration process called ``chaotic
 iterations'' on which the post-treatment is based. 
 The proposed PRNG and its proof of chaos are given in  Section~\ref{sec:pseudorandom}.
 The remainder of this paper  is organized as follows. In Section~\ref{section:related
   works} we  review some GPU implementations  of PRNGs.  Section~\ref{section:BASIC
   RECALLS} gives some basic recalls  on the well-known Devaney's formulation of chaos, 
   and on an iteration process called ``chaotic
 iterations'' on which the post-treatment is based. 
 The proposed PRNG and its proof of chaos are given in  Section~\ref{sec:pseudorandom}.

-Section~\ref{sec:efficient    PRNG}   presents   an   efficient
-implementation of  this chaotic PRNG  on a CPU, whereas   Section~\ref{sec:efficient PRNG
+
+Section~\ref{The generation of pseudorandom sequence} illustrates the statistical
+improvement related to the chaotic iteration based post-treatment, for
+our previously released PRNGs and a new efficient 
+implementation on CPU.
+
+ Section~\ref{sec:efficient PRNG

   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 
 Section~\ref{sec:experiments}.
 We show in Section~\ref{sec:security analysis} that, if the inputted
 generator is cryptographically secure, then it is the case too for the
 generator provided by the post-treatment.
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 
 Section~\ref{sec:experiments}.
 We show in Section~\ref{sec:security analysis} that, if the inputted
 generator is cryptographically secure, then it is the case too for the
 generator provided by the post-treatment.

+A practical
+security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.

 Such a proof leads to the proposition of a cryptographically secure and
 chaotic generator on GPU based on the famous Blum Blum Shub
 Such a proof leads to the proposition of a cryptographically secure and
 chaotic generator on GPU based on the famous Blum Blum Shub

-in Section~\ref{sec:CSGPU}, and to an improvement of the
+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
 summarized and intended future work is presented.
 Blum-Goldwasser protocol in Sect.~\ref{Blum-Goldwasser}.
 This research work ends by a conclusion section, in which the contribution is
 summarized and intended future work is presented.


@@ -184,7 +213,7 @@ summarized and intended future work is presented.
 
 
 
 
 
 

-\section{Related works on GPU based PRNGs}
+\section{Related work on GPU based PRNGs}

 \label{section:related works}
 
 Numerous research works on defining GPU based PRNGs have already been proposed  in the
 \label{section:related works}
 
 Numerous research works on defining GPU based PRNGs have already been proposed  in the


@@ -243,7 +272,7 @@ with basic notions on topology (see for instance~\cite{Devaney}).
 
 
 \subsection{Devaney's Chaotic Dynamical Systems}
 
 
 \subsection{Devaney's Chaotic Dynamical Systems}

-
+\label{subsec:Devaney}

 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
 denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$
 is for the $k^{th}$ composition of a function $f$. Finally, the following
 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
 denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$
 is for the $k^{th}$ composition of a function $f$. Finally, the following


@@ -488,11 +517,9 @@ 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 
 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 

-\begin{color}{red}

 should improve the statistical properties of each
 generator taken alone. 
 should improve 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.
+Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as present input.

 
 
 
 
 
 


@@ -558,14 +585,13 @@ return $y$\;
 
 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 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.
+In this ``New CI PRNG'', we prevent a given bit from changing twice 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. 
 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 
+Some of these vectors will be randomly extracted and our pseudorandom 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 
 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 


@@ -575,17 +601,17 @@ 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 $x_i^n = \overline{x_i^{n-1}}$.
 
 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
+Such a procedure is equivalent to achieving chaotic iterations with

 the Boolean vectorial negation $f_0$ and some well-chosen strategies.
 Finally, some $x^n$ are selected
 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.
+by a sequence $m^n$ as the pseudorandom 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}.
 $(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ 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).
+This function must be chosen such that the outputs of the resulted PRNG are uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$. Function of \eqref{Formula} achieves this
+goal (other candidates and more information can be found in ~\cite{bg10:ip}).

 
 \begin{equation}
 \label{Formula}
 
 \begin{equation}
 \label{Formula}


@@ -611,8 +637,7 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
 }
 \ENDFOR
 \STATE$a\leftarrow{PRNG_1()}$\;
 }
 \ENDFOR
 \STATE$a\leftarrow{PRNG_1()}$\;

-\STATE$m\leftarrow{g(a)}$\;
-\STATE$k\leftarrow{m}$\;
+\STATE$k\leftarrow{g(a)}$\;

 \WHILE{$i=0,\dots,k$}
 
 \STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
 \WHILE{$i=0,\dots,k$}
 
 \STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;


@@ -635,12 +660,11 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
 \label{Chaotic iteration1}
 \end{algorithmic}
 \end{algorithm}
 \label{Chaotic iteration1}
 \end{algorithmic}
 \end{algorithm}

-\end{color}

 
 \subsection{Improving the Speed of the Former Generator}
 
 
 \subsection{Improving the Speed of the Former Generator}
 

-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}
+Instead of updating only one cell at each iteration, we now propose to choose a
+subset of components and to update them together, for speed improvement. Such a proposition leads 

 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:
 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:


@@ -652,7 +676,7 @@ 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.
 \forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
 \end{array}
 \right.

-\label{equation Oplus0}
+\label{equation Oplus}

 \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
 \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


@@ -662,7 +686,7 @@ 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.
 
 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 Oplus0} is of 
+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 following discrete dynamical system in chaotic iterations:
 
 ordinary use as a good elementary brick in various PRNGs. It corresponds
 to the following discrete dynamical system in chaotic iterations:
 


@@ -684,7 +708,7 @@ we select a subset of components to change.
 
 
 Obviously, replacing the previous CI PRNG Algorithms by 
 
 
 Obviously, replacing the previous CI PRNG Algorithms by 

-Equation~\ref{equation Oplus0}, which is possible when the iteration function is
+Equation~\ref{equation Oplus}, 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).
 the vectorial negation, leads to a speed improvement 
 (the resulting generator will be referred as ``Xor CI PRNG''
 in what follows).


@@ -867,6 +891,8 @@ the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
 
 In conclusion,
 %%RAPH : ici j'ai rajouté une ligne
 
 In conclusion,
 %%RAPH : ici j'ai rajouté une ligne

+%%TOF : ici j'ai rajouté un commentaire
+%%TOF : ici aussi

 $
 \forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
 ,$ $\forall n\geqslant N_{0},$
 $
 \forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
 ,$ $\forall n\geqslant N_{0},$


@@ -941,41 +967,123 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 \end{proof}
 
 
 \end{proof}
 
 

-\begin{color}{red}

 \section{Statistical Improvements Using Chaotic Iterations}
 
 \section{Statistical Improvements Using Chaotic Iterations}
 

-\label{The generation of pseudo-random sequence}
+\label{The generation of pseudorandom sequence}

 
 
 
 

-Let us now explain why we are reasonable grounds to believe that chaos 
+Let us now explain why we have reasonable ground to believe that chaos 

 can improve statistical properties.
 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}).
+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.
 
 
 \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:
+Firstly, the simple linear congruency generators (LCGs) will be used. 
+They are defined by the following recurrence:

 \begin{equation}
 \begin{equation}

-x^n = (ax^{n-1} + c)~mod~m
+x^n = (ax^{n-1} + c)~mod~m,

 \label{LCG}
 \end{equation}
 \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) 
+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}.
 
 combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
 

-Secondly, the multiple recursive generators (MRGs) will be used too, which
+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}
 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
+x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m .

 \label{MRG}
 \end{equation}
 \label{MRG}
 \end{equation}

-Combination of two MRGs (referred as 2MRGs) is also used in these experimentations.
+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:
 
 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:


@@ -993,7 +1101,7 @@ 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}
 \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:
+and the SWC generator, which is based on the following recurrence:

 \begin{equation}
 \label{SWC}
 \begin{array}{l}
 \begin{equation}
 \label{SWC}
 \begin{array}{l}


@@ -1002,12 +1110,12 @@ c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end
 
 Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
 \begin{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}
+x^n = x^{n-r} \oplus x^{n-k} .

 \label{GFSR}
 \end{equation}
 
 
 \label{GFSR}
 \end{equation}
 
 

-Finally, the nonlinear inversive generator~\cite{LEcuyerS07} has been regarded too, which is:
+Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:

 
 \begin{equation}
 \label{INV}
 
 \begin{equation}
 \label{INV}


@@ -1021,7 +1129,7 @@ a^1 & \text{if}~  z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
 
 \begin{table}
 \renewcommand{\arraystretch}{1.3}
 
 \begin{table}
 \renewcommand{\arraystretch}{1.3}

-\caption{TestU01 Statistical Test}
+\caption{TestU01 Statistical Test Failures}

 \label{TestU011}
 \centering
   \begin{tabular}{lccccc}
 \label{TestU011}
 \centering
   \begin{tabular}{lccccc}


@@ -1043,7 +1151,7 @@ Failures          &       &261            &146    &0       \\
 
 \begin{table}
 \renewcommand{\arraystretch}{1.3}
 
 \begin{table}
 \renewcommand{\arraystretch}{1.3}

-\caption{TestU01 Statistical Test for Old CI algorithms ($\mathsf{N}=4$)}
+\caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}

 \label{TestU01 for Old CI}
 \centering
   \begin{tabular}{lcccc}
 \label{TestU01 for Old CI}
 \centering
   \begin{tabular}{lcccc}


@@ -1071,7 +1179,7 @@ Failures                          &138    &9      &0      &0       \\
 \subsection{Statistical tests}
 \label{Security analysis}
 
 \subsection{Statistical tests}
 \label{Security analysis}
 

-Three batteries of tests are reputed and usually used
+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
 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


@@ -1095,7 +1203,7 @@ DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18
 \end{table*}
 
 Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the 
 \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
+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
 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


@@ -1120,9 +1228,10 @@ 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.
 \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.
+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*}
 
 
 \begin{table*}


@@ -1147,10 +1256,11 @@ DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18}  & \textbf{
 \end{table*}
 
 
 \end{table*}
 
 

-We have then investigate in~\cite{bfg12a:ip} if it is possible to improve
+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 
 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 
+$\oplus$ 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*}
 using chaotic iterations on defective generators.
 
 \begin{table*}


@@ -1165,19 +1275,23 @@ Threshold  value $m$& 19 & 7  & 2& 1 & 11& 9& 3& 4\\ \hline\hline
 \end{tabular}
 \end{table*}
 
 \end{tabular}
 \end{table*}
 

-Finally, the TestU01 battery as been launched on three well-known generators 
+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.
 (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.

 
 
 
 

-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
+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.
 raise ambiguity.

-\end{color}

 
 

-\subsection{Efficient Implementation of a PRNG based on Chaotic Iterations}
+
+\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}

 \label{sec:efficient PRNG}
 %
 %Based on the proof presented in the previous section, it is now possible to 
 \label{sec:efficient PRNG}
 %
 %Based on the proof presented in the previous section, it is now possible to 


@@ -1222,7 +1336,7 @@ raise ambiguity.
 
 
 
 
 
 

-\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label=algo:seqCIPRNG}
+\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label={algo:seqCIPRNG}}

 \begin{small}
 \begin{lstlisting}
 
 \begin{small}
 \begin{lstlisting}
 


@@ -1256,7 +1370,13 @@ works with 32-bits, we use the command \texttt{(unsigned int)}, that selects the
 
 Thus producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers
 that  are provided by  3 64-bits  PRNGs.  This  version successfully  passes the
 
 Thus producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers
 that  are provided by  3 64-bits  PRNGs.  This  version successfully  passes the

-stringent BigCrush battery of tests~\cite{LEcuyerS07}.
+stringent BigCrush battery of tests~\cite{LEcuyerS07}. 
+At this point, we thus
+have defined an efficient and statistically unbiased generator. Its speed is
+directly related to the use of linear operations, but for the same reason,
+this fast generator cannot be proven as secure.
+
+

 
 \section{Efficient PRNGs based on Chaotic Iterations on GPU}
 \label{sec:efficient PRNG gpu}
 
 \section{Efficient PRNGs based on Chaotic Iterations on GPU}
 \label{sec:efficient PRNG gpu}


@@ -1392,7 +1512,7 @@ version\label{IR}}
 \label{algo:gpu_kernel2} 
 \end{algorithm}
 
 \label{algo:gpu_kernel2} 
 \end{algorithm}
 

-\subsection{Theoretical Evaluation of the Improved Version}
+\subsection{Chaos Evaluation of the Improved Version}

 
 A run of Algorithm~\ref{algo:gpu_kernel2} consists in an operation ($x=x\oplus t$) having 
 the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
 
 A run of Algorithm~\ref{algo:gpu_kernel2} consists in an operation ($x=x\oplus t$) having 
 the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative


@@ -1490,9 +1610,25 @@ as it is shown in the next sections.
 
 
 \section{Security Analysis}
 
 
 \section{Security Analysis}

-\label{sec:security analysis}

 
 
 
 

+This section is dedicated to the security analysis of the
+  proposed PRNGs, both from a theoretical and from a practical point of view.
+
+\subsection{Theoretical Proof of Security}
+\label{sec:security analysis}
+
+The standard definition
+  of {\it indistinguishability} used is the classical one as defined for
+  instance in~\cite[chapter~3]{Goldreich}. 
+  This property shows that predicting the future results of the PRNG
+  cannot be done in a reasonable time compared to the generation time. It is important to emphasize that this
+  is a relative notion between breaking time and the sizes of the
+  keys/seeds. Of course, if small keys or seeds are chosen, the system can
+  be broken in practice. But it also means that if the keys/seeds are large
+  enough, the system is secured.
+As a complement, an example of a concrete practical evaluation of security
+is outlined in the next subsection.

 
 In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.
 
 In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.


@@ -1512,12 +1648,14 @@ probabilities are taken over $U_m$, $U_{\ell_G(m)}$ as well as over the
 internal coin tosses of $D$. 
 \end{definition}
 
 internal coin tosses of $D$. 
 \end{definition}
 

-Intuitively, it means that there is no polynomial time algorithm that can
-distinguish a perfect uniform random generator from $G$ with a non
-negligible probability. The interested reader is referred
-to~\cite[chapter~3]{Goldreich} for more information. Note that it is
-quite easily possible to change the function $\ell$ into any polynomial
-function $\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}.
+Intuitively,  it means  that  there is  no  polynomial time  algorithm that  can
+distinguish a  perfect uniform random generator  from $G$ with  a non negligible
+probability.   An equivalent  formulation of  this well-known  security property
+means that  it is  possible \emph{in practice}  to predict  the next bit  of the
+generator, knowing all  the previously produced ones.  The  interested reader is
+referred to~\cite[chapter~3]{Goldreich}  for more  information. Note that  it is
+quite easily possible to change the function $\ell$ into any polynomial function
+$\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}.

 
 The generation schema developed in (\ref{equation Oplus}) is based on a
 pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,
 
 The generation schema developed in (\ref{equation Oplus}) is based on a
 pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,


@@ -1539,7 +1677,7 @@ PRNG too.
 \end{proposition}
 
 \begin{proof}
 \end{proposition}
 
 \begin{proof}

-The proposition is proved by contraposition. Assume that $X$ is not
+The proposition is proven by contraposition. Assume that $X$ is not

 secure. By Definition, there exists a polynomial time probabilistic
 algorithm $D$, a positive polynomial $p$, such that for all $k_0$ there exists
 $N\geq \frac{k_0}{2}$ satisfying 
 secure. By Definition, there exists a polynomial time probabilistic
 algorithm $D$, a positive polynomial $p$, such that for all $k_0$ there exists
 $N\geq \frac{k_0}{2}$ satisfying 


@@ -1602,6 +1740,98 @@ proving that $H$ is not secure, which is a contradiction.
 \end{proof}
 
 
 \end{proof}
 
 

+
+\subsection{Practical Security Evaluation}
+\label{sec:Practicak evaluation}
+
+Pseudorandom generators based on Eq.~\eqref{equation Oplus} are thus cryptographically secure when
+they are XORed with an already cryptographically
+secure PRNG. But, as stated previously,
+such a property does not mean that, whatever the
+key size, no attacker can predict the next bit
+knowing all the previously released ones.
+However, given a key size, it is possible to 
+measure in practice the minimum duration needed
+for an attacker to break a cryptographically
+secure PRNG, if we know the power of his/her
+machines. Such a concrete security evaluation 
+is related to the $(T,\varepsilon)-$security
+notion, which is recalled and evaluated in what 
+follows, for the sake of completeness.
+
+Let us firstly recall that,
+\begin{definition}
+Let $\mathcal{D} : \mathds{B}^M \longrightarrow \mathds{B}$ be a probabilistic algorithm that runs
+in time $T$. 
+Let $\varepsilon > 0$. 
+$\mathcal{D}$ is called a $(T,\varepsilon)-$distinguishing attack on pseudorandom
+generator $G$ if
+
+\begin{flushleft}
+$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right.$
+\end{flushleft}
+
+\begin{flushright}
+$ - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$
+\end{flushright}
+
+\noindent where the probability is taken over the internal coin flips of $\mathcal{D}$, and the notation
+``$\in_R$'' indicates the process of selecting an element at random and uniformly over the
+corresponding set.
+\end{definition}
+
+Let us recall that the running time of a probabilistic algorithm is defined to be the
+maximum of the expected number of steps needed to produce an output, maximized
+over all inputs; the expected number is averaged over all coin flips made by the algorithm~\cite{Knuth97}.
+We are now able to define the notion of cryptographically secure PRNGs:
+
+\begin{definition}
+A pseudorandom generator is $(T,\varepsilon)-$secure if there exists no $(T,\varepsilon)-$distinguishing attack on this pseudorandom generator.
+\end{definition}
+
+
+
+
+
+
+
+Suppose now that the PRNG of Eq.~\eqref{equation Oplus} 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 a truly random one, with a probability
+greater than $\varepsilon = 0.2$.
+We consider that $N$ has 900 bits.
+
+Predicting the next generated bit knowing all the
+previously released ones by Eq.~\eqref{equation Oplus} is obviously equivalent to predicting the
+next bit in 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}
+\begin{equation}
+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)
+\label{mesureConcrete}
+\end{equation}
+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.
+
+
+

 \section{Cryptographical Applications}
 
 \subsection{A Cryptographically Secure PRNG for GPU}
 \section{Cryptographical Applications}
 
 \subsection{A Cryptographically Secure PRNG for GPU}


@@ -1725,44 +1955,38 @@ It should  be noticed that this generator has once more the form $x^{n+1} = x^n
 where $S^n$ is referred in this algorithm as $t$: each iteration of this
 PRNG ends with $x = x \wedge t$. This $S^n$ is only constituted
 by secure bits produced by the BBS generator, and thus, due to
 where $S^n$ is referred in this algorithm as $t$: each iteration of this
 PRNG ends with $x = x \wedge t$. This $S^n$ is only constituted
 by secure bits produced by the BBS generator, and thus, due to

-Proposition~\ref{cryptopreuve}, the resulted PRNG is cryptographically
-secure.
-
-
-
-\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.
+Proposition~\ref{cryptopreuve}, the resulted PRNG is 
+cryptographically secure.
+
+As stated before, even if the proposed PRNG is cryptocaphically
+secure, it does not mean that such a generator
+can be used as described here when attacks are
+awaited. The problem is to determine the minimum 
+time required for an attacker, with a given 
+computational power, to predict under a probability
+lower than 0.5 the $n+1$th bit, knowing the $n$
+previous ones. The proposed GPU generator will be
+useful in a security context, at least in some 
+situations where a secret protected by a pseudorandom
+keystream is rapidly obsolete, if this time to 
+predict the next bit is large enough when compared
+to both the generation and transmission times.
+It is true that the prime numbers used in the last
+section are very small compared to up-to-date 
+security recommendations. However the attacker has not
+access to each BBS, but to the output produced 
+by Algorithm~\ref{algo:bbs_gpu}, which is far
+more complicated than a simple BBS. Indeed, to
+determine if this cryptographically secure PRNG
+on GPU can be useful in security context with the 
+proposed parameters, or if it is only a very fast
+and statistically perfect generator on GPU, its
+$(T,\varepsilon)-$security must be determined, and
+a formulation similar to Eq.\eqref{mesureConcrete}
+must be established. Authors
+hope to achieve this difficult task in a future
+work.

 
 

-\end{color}

 
 \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
 \label{Blum-Goldwasser}
 
 \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
 \label{Blum-Goldwasser}


@@ -1846,8 +2070,8 @@ namely the BigCrush.
 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.
 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.

-\begin{color}{red} An improvement of the Blum-Goldwasser cryptosystem, making it 
-behaves chaotically, has finally been proposed. \end{color}
+An improvement of the Blum-Goldwasser cryptosystem, making it 
+behave chaotically, has finally been proposed. 

 
 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,
 
 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,