Merge branch 'master' of ssh://info.iut-bm.univ-fcomte.fr/prng_gpu

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index ed7e927f4472ce25eb079ffbc4b004c94fca63bb..55fc756560f07eab5d8a70a102c4798e8551a695 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -1,4 +1,5 @@
-\documentclass{article}
+%\documentclass{article}
+\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}

 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{fullpage}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{fullpage}


@@ -38,10 +39,10 @@
 \begin{document}
 
 \author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe
 \begin{document}
 
 \author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe

-Guyeux, and Pierre-Cyrille Heam\thanks{Authors in alphabetic order}}
+Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}

    
    

-\maketitle

 
 

+\IEEEcompsoctitleabstractindextext{

 \begin{abstract}
 In this paper we present a new pseudorandom number generator (PRNG) on
 graphics processing units  (GPU). This PRNG is based  on the so-called chaotic iterations.  It
 \begin{abstract}
 In this paper we present a new pseudorandom number generator (PRNG) on
 graphics processing units  (GPU). This PRNG is based  on the so-called chaotic iterations.  It


@@ -56,6 +57,13 @@ A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is fin
 
 
 \end{abstract}
 
 
 \end{abstract}

+}
+
+\maketitle
+
+\IEEEdisplaynotcompsoctitleabstractindextext
+\IEEEpeerreviewmaketitle
+

 
 \section{Introduction}
 
 
 \section{Introduction}
 


@@ -135,7 +143,7 @@ allows us to generate almost 20 billion of pseudorandom numbers per second.
 Furthermore, we show that the proposed post-treatment preserves the
 cryptographical security of the inputted PRNG, when this last has such a 
 property.
 Furthermore, we show that the proposed post-treatment preserves the
 cryptographical security of the inputted PRNG, when this last has such a 
 property.

-Last, but not least, we propose a rewritting of the Blum-Goldwasser asymmetric
+Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric

 key encryption protocol by using the proposed method.
 
 The remainder of this paper  is organized as follows. In Section~\ref{section:related
 key encryption protocol by using the proposed method.
 
 The remainder of this paper  is organized as follows. In Section~\ref{section:related


@@ -216,7 +224,10 @@ We can finally remark that, to the best of our knowledge, no GPU implementation
 \label{section:BASIC RECALLS}
 
 This section is devoted to basic definitions and terminologies in the fields of
 \label{section:BASIC RECALLS}
 
 This section is devoted to basic definitions and terminologies in the fields of

-topological chaos and chaotic iterations.
+topological chaos and chaotic iterations. We assume the reader is familiar
+with basic notions on topology (see for instance~\cite{Devaney}).
+
+

 \subsection{Devaney's Chaotic Dynamical Systems}
 
 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
 \subsection{Devaney's Chaotic Dynamical Systems}
 
 In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$


@@ -229,7 +240,7 @@ Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f :
 \mathcal{X} \rightarrow \mathcal{X}$.
 
 \begin{definition}
 \mathcal{X} \rightarrow \mathcal{X}$.
 
 \begin{definition}

-$f$ is said to be \emph{topologically transitive} if, for any pair of open sets
+The function $f$ is said to be \emph{topologically transitive} if, for any pair of open sets

 $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq
 \varnothing$.
 \end{definition}
 $U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq
 \varnothing$.
 \end{definition}


@@ -248,7 +259,7 @@ necessarily the same period).
 
 
 \begin{definition}[Devaney's formulation of chaos~\cite{Devaney}]
 
 
 \begin{definition}[Devaney's formulation of chaos~\cite{Devaney}]

-$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and
+The function $f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and

 topologically transitive.
 \end{definition}
 
 topologically transitive.
 \end{definition}
 


@@ -256,12 +267,12 @@ The chaos property is strongly linked to the notion of ``sensitivity'', defined
 on a metric space $(\mathcal{X},d)$ by:
 
 \begin{definition}
 on a metric space $(\mathcal{X},d)$ by:
 
 \begin{definition}

-\label{sensitivity} $f$ has \emph{sensitive dependence on initial conditions}
+\label{sensitivity} The function $f$ has \emph{sensitive dependence on initial conditions}

 if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any
 neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that
 $d\left(f^{n}(x), f^{n}(y)\right) >\delta $.
 
 if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any
 neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that
 $d\left(f^{n}(x), f^{n}(y)\right) >\delta $.
 

-$\delta$ is called the \emph{constant of sensitivity} of $f$.
+The constant $\delta$ is called the \emph{constant of sensitivity} of $f$.

 \end{definition}
 
 Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is
 \end{definition}
 
 Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is


@@ -321,11 +332,13 @@ 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:
 
 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}
 \begin{array}{lrll}
 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)+f(E)_{k}.\overline{\delta
+& (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},%
 \end{array}%
 \end{equation}%
 (k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
 \end{array}%
 \end{equation}%


@@ -467,8 +480,9 @@ generator taken alone. Furthermore, our generator
 possesses various chaos properties that none of the generators used as input
 present.
 
 possesses various chaos properties that none of the generators used as input
 present.
 

+

 \begin{algorithm}[h!]
 \begin{algorithm}[h!]

-%\begin{scriptsize}
+\begin{small}

 \KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$
 ($n$ bits)}
 \KwOut{a configuration $x$ ($n$ bits)}
 \KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$
 ($n$ bits)}
 \KwOut{a configuration $x$ ($n$ bits)}


@@ -480,12 +494,16 @@ $s\leftarrow{\textit{XORshift}(n)}$\;
 $x\leftarrow{F_f(s,x)}$\;
 }
 return $x$\;
 $x\leftarrow{F_f(s,x)}$\;
 }
 return $x$\;

-%\end{scriptsize}
+\end{small}

 \caption{PRNG with chaotic functions}
 \label{CI Algorithm}
 \end{algorithm}
 
 \caption{PRNG with chaotic functions}
 \label{CI Algorithm}
 \end{algorithm}
 

+
+
+

 \begin{algorithm}[h!]
 \begin{algorithm}[h!]

+\begin{small}

 \KwIn{the internal configuration $z$ (a 32-bit word)}
 \KwOut{$y$ (a 32-bit word)}
 $z\leftarrow{z\oplus{(z\ll13)}}$\;
 \KwIn{the internal configuration $z$ (a 32-bit word)}
 \KwOut{$y$ (a 32-bit word)}
 $z\leftarrow{z\oplus{(z\ll13)}}$\;


@@ -493,7 +511,7 @@ $z\leftarrow{z\oplus{(z\gg17)}}$\;
 $z\leftarrow{z\oplus{(z\ll5)}}$\;
 $y\leftarrow{z}$\;
 return $y$\;
 $z\leftarrow{z\oplus{(z\ll5)}}$\;
 $y\leftarrow{z}$\;
 return $y$\;

-\medskip
+\end{small}

 \caption{An arbitrary round of \textit{XORshift} algorithm}
 \label{XORshift}
 \end{algorithm}
 \caption{An arbitrary round of \textit{XORshift} algorithm}
 \label{XORshift}
 \end{algorithm}


@@ -536,7 +554,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N
 \label{equation Oplus}
 \end{equation}
 where $\oplus$ is for the bitwise exclusive or between two integers. 
 \label{equation Oplus}
 \end{equation}
 where $\oplus$ is for the bitwise exclusive or between two integers. 

-This rewritting can be understood as follows. The $n-$th term $S^n$ of the
+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 
 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 


@@ -605,12 +623,13 @@ Let us introduce the following function:
 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:
 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}} \\
 \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)+f(E)_{j}.\overline{\chi
-(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+& (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},%

 \end{array}%
 \end{equation}%
 where + and . are the Boolean addition and product operations, and $\overline{x}$ 
 \end{array}%
 \end{equation}%
 where + and . are the Boolean addition and product operations, and $\overline{x}$ 


@@ -622,7 +641,7 @@ Consider the phase space:
 \end{equation}
 \noindent and the map defined on $\mathcal{X}$:
 \begin{equation}
 \end{equation}
 \noindent and the map defined on $\mathcal{X}$:
 \begin{equation}

-G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), \label{Gf}
+G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), %\label{Gf} %%RAPH, j'ai viré ce label qui existe déjà avant...

 \end{equation}
 \noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
 (S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in
 \end{equation}
 \noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
 (S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in


@@ -649,17 +668,21 @@ Let us introduce:
 d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
 \label{nouveau d}
 \end{equation}
 d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
 \label{nouveau d}
 \end{equation}

-\noindent where
-\begin{equation}
-\left\{
-\begin{array}{lll}
-\displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
-}\delta (E_{k},\check{E}_{k})}\textrm{ is once more the Hamming distance}, \\
-\displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
-\sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.%
-\end{array}%
-\right.
-\end{equation}
+\noindent where $ \displaystyle{d_{e}(E,\check{E})} = \displaystyle{\sum_{k=1}^{\mathsf{N}%
+ }\delta (E_{k},\check{E}_{k})}$  is once more the Hamming distance, and
+$  \displaystyle{d_{s}(S,\check{S})}  =  \displaystyle{\dfrac{9}{\mathsf{N}}%
+ \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}$,
+%%RAPH : ici, j'ai supprimé tous les sauts à la ligne
+%% \begin{equation}
+%% \left\{
+%% \begin{array}{lll}
+%% \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
+%% }\delta (E_{k},\check{E}_{k})} \textrm{ is once more the Hamming distance}, \\
+%% \displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
+%% \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.%
+%% \end{array}%
+%% \right.
+%% \end{equation}

 where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
 $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
 
 where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
 $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
 


@@ -740,12 +763,14 @@ G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies)
 the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
 10^{-(k+1)}\leqslant \varepsilon $.\bigskip \newline
 In conclusion,
 the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
 10^{-(k+1)}\leqslant \varepsilon $.\bigskip \newline
 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 \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 n\geqslant N_{0},$$
+$$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)

 \leqslant \varepsilon .
 $$
 \leqslant \varepsilon .
 $$

+\end{flushleft}

 $G_{f}$ is consequently continuous.
 \end{proof}
 
 $G_{f}$ is consequently continuous.
 \end{proof}
 


@@ -785,7 +810,11 @@ where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
 claimed in the lemma.
 \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}...
 We can now prove Theorem~\ref{t:chaos des general}...

+>>>>>>> e55d237aba022a66cc2d7650d295b29169878f45

 
 \begin{proof}[Theorem~\ref{t:chaos des general}]
 Firstly, strong transitivity implies transitivity.
 
 \begin{proof}[Theorem~\ref{t:chaos des general}]
 Firstly, strong transitivity implies transitivity.


@@ -803,8 +832,10 @@ and $t_2\in\mathds{N}$ such
 that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
 
 Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
 that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
 
 Consider the strategy $\tilde S$ that alternates the first $t_1$ terms

-of $S$ and the first $t_2$ terms of $S'$: $$\tilde
-S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
+of $S$ and the first $t_2$ terms of $S'$: 
+%%RAPH : j'ai coupé la ligne en 2
+$$\tilde
+S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,$$$$\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It

 is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
 $t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
 point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
 is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
 $t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
 point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we


@@ -824,41 +855,44 @@ given PRNG.
 An iteration of the system is simply the bitwise exclusive or between
 the last computed state and the current strategy.
 Topological properties of disorder exhibited by chaotic 
 An iteration of the system is simply the bitwise exclusive or between
 the last computed state and the current strategy.
 Topological properties of disorder exhibited by chaotic 

-iterations can be inherited by the inputted generator, hoping by doing so to 
+iterations can be inherited by the inputted generator, we hope by doing so to 

 obtain some statistical improvements while preserving speed.
 
 obtain some statistical improvements while preserving speed.
 

-
-Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
-are
-done.  
-Suppose  that $x$ and the  strategy $S^i$ are given as
-binary vectors.
-Table~\ref{TableExemple} shows the result of $x \oplus S^i$.
-
-\begin{table}
-$$
-\begin{array}{|cc|cccccccccccccccc|}
-\hline
-x      &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
-\hline
-S^i      &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
-\hline
-x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
-\hline
-
-\hline
- \end{array}
-$$
-\caption{Example of an arbitrary round of the proposed generator}
-\label{TableExemple}
-\end{table}
-
-
-
-
-\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iteration\
-s},label=algo:seqCIPRNG}
+%%RAPH : j'ai viré tout ca
+%% Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
+%% are
+%% done.  
+%% Suppose  that $x$ and the  strategy $S^i$ are given as
+%% binary vectors.
+%% Table~\ref{TableExemple} shows the result of $x \oplus S^i$.
+
+%% \begin{table}
+%% \begin{scriptsize}
+%% $$
+%% \begin{array}{|cc|cccccccccccccccc|}
+%% \hline
+%% x      &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
+%% \hline
+%% S^i      &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
+%% \hline
+%% x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
+%% \hline
+
+%% \hline
+%%  \end{array}
+%% $$
+%% \end{scriptsize}
+%% \caption{Example of an arbitrary round of the proposed generator}
+%% \label{TableExemple}
+%% \end{table}
+
+
+
+
+\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label=algo:seqCIPRNG}
+\begin{small}

 \begin{lstlisting}
 \begin{lstlisting}

+

 unsigned int CIPRNG() {
   static unsigned int x = 123123123;
   unsigned long t1 = xorshift();
 unsigned int CIPRNG() {
   static unsigned int x = 123123123;
   unsigned long t1 = xorshift();


@@ -873,7 +907,7 @@ unsigned int CIPRNG() {
   return x;
 }
 \end{lstlisting}
   return x;
 }
 \end{lstlisting}

-
+\end{small}

 
 
 
 
 
 


@@ -902,7 +936,7 @@ used  (if,  while,  ...),  the  better the  performances  on  GPU  is.
 Obviously, having these requirements in  mind, it is possible to build
 a   program    similar   to    the   one   presented    in  Listing 
 \ref{algo:seqCIPRNG}, which computes  pseudorandom numbers on GPU.  To
 Obviously, having these requirements in  mind, it is possible to build
 a   program    similar   to    the   one   presented    in  Listing 
 \ref{algo:seqCIPRNG}, which computes  pseudorandom numbers on GPU.  To

-do  so,  we  must   firstly  remind  that  in  the  CUDA~\cite{Nvid10}
+do  so,  we  must   firstly  recall  that  in  the  CUDA~\cite{Nvid10}

 environment,    threads    have     a    local    identifier    called
 \texttt{ThreadIdx},  which   is  relative  to   the  block  containing
 them. Furthermore, in  CUDA, parts of  the code that are executed by the  GPU, are
 environment,    threads    have     a    local    identifier    called
 \texttt{ThreadIdx},  which   is  relative  to   the  block  containing
 them. Furthermore, in  CUDA, parts of  the code that are executed by the  GPU, are


@@ -916,7 +950,7 @@ It is possible to deduce from the CPU version a quite similar version adapted to
 The simple principle consists in making each thread of the GPU computing the CPU version of our PRNG.  
 Of course,  the  three xor-like
 PRNGs  used in these computations must have different  parameters. 
 The simple principle consists in making each thread of the GPU computing the CPU version of our PRNG.  
 Of course,  the  three xor-like
 PRNGs  used in these computations must have different  parameters. 

-In a given thread, these lasts are
+In a given thread, these parameters are

 randomly picked from another PRNGs. 
 The  initialization stage is performed by  the CPU.
 To do it, the  ISAAC  PRNG~\cite{Jenkins96} is used to  set  all  the
 randomly picked from another PRNGs. 
 The  initialization stage is performed by  the CPU.
 To do it, the  ISAAC  PRNG~\cite{Jenkins96} is used to  set  all  the


@@ -929,8 +963,9 @@ number  $x$  that saves  the  last  generated  pseudorandom number. Additionally
 implementation of the  xor128, the xorshift, and the  xorwow respectively require
 4, 5, and 6 unsigned long as internal variables.
 
 implementation of the  xor128, the xorshift, and the  xorwow respectively require
 4, 5, and 6 unsigned long as internal variables.
 

-\begin{algorithm}

 
 

+\begin{algorithm}
+\begin{small}

 \KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like
 PRNGs in global memory\;
 NumThreads: number of threads\;}
 \KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like
 PRNGs in global memory\;
 NumThreads: number of threads\;}


@@ -943,14 +978,16 @@ NumThreads: number of threads\;}
   }
   store internal variables in InternalVarXorLikeArray[threadIdx]\;
 }
   }
   store internal variables in InternalVarXorLikeArray[threadIdx]\;
 }

-
+\end{small}

 \caption{Main kernel of the GPU ``naive'' version of the PRNG based on chaotic iterations}
 \label{algo:gpu_kernel}
 \end{algorithm}
 
 \caption{Main kernel of the GPU ``naive'' version of the PRNG based on chaotic iterations}
 \label{algo:gpu_kernel}
 \end{algorithm}
 

+
+

 Algorithm~\ref{algo:gpu_kernel}  presents a naive  implementation of the proposed  PRNG on
 GPU.  Due to the available  memory in the  GPU and the number  of threads
 Algorithm~\ref{algo:gpu_kernel}  presents a naive  implementation of the proposed  PRNG on
 GPU.  Due to the available  memory in the  GPU and the number  of threads

-used simultenaously,  the number  of random numbers  that a thread  can generate
+used simultaneously,  the number  of random numbers  that a thread  can generate

 inside   a    kernel   is   limited  (\emph{i.e.},    the    variable   \texttt{n}   in
 algorithm~\ref{algo:gpu_kernel}). For instance, if  $100,000$ threads are used and
 if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)},
 inside   a    kernel   is   limited  (\emph{i.e.},    the    variable   \texttt{n}   in
 algorithm~\ref{algo:gpu_kernel}). For instance, if  $100,000$ threads are used and
 if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)},


@@ -994,7 +1031,7 @@ bits).
 This version  can also pass the whole {\it BigCrush} battery of tests.
 
 \begin{algorithm}
 This version  can also pass the whole {\it BigCrush} battery of tests.
 
 \begin{algorithm}

-
+\begin{small}

 \KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs
 in global memory\;
 NumThreads: Number of threads\;
 \KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs
 in global memory\;
 NumThreads: Number of threads\;


@@ -1016,7 +1053,7 @@ array\_comb1, array\_comb2: Arrays containing combinations of size combination\_
   }
   store internal variables in InternalVarXorLikeArray[threadId]\;
 }
   }
   store internal variables in InternalVarXorLikeArray[threadId]\;
 }

-
+\end{small}

 \caption{Main kernel for the chaotic iterations based PRNG GPU efficient
 version\label{IR}}
 \label{algo:gpu_kernel2} 
 \caption{Main kernel for the chaotic iterations based PRNG GPU efficient
 version\label{IR}}
 \label{algo:gpu_kernel2} 


@@ -1081,7 +1118,7 @@ As a  comparison,   Listing~\ref{algo:seqCIPRNG}  leads   to the  generation of
 
 \begin{figure}[htbp]
 \begin{center}
 
 \begin{figure}[htbp]
 \begin{center}

-  \includegraphics[scale=.7]{curve_time_xorlike_gpu.pdf}
+  \includegraphics[width=\columnwidth]{curve_time_xorlike_gpu.pdf}

 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG}
 \label{fig:time_xorlike_gpu}
 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG}
 \label{fig:time_xorlike_gpu}


@@ -1095,12 +1132,12 @@ In Figure~\ref{fig:time_bbs_gpu} we highlight  the performances of the optimized
 BBS-based PRNG on GPU.  On  the Tesla C1060 we obtain approximately 700MSample/s
 and  on the  GTX 280  about  670MSample/s, which  is obviously  slower than  the
 xorlike-based PRNG on GPU. However, we  will show in the next sections that this
 BBS-based PRNG on GPU.  On  the Tesla C1060 we obtain approximately 700MSample/s
 and  on the  GTX 280  about  670MSample/s, which  is obviously  slower than  the
 xorlike-based PRNG on GPU. However, we  will show in the next sections that this

-new PRNG  has a strong  level of  security, which is  necessary paid by  a speed
+new PRNG  has a strong  level of  security, which is  necessarily paid by  a speed

 reduction.
 
 \begin{figure}[htbp]
 \begin{center}
 reduction.
 
 \begin{figure}[htbp]
 \begin{center}

-  \includegraphics[scale=.7]{curve_time_bbs_gpu.pdf}
+  \includegraphics[width=\columnwidth]{curve_time_bbs_gpu.pdf}

 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second using the BBS-based PRNG}
 \label{fig:time_bbs_gpu}
 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second using the BBS-based PRNG}
 \label{fig:time_bbs_gpu}


@@ -1108,7 +1145,7 @@ reduction.
 
 All  these  experiments allow  us  to conclude  that  it  is possible  to
 generate a very large quantity of pseudorandom  numbers statistically perfect with the  xor-like version.
 
 All  these  experiments allow  us  to conclude  that  it  is possible  to
 generate a very large quantity of pseudorandom  numbers statistically perfect with the  xor-like version.

-In a certain extend, it is the case too with the secure BBS-based version, the speed deflation being
+To a certain extend, it is also the case with the secure BBS-based version, the speed deflation being

 explained by the fact that the former  version has ``only''
 chaotic properties and statistical perfection, whereas the latter is also cryptographically secure,
 as it is shown in the next sections.
 explained by the fact that the former  version has ``only''
 chaotic properties and statistical perfection, whereas the latter is also cryptographically secure,
 as it is shown in the next sections.


@@ -1128,17 +1165,17 @@ In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.
 In a cryptographic context, a pseudorandom generator is a deterministic
 algorithm $G$ transforming strings  into strings and such that, for any
 denoted by $uv$.
 In a cryptographic context, a pseudorandom generator is a deterministic
 algorithm $G$ transforming strings  into strings and such that, for any

-seed $k$ of length $k$, $G(k)$ (the output of $G$ on the input $k$) has size
-$\ell_G(k)$ with $\ell_G(k)>k$.
+seed $s$ of length $m$, $G(s)$ (the output of $G$ on the input $s$) has size
+$\ell_G(m)$ with $\ell_G(m)>m$.

 The notion of {\it secure} PRNGs can now be defined as follows. 
 
 \begin{definition}
 A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time
 algorithm $D$, for any positive polynomial $p$, and for all sufficiently
 The notion of {\it secure} PRNGs can now be defined as follows. 
 
 \begin{definition}
 A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time
 algorithm $D$, for any positive polynomial $p$, and for all sufficiently

-large $k$'s,
-$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(k)},$$
+large $m$'s,
+$$| \mathrm{Pr}[D(G(U_m))=1]-Pr[D(U_{\ell_G(m)})=1]|< \frac{1}{p(m)},$$

 where $U_r$ is the uniform distribution over $\{0,1\}^r$ and the
 where $U_r$ is the uniform distribution over $\{0,1\}^r$ and the

-probabilities are taken over $U_N$, $U_{\ell_G(N)}$ as well as over the
+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}
 


@@ -1147,7 +1184,7 @@ 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
 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 $\ell^\prime(N)>N)$~\cite[Chapter 3.3]{Goldreich}.
+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,


@@ -1158,7 +1195,7 @@ strings of length $N$ such that $H(S_0)=S_1 \ldots S_k$ ($H(S_0)$ is the concate
 the $S_i$'s). The cryptographic PRNG $X$ defined in (\ref{equation Oplus})
 is the algorithm mapping any string of length $2N$ $x_0S_0$ into the string
 $(x_0\oplus S_0 \oplus S_1)(x_0\oplus S_0 \oplus S_1\oplus S_2)\ldots
 the $S_i$'s). The cryptographic PRNG $X$ defined in (\ref{equation Oplus})
 is the algorithm mapping any string of length $2N$ $x_0S_0$ into the string
 $(x_0\oplus S_0 \oplus S_1)(x_0\oplus S_0 \oplus S_1\oplus S_2)\ldots

-(x_o\bigoplus_{i=0}^{i=k}S_i)$. Particularly one has $\ell_{X}(2N)=kN=\ell_H(N)$. 
+(x_o\bigoplus_{i=0}^{i=k}S_i)$. One in particular has $\ell_{X}(2N)=kN=\ell_H(N)$. 

 We claim now that if this PRNG is secure,
 then the new one is secure too.
 
 We claim now that if this PRNG is secure,
 then the new one is secure too.
 


@@ -1202,8 +1239,10 @@ $y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. It follows,
 by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$
 is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}),
 one has
 by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$
 is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}),
 one has

+$\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]$ and,
+therefore, 

 \begin{equation}\label{PCH-2}
 \begin{equation}\label{PCH-2}

-\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]=\mathrm{Pr}[D(U_{kN})=1].
+\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(U_{kN})=1].

 \end{equation}
 
 Now, using (\ref{PCH-1}) again, one has  for every $x$,
 \end{equation}
 
 Now, using (\ref{PCH-1}) again, one has  for every $x$,


@@ -1212,7 +1251,7 @@ D^\prime(H(x))=D(\varphi_y(H(x))),
 \end{equation}
 where $y$ is randomly generated. By construction, $\varphi_y(H(x))=X(yx)$,
 thus
 \end{equation}
 where $y$ is randomly generated. By construction, $\varphi_y(H(x))=X(yx)$,
 thus

-\begin{equation}\label{PCH-3}
+\begin{equation}%\label{PCH-3}      %%RAPH : j'ai viré ce label qui existe déjà, il est 3 ligne avant

 D^\prime(H(x))=D(yx),
 \end{equation}
 where $y$ is randomly generated. 
 D^\prime(H(x))=D(yx),
 \end{equation}
 where $y$ is randomly generated. 


@@ -1222,11 +1261,11 @@ It follows that
 \mathrm{Pr}[D^\prime(H(U_{N}))=1]=\mathrm{Pr}[D(U_{2N})=1].
 \end{equation}
  From (\ref{PCH-2}) and (\ref{PCH-4}), one can deduce that
 \mathrm{Pr}[D^\prime(H(U_{N}))=1]=\mathrm{Pr}[D(U_{2N})=1].
 \end{equation}
  From (\ref{PCH-2}) and (\ref{PCH-4}), one can deduce that

-there exist a polynomial time probabilistic
+there exists a polynomial time probabilistic

 algorithm $D^\prime$, a positive polynomial $p$, such that for all $k_0$ there exists
 $N\geq \frac{k_0}{2}$ satisfying 
 $$| \mathrm{Pr}[D(H(U_{N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)},$$
 algorithm $D^\prime$, a positive polynomial $p$, such that for all $k_0$ there exists
 $N\geq \frac{k_0}{2}$ satisfying 
 $$| \mathrm{Pr}[D(H(U_{N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)},$$

-proving that $H$ is not secure, a contradiction. 
+proving that $H$ is not secure, which is a contradiction. 

 \end{proof}
 
 
 \end{proof}
 
 


@@ -1255,21 +1294,21 @@ lesser than $2^{16}$.  So in practice we can choose prime numbers around
 indistinguishable    bits    is    lesser    than   or    equals    to
 $log_2(log_2(M))$). In other words, to generate a  32-bits number, we need to use
 8 times  the BBS  algorithm with possibly different  combinations of  $M$. This
 indistinguishable    bits    is    lesser    than   or    equals    to
 $log_2(log_2(M))$). In other words, to generate a  32-bits number, we need to use
 8 times  the BBS  algorithm with possibly different  combinations of  $M$. This

-approach is  not sufficient to be able to pass  all the TestU01,
+approach is  not sufficient to be able to pass  all the tests of TestU01,

 as small values of  $M$ for the BBS  lead to
 as small values of  $M$ for the BBS  lead to

-  small periods. So, in  order to add randomness  we proceed with
+  small periods. So, in  order to add randomness  we have proceeded with

 the followings  modifications. 
 \begin{itemize}
 \item
 Firstly, we  define 16 arrangement arrays  instead of 2  (as described in
 Algorithm \ref{algo:gpu_kernel2}), but only 2 of them are used at each call of
 the followings  modifications. 
 \begin{itemize}
 \item
 Firstly, we  define 16 arrangement arrays  instead of 2  (as described in
 Algorithm \ref{algo:gpu_kernel2}), but only 2 of them are used at each call of

-the  PRNG kernels. In  practice, the  selection of   combinations
+the  PRNG kernels. In  practice, the  selection of   combination

 arrays to be used is different for all the threads. It is determined
 by using  the three last bits  of two internal variables  used by BBS.
 %This approach  adds more randomness.   
 In Algorithm~\ref{algo:bbs_gpu},
 character  \& is for the  bitwise AND. Thus using  \&7 with  a number
 arrays to be used is different for all the threads. It is determined
 by using  the three last bits  of two internal variables  used by BBS.
 %This approach  adds more randomness.   
 In Algorithm~\ref{algo:bbs_gpu},
 character  \& is for the  bitwise AND. Thus using  \&7 with  a number

-gives the last 3 bits, providing so a number between 0 and 7.
+gives the last 3 bits, thus providing a number between 0 and 7.

 \item
 Secondly, after the  generation of the 8 BBS numbers  for each thread, we
 have a 32-bits number whose period is possibly quite small. So
 \item
 Secondly, after the  generation of the 8 BBS numbers  for each thread, we
 have a 32-bits number whose period is possibly quite small. So


@@ -1277,7 +1316,7 @@ to add randomness,  we generate 4 more BBS numbers   to
 shift  the 32-bits  numbers, and  add up to  6 new  bits.  This  improvement is
 described  in Algorithm~\ref{algo:bbs_gpu}.  In  practice, the last 2 bits
 of the first new BBS number are  used to make a left shift of at most
 shift  the 32-bits  numbers, and  add up to  6 new  bits.  This  improvement is
 described  in Algorithm~\ref{algo:bbs_gpu}.  In  practice, the last 2 bits
 of the first new BBS number are  used to make a left shift of at most

-3 bits. The  last 3 bits of the  second new BBS number are  add to the
+3 bits. The  last 3 bits of the  second new BBS number are  added to the

 strategy whatever the value of the first left shift. The third and the
 fourth new BBS  numbers are used similarly to apply  a new left shift
 and add 3 new bits.
 strategy whatever the value of the first left shift. The third and the
 fourth new BBS  numbers are used similarly to apply  a new left shift
 and add 3 new bits.


@@ -1290,7 +1329,7 @@ variable for BBS number 8 is stored in place 1.
 \end{itemize}
 
 \begin{algorithm}
 \end{itemize}
 
 \begin{algorithm}

-
+\begin{small}

 \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS
 in global memory\;
 NumThreads: Number of threads\;
 \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS
 in global memory\;
 NumThreads: Number of threads\;


@@ -1326,7 +1365,7 @@ array\_shift[4]=\{0,1,3,7\}\;
   }
   store internal variables in InternalVarXorLikeArray[threadId] using a rotation\;
 }
   }
   store internal variables in InternalVarXorLikeArray[threadId] using a rotation\;
 }

-
+\end{small}

 \caption{main kernel for the BBS based PRNG GPU}
 \label{algo:bbs_gpu}
 \end{algorithm}
 \caption{main kernel for the BBS based PRNG GPU}
 \label{algo:bbs_gpu}
 \end{algorithm}


@@ -1344,7 +1383,7 @@ variability.  In these operations, we make twice a left shift of $t$ of \emph{at
   most}  3 bits,  represented by  \texttt{shift} in  the algorithm,  and  we put
 \emph{exactly} the \texttt{shift}  last bits from a BBS  into the \texttt{shift}
 last bits of $t$. For this, an array named \texttt{array\_shift}, containing the
   most}  3 bits,  represented by  \texttt{shift} in  the algorithm,  and  we put
 \emph{exactly} the \texttt{shift}  last bits from a BBS  into the \texttt{shift}
 last bits of $t$. For this, an array named \texttt{array\_shift}, containing the

-correspondance between the  shift and the number obtained  with \texttt{shift} 1
+correspondence between the  shift and the number obtained  with \texttt{shift} 1

 to make the \texttt{and} operation is used. For example, with a left shift of 0,
 we  make an  and operation  with 0,  with  a left  shift of  3, we  make an  and
 operation with 7 (represented by 111 in binary mode).
 to make the \texttt{and} operation is used. For example, with a left shift of 0,
 we  make an  and operation  with 0,  with  a left  shift of  3, we  make an  and
 operation with 7 (represented by 111 in binary mode).


@@ -1401,7 +1440,7 @@ When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$
 \item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$.
 \item The initial seed can be obtained using the following procedure: $x_0=q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q~{mod}~N$.
 \item She recomputes the bit-vector $b$ by using BBS and $x_0$.
 \item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$.
 \item The initial seed can be obtained using the following procedure: $x_0=q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q~{mod}~N$.
 \item She recomputes the bit-vector $b$ by using BBS and $x_0$.

-\item Alice computes finally the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus  b$.
+\item Alice finally computes the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus  b$.

 \end{enumerate}
 
 
 \end{enumerate}
 
 


@@ -1414,14 +1453,16 @@ Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too
 her new public key will be $(S^0, N)$.
 
 To encrypt his message, Bob will compute
 her new public key will be $(S^0, N)$.
 
 To encrypt his message, Bob will compute

-\begin{equation}
-c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)
-\end{equation}
+%%RAPH : ici, j'ai mis un simple $
+%\begin{equation}
+$c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, \right.$
+$ \left. m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)$
+%%\end{equation}

 instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$. 
 
 The same decryption stage as in Blum-Goldwasser leads to the sequence 
 $\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$.
 instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$. 
 
 The same decryption stage as in Blum-Goldwasser leads to the sequence 
 $\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$.

-Thus, with a simple use of $S^0$, Alice can obtained the plaintext.
+Thus, with a simple use of $S^0$, Alice can obtain the plaintext.

 By doing so, the proposed generator is used in place of BBS, leading to
 the inheritance of all the properties presented in this paper.
 
 By doing so, the proposed generator is used in place of BBS, leading to
 the inheritance of all the properties presented in this paper.
 


@@ -1432,7 +1473,7 @@ In  this  paper, a formerly proposed PRNG based on chaotic iterations
 has been generalized to improve its speed. It has been proven to be
 chaotic according to Devaney.
 Efficient implementations on  GPU using xor-like  PRNGs as input generators
 has been generalized to improve its speed. It has been proven to be
 chaotic according to Devaney.
 Efficient implementations on  GPU using xor-like  PRNGs as input generators

-shown that a very large quantity of pseudorandom numbers can be generated per second (about
+have shown that a very large quantity of pseudorandom numbers can be generated per second (about

 20Gsamples/s), and that these proposed PRNGs succeed to pass the hardest battery in TestU01,
 namely the BigCrush.
 Furthermore, we have shown that when the inputted generator is cryptographically
 20Gsamples/s), and that these proposed PRNGs succeed to pass the hardest battery in TestU01,
 namely the BigCrush.
 Furthermore, we have shown that when the inputted generator is cryptographically