X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/blobdiff_plain/013c063fb5cb99cc33b413d937f330acc42ca73e..16b4107f3a70f199b9c04d0346050f4f52b5114b:/BookGPU/Chapters/chapter18/ch18.tex?ds=sidebyside

diff --git a/BookGPU/Chapters/chapter18/ch18.tex b/BookGPU/Chapters/chapter18/ch18.tex
index 3808995..71d2b84 100755
--- a/BookGPU/Chapters/chapter18/ch18.tex
+++ b/BookGPU/Chapters/chapter18/ch18.tex
@@ -1,47 +1,45 @@
-\chapterauthor{RaphaÃ«l Couturier}{Femto-ST Institute, University of Franche-Comte}
-\chapterauthor{Christophe Guyeux}{Femto-ST Institute, University of Franche-Comte}
+\chapterauthor{RaphaÃ«l Couturier and Christophe Guyeux}{Femto-ST Institute, University of Franche-Comte, France}
+%\chapterauthor{Christophe Guyeux}{Femto-ST Institute, University of Franche-Comt\'{e}}
 
 
-\chapter{Pseudo Random Number Generator on GPU}
+\chapter{Pseudorandom number generator on GPU}
 \label{chapter18}
 
 \section{Introduction}
 
 Randomness is of importance in many fields such as scientific
-simulations or cryptography.  ``Random numbers'' can mainly be
-generated either by a deterministic and reproducible algorithm called
-a pseudorandom number generator (PRNG). In this paper, we focus on
-reproducible generators, useful for instance in Monte-Carlo based
+simulations or cryptography.  Random numbers can mainly be
+generated by either a deterministic and reproducible algorithm called
+a pseudorandom number generator (PRNG)\index{PRNG}, or by a physical nondeterministic 
+process having all the characteristics of a random noise, called a truly random number
+generator (TRNG). In this chapter, we focus on
+reproducible generators, useful for instance in MonteCarlo-based
 simulators.  These domains need PRNGs that are statistically
 irreproachable.  In some fields such as in numerical simulations,
 speed is a strong requirement that is usually attained by using
 parallel architectures. In that case, a recurrent problem is that a
 deflation of the statistical qualities is often reported, when the
-parallelization of a good PRNG is realized. In some fields such as in
-numerical simulations, speed is a strong requirement that is usually
-attained by using parallel architectures. In that case, a recurrent
-problem is that a deflation of the statistical qualities is often
-reported, when the parallelization of a good PRNG is realized.  This
-is why ad-hoc PRNGs for each possible architecture must be found to
-achieve both speed and randomness.  On the other side, speed is not
-the main requirement in cryptography: the great need is to
+parallelization of a good PRNG is realized.  This
+is why adhoc PRNGs for each possible architecture must be found to
+achieve both speed and randomness.  On the other hand, speed is not
+the main requirement in cryptography: the most important point 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.  Or, 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''
+digits that have been already released~\cite{Goldreich}. ``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
+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
-generator that is unpredictable, disordered, sensible to its seed, or in other word chaotic.
-Their desire is to map a given chaotic dynamics into a sequence that seems random 
+third requirement: to define chaotic generators~\cite{kellert1994wake,  Wu20051195,gleick2011chaos}.
+The main idea is to  benefit from a chaotic dynamical system to obtain a
+generator that is unpredictable, disordered, sensible to its seed, or in other words, chaotic.
+These scientists' desire is to map a given chaotic dynamics into a sequence that seems random 
 and unassailable due to chaos.
 However, the chaotic maps used as a pattern are defined in the real line 
 whereas computers deal with finite precision numbers.
@@ -50,13 +48,40 @@ Furthermore, authors of such chaotic generators often claim their PRNG
 as secure due to their chaos properties, but there is no obvious relation
 between chaos and security as it is understood in cryptography.
 This is why the use of chaos for PRNG still remains marginal and disputable.
-
-
-The remainder of this paper is organized as follows. 
-A COMPLETER
-
-
-\section{Basic Recalls}
+However, we have established in previous researches that these flaws can 
+be circumvented by using a tool called choatic iterations. 
+Such investigations have led to the definition of a new
+family of PRNGs that are chaotic while being fast and statistically perfect,
+or cryptographically secure~\cite{bgw09:ip,bgw10:ip,bfgw11:ij,bfg12a:ip}. This family is improved and adapted for GPU 
+architectures in this chapter.
+
+
+Let us finish this introduction by noticing that, in this chapter, 
+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}\index{TestU01}.
+More precisely, each time we performed a test on a PRNG, we ran it
+twice in order to observe if all $p$-values were inside [0.01, 0.99]. In
+fact, we observed that few $p$-values (fewer than 10 out of 160) are sometimes
+outside this interval but inside [0.001, 0.999], so that is why a
+second run has allowed 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
+[0.01, 0.99].
+Chaos, for its part, refers to the well-established definition of a
+chaotic dynamical system defined by Devaney~\cite{Devaney}.
+
+The remainder of this chapter is organized as follows. 
+Basic definitions and terminologies about both topological chaos
+and chaotic iterations are provided in the next section. Various 
+chaotic iterations based on pseudorandom number generators are then
+presented in Section~\ref{sec:efficient PRNG}. They encompass
+naive and improved efficient generators for CPU and for GPU. 
+These generators are finally experimented in Section~\ref{sec:experiments}.
+
+
+\section{Basic remindees}
 \label{section:BASIC RECALLS}
 
 This section is devoted to basic definitions and terminologies in the fields of
@@ -64,17 +89,121 @@ 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}
+\subsection{A short presentation of chaos}
+
+
+Chaos theory studies the behavior of dynamical systems that are perfectly predictable, yet appear to be wildly amorphous and meaningless. 
+Chaotic systems\index{chaotic systems} are highly sensitive to initial conditions, 
+which is popularly referred to as the butterfly effect. 
+In other words, small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes, 
+in general rendering long-term prediction impossible  \cite{kellert1994wake}. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved \cite{kellert1994wake}. That is, the deterministic nature of these systems does not make them predictable \cite{kellert1994wake,Werndl01032009}. This behavior is known as deterministic chaos, or simply chaos. It has been well-studied in mathematics and
+physics, leading among other things to the well-established definition of Devaney which can be found next.
+
+
+
+
+
+\subsection{On Devaney's definition of chaos}\index{chaos}
+\label{sec:dev}
+Consider a metric space $(\mathcal{X},d)$ and a continuous function $f:\mathcal{X}\longrightarrow \mathcal{X}$, for one-dimensional dynamical systems of the form:
+\begin{equation}
+x^0 \in \mathcal{X} \textrm{  and } \forall n \in \mathds{N}^*, x^n=f(x^{n-1}),
+\label{Devaney}
+\end{equation}
+the following definition of chaotic behavior, formulated by Devaney~\cite{Devaney}, is widely accepted.
+
+\begin{definition}
+ A dynamical system of Form~(\ref{Devaney}) is said to be chaotic if the following conditions hold.
+\begin{itemize}
+\item Topological transitivity\index{topological transitivity}:
+
+\begin{equation}
+\forall U,V \textrm{ open sets of } \mathcal{X},~\exists k>0, f^k(U) \cap V \neq \varnothing .
+\end{equation}
+
+Intuitively, a topologically transitive map has points that eventually move under iteration from one arbitrarily small neighborhood to any other. Consequently, the dynamical system cannot be decomposed into two disjoint open sets that are invariant under the map. Note that if a map possesses a dense orbit, then it is clearly topologically transitive.
+\item Density of periodic points in $\mathcal{X}$\index{density of periodic points}:
+
+Let $P=\{p\in \mathcal{X}|\exists n \in \mathds{N}^{\ast}:f^n(p)=p\}$ the set of periodic points of $f$. Then $P$ is dense in $\mathcal{X}$:
+
+\begin{equation}
+ \overline{P}=\mathcal{X} .
+\end{equation}
+
+The density of periodic orbits means that every point in  space is closely approached  by periodic orbits in an arbitrary way. Topologically mixing systems failing this condition may not display sensitivity to initial conditions presented below and, hence,may not be chaotic.
+\item Sensitive dependence on initial conditions\index{sensitive dependence on initial conditions}:
+
+$\exists \varepsilon>0,$ $\forall x \in \mathcal{X},$ $\forall \delta >0,$ $\exists y \in \mathcal{X},$ $\exists n \in \mathbb{N},$ $d(x,y)<\delta$ and $d\left(f^n(x),f^n(y)\right) \geqslant \varepsilon.$
+
+Intuitively, a map possesses sensitive dependence on initial conditions if there exist points arbitrarily close to $x$ that eventually separate from $x$ by at least $\varepsilon$ under the iteration of $f$. Not all points near $x$ need to be eventually separate from $x$ under iteration, but there must be at least one such point in every neighborhood of $x$. If a map possesses sensitive dependence on initial conditions, then for all practical purposes, the dynamics of the map defy numerical computation. Small errors in computation that are introduced by round-off may become magnified upon iteration. The results of numerical computation of an orbit, no matter how accurate, may bear no resemblance whatsoever with the real orbit.
+\end{itemize}
+
+\end{definition}
+
+When $f$ is chaotic, then the system $(\mathcal{X}, f)$ is chaotic and quoting Devaney~\cite[p. 50]{Devaney}: ``it is unpredictable because of the sensitive dependence on initial conditions. It cannot be broken down or decomposed into two subsystems which do not interact because of topological transitivity. And, in the midst of this random behavior, we nevertheless have an element of regularity.'' Fundamentally different behaviors are consequently possible and occur in an unpredictable way.
+
+
 
 
-A COMPLETER
 
 
 
-\section{Toward Efficiency and Improvement for CI PRNG}
+
+\subsection{Chaotic iterations}\index{chaotic iterations}
+\label{subsection:Chaotic iterations}
+
+Let us now introduce an example of a dynamical systems family that has
+the potentiality to become chaotic, depending on the choice of the iteration 
+function. This family is the basis of the PRNGs we will
+develop during this chapter.
+
+\begin{definition}
+\label{Chaotic iterations}
+The set $\mathds{B}$ denoting $\{0,1\}$, let $f:\mathds{B}^{\mathsf{N}%
+}\longrightarrow \mathds{B}^{\mathsf{N}}$ be an ``iteration'' function and $S$ be a 
+sequence of subsets of $\llbracket 1, \mathsf{N}\rrbracket$ called a chaotic strategy. Then, the so-called \emph{chaotic iterations} are defined by~\cite{Robert1986}:
+
+\begin{equation}
+\label{eq:generalIC}
+\left\{\begin{array}{l}
+x^0\in \mathds{B}^{\mathsf{N}}, \\
+\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket%
+,x_i^n=
+\left\{
+\begin{array}{ll}
+x_i^{n-1} & \text{if}~ i\notin S^n  \\
+f(x^{n-1})_{i}  & \text{if}~ i \in S^n.
+\end{array} 
+\right. 
+\end{array}
+\right.
+\end{equation}
+\end{definition}
+
+In other words, at the $nth$ iteration, only the  cells of $S^{n}$ are
+iterated. 
+Chaotic iterations generate a set of vectors;
+they are defined by an initial state $x^{0}$, an iteration function $f$, and a chaotic strategy $S$~\cite{bg10:ij}.
+These ``chaotic iterations'' can behave chaotically as defined by Devaney, 
+depending on the choice of $f$~\cite{bg10:ij}. For instance,
+chaos is obtained when $f$ is the vectorial negation.
+Note that, with this example of function, chaotic iterations
+defined above can be rewritten as
+\begin{equation}
+\label{equation Oplus}
+x^0 \in \llbracket 0,2^\mathsf{N}-1\rrbracket,~\mathcal{S}^n \in \mathcal{P}\left(\llbracket 1,2^\mathsf{N}-1\rrbracket\right)^\mathds{N},~~ x^{n+1}=x^n \oplus \mathcal{S}^n,
+\end{equation}
+where $\mathcal{P}(X)$ stands for the set of subsets of $X$, whereas 
+$a\oplus b$ is the bitwise exclusive or operation between the binary
+representation of the integers $a$ and $b$. Note that the term
+ $\mathcal{S}^n$ is directly and obviously linked to the $S^n$ of
+Eq.~\ref{eq:generalIC}. Such an iterative sequence 
+satisfies the Devaney's definition of chaos.
+
+\section{Toward efficiency and improvement for CI PRNG}
 \label{sec:efficient PRNG}
 
-\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
+\subsection{First efficient implementation of a PRNG based on chaotic iterations}
 %
 %Based on the proof presented in the previous section, it is now possible to 
 %improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}. 
@@ -141,61 +270,61 @@ unsigned int CIPRNG() {
 
 
 In Listing~\ref{algo:seqCIPRNG} a sequential  version of the proposed PRNG based
-on  chaotic  iterations  is  presented.   The xor  operator  is  represented  by
-\textasciicircum.  This function uses  three classical 64-bits PRNGs, namely the
+on  chaotic  iterations  is  presented, which extends the generator family 
+formerly presented in~\cite{bgw09:ip,guyeux10}.   The xor  operator  is  represented  by
+\textasciicircum.  This function uses  three classical 64-bit PRNGs, namely the
 \texttt{xorshift},         the          \texttt{xor128},         and         the
 \texttt{xorwow}~\cite{Marsaglia2003}.  In the following, we call them ``xor-like
 PRNGs''.   As each  xor-like PRNG  uses 64-bits  whereas our  proposed generator
-works with 32-bits, we use the command \texttt{(unsigned int)}, that selects the
+works with 32-bits, we use the command \texttt{(unsigned int)}, which selects the
 32 least  significant bits  of a given  integer, and the  code \texttt{(unsigned
   int)(t$>>$32)} in order to obtain the 32 most significant bits of \texttt{t}.
 
-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}. 
-At this point, we thus
-have defined an efficient and statistically unbiased generator. Its speed is
+Thus producing a pseudorandom number needs 6 xor operations with 6 32-bit numbers
+that  are provided by  3 64-bit  PRNGs.  This  version successfully  passes the
+stringent {\it BigCrush} battery of tests~\cite{LEcuyerS07}. 
+At this point, we 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.
 
 
 
-\subsection{Efficient PRNGs based on Chaotic Iterations on GPU}
+\subsection{Efficient PRNGs based on chaotic iterations on GPU}
 \label{sec:efficient PRNG gpu}
 
-In order to  take benefits from the computing power  of GPU, a program
+In order to  benefit from the computing power  of GPU, a program
 needs  to have  independent blocks  of  threads that  can be  computed
 simultaneously. In general,  the larger the number of  threads is, the
 more local  memory is  used, and the  less branching  instructions are
-used  (if,  while,  ...),  the  better the  performances  on  GPU  is.
+used  (if,  while,  etc.) and so,  the  better the  performances  on  GPU  are.
 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  recall  that  in  the  CUDA~\cite{Nvid10}
+do  so,  we  must   first  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
+them. Furthermore, in  CUDA, parts of  the code that are executed by the  GPU are
 called {\it kernels}.
 
 
-\subsection{Naive Version for GPU}
+\subsection{Naive version for GPU}
 
  
 It is possible to deduce from the CPU version a quite similar version adapted to GPU.
-The simple principle consists in making each thread of the GPU computing the CPU version of our PRNG.  
+The simple principle consists of making each thread of the GPU compute 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 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
+To do this, the   Indirection, Shift, Accumulate, Add, and Count (ISAAC)  PRNG~\cite{Jenkins96} is used to  set  all  the
 parameters embedded into each thread.   
 
 The implementation of  the three
 xor-like  PRNGs  is  straightforward  when  their  parameters  have  been
 allocated in  the GPU memory.  Each xor-like  works with  an internal
 number  $x$  that saves  the  last  generated  pseudorandom number. Additionally,  the
-implementation of the  xor128, the xorshift, and the  xorwow respectively require
+implementation of the  {\it xor128}, the {\it xorshift}, and the  {\it xorwow}, respectively, require
 4, 5, and 6 unsigned long as internal variables.
 
 
@@ -214,7 +343,7 @@ NumThreads: number of threads\;}
   store internal variables in InternalVarXorLikeArray[threadIdx]\;
 }
 \end{small}
-\caption{Main kernel of the GPU ``naive'' version of the PRNG based on chaotic iterations}
+\caption{main kernel of the GPU ``naive'' version of the PRNG based on chaotic iterations}
 \label{algo:gpu_kernel}
 \end{algorithm}
 
@@ -223,35 +352,35 @@ NumThreads: 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 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)},
-then   the  memory   required   to  store all of the  internals   variables  of both the  xor-like
-PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits numbers}
+inside   a    kernel   is   limited  (i.e.,    the    variable   {\it 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-bit number).},
+then   the  memory   required   to  store all of the  internal   variables  of both the  xor-like
+PRNGs\footnote{We multiply this number by $2$ in order to count 32-bit numbers.}
 and  the pseudorandom  numbers generated by  our  PRNG,  is  equal to  $100,000\times  ((4+5+6)\times
-2+(1+100))=1,310,000$ 32-bits numbers, that is, approximately $52$Mb.
+2+(1+100))=1,310,000$ 32-bit numbers, that is, approximately $52$Mb.
 
-This generator is able to pass the whole BigCrush battery of tests, for all
-the versions that have been tested depending on their number of threads 
-(called \texttt{NumThreads} in our algorithm, tested up to $5$ million).
+This generator is able to pass the whole {\it BigCrush} battery of tests, for all
+the versions that have been tested depend on their number of threads 
+(called NumThreads in our algorithm, tested up to $5$ million).
 
 
-\subsection{Improved Version for GPU}
+\subsection{Improved version for GPU}
 
 As GPU cards using CUDA have shared memory between threads of the same block, it
 is possible  to use this  feature in order  to simplify the  previous algorithm,
-i.e., to use less  than 3 xor-like PRNGs. The solution  consists in computing only
-one xor-like PRNG by thread, saving  it into the shared memory, and then to use the results
+i.e., to use fewer  than 3 xor-like PRNGs. The solution  consists in computing only
+one xor-like PRNG by thread, saving  it into the shared memory, and then  using the results
 of some  other threads in the  same block of  threads. In order to  define which
 thread uses the result of which other  one, we can use a combination array that
 contains  the indexes  of  all threads  and  for which  a combination has  been
 performed. 
 
 In  Algorithm~\ref{algo:gpu_kernel2},  two  combination  arrays are  used.   The
-variable     \texttt{offset}    is     computed    using     the     value    of
-\texttt{combination\_size}.   Then we  can compute  \texttt{o1}  and \texttt{o2}
+variable     offset    is     computed    using     the     value    of
+combination\_size.   Then we  can compute  o1  and o2
 representing the  indexes of  the other  threads whose results  are used  by the
-current one.   In this algorithm, we  consider that a 32-bits  xor-like PRNG has
+current one.   In this algorithm, we  consider that a 32-bit  xor-like PRNG has
 been chosen. In practice, we  use the xor128 proposed in~\cite{Marsaglia2003} in
 which  unsigned longs  (64 bits)  have been  replaced by  unsigned  integers (32
 bits).
@@ -267,7 +396,7 @@ array\_comb1, array\_comb2: Arrays containing combinations of size combination\_
 
 \KwOut{NewNb: array containing random numbers in global memory}
 \If{threadId is concerned} {
-  retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory and x\;
+  retrieve data from InternalVarXorLikeArray[threadIdx] in local variables including shared memory and x\;
   offset = threadIdx\%combination\_size\;
   o1 = threadIdx-offset+array\_comb1[offset]\;
   o2 = threadIdx-offset+array\_comb2[offset]\;
@@ -277,9 +406,9 @@ array\_comb1, array\_comb2: Arrays containing combinations of size combination\_
     shared\_mem[threadId]=t\;
     x = x\textasciicircum t\;
 
-    store the new PRNG in NewNb[NumThreads*threadId+i]\;
+    store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
   }
-  store internal variables in InternalVarXorLikeArray[threadId]\;
+  store internal variables in InternalVarXorLikeArray[threadIdx]\;
 }
 \end{small}
 \caption{Main kernel for the chaotic iterations based PRNG GPU efficient
@@ -287,55 +416,64 @@ version\label{IR}}
 \label{algo:gpu_kernel2} 
 \end{algorithm}
 
-\subsection{Chaos 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 
+A run of Algorithm~\ref{algo:gpu_kernel2} consists of an operation ($x=x\oplus t$) having 
 the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
 system of Eq.~\ref{eq:generalIC}. That is, an iteration of the general chaotic
 iterations is realized between the last stored value $x$ of the thread and a strategy $t$
 (obtained by a bitwise exclusive or between a value provided by a xor-like() call
 and two values previously obtained by two other threads).
-To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
+To be certain that such iterations correspond to the chaotic one recalled at the
+end of  Section~\ref{sec:dev},
 we must guarantee that this dynamical system iterates on the space 
-$\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
+$\mathcal{X} =\mathds{B}^\mathsf{N} \times \mathcal{P}\left(\llbracket 1, 2^\mathsf{N} \rrbracket\right)^\mathds{N}$.
 The left term $x$ obviously belongs to $\mathds{B}^ \mathsf{N}$.
-To prevent from any flaws of chaotic properties, we must check that the right 
-term (the last $t$), corresponding to the strategies,  can possibly be equal to any
-integer of $\llbracket 1, \mathsf{N} \rrbracket$. 
+To prevent any flaws of chaotic properties, we must check that the right 
+term (the last $t$ in Algorithm~\ref{algo:gpu_kernel2}), corresponding to the strategies,  can possibly be equal to any
+integer of $\llbracket 1, 2^\mathsf{N} \rrbracket$. 
 
-Such a result is obvious, as for the xor-like(), all the
-integers belonging into its interval of definition can occur at each iteration, and thus the 
+Such a result is obvious; as for the xor-like(), all the
+integers belonging to its interval of definition can occur at each iteration, and thus the 
 last $t$ respects the requirement. Furthermore, it is possible to
-prove by an immediate mathematical induction that, as the initial $x$
-is uniformly distributed (it is provided by a cryptographically secure PRNG),
-the two other stored values shmem[o1] and shmem[o2] are uniformly distributed too,
+prove by an immediate mathematical induction that, supposing that the initial $x$
+is uniformly distributed, %(it is provided by a cryptographically secure PRNG),
+the two other stored values shmem[o1] and shmem[o2] are uniformly distributed too
 (this is the induction hypothesis), and thus the next $x$ is finally uniformly distributed.
 
 Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
 chaotic iterations presented previously, and for this reason, it satisfies the 
-Devaney's formulation of a chaotic behavior.
+Devaney's formulation of  chaotic behavior.
 
 \section{Experiments}
 \label{sec:experiments}
 
 Different experiments  have been  performed in order  to measure  the generation
-speed. We have used a first computer equipped with a Tesla C1060 NVidia  GPU card
+speed. We have used one computer equipped with a Tesla C1060 NVIDIA  GPU card
 and an
 Intel  Xeon E5530 cadenced  at 2.40  GHz,  and 
 a second computer  equipped with a smaller  CPU and  a GeForce GTX  280. 
 All the
 cards have 240 cores.
 
+\begin{figure}[htpb]
+\begin{center}
+  \includegraphics[scale=0.65]{Chapters/chapter18/figures/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{figure}
+
 In  Figure~\ref{fig:time_xorlike_gpu} we  compare the  quantity of  pseudorandom numbers
 generated per second with various xor-like based PRNGs. In this figure, the optimized
-versions use the {\it xor64} described in~\cite{Marsaglia2003}, whereas the naive versions
+versions use the  xor64 described in~\cite{Marsaglia2003}, whereas the naive versions
 embed  the three  xor-like  PRNGs described  in Listing~\ref{algo:seqCIPRNG}.   In
 order to obtain the optimal performances, the storage of pseudorandom numbers
-into the GPU memory has been removed. This step is time consuming and slows down the numbers
+to the GPU memory has been removed. This step is time-consuming and slows down the numbers
 generation.  Moreover this   storage  is  completely
 useless, in case of applications that consume the pseudorandom
-numbers  directly   after generation. We can see  that when the number of  threads is greater
-than approximately 30,000 and lower than 5 million, the number of pseudorandom numbers generated
+numbers  directly   after they have been generated. We can see  that when the number of  threads is greater
+than approximately 30,000 and less than 5 million, the number of pseudorandom numbers generated
 per second  is almost constant.  With the  naive version, this value ranges from 2.5 to
 3GSamples/s.   With  the  optimized   version,  it  is  approximately  equal to
 20GSamples/s. Finally  we can remark  that both GPU  cards are quite  similar, but in
@@ -344,49 +482,30 @@ should be of better quality.
 As a  comparison,   Listing~\ref{algo:seqCIPRNG}  leads   to the  generation of  about
 138MSample/s when using one core of the Xeon E5530.
 
-\begin{figure}[htbp]
-\begin{center}
-  \includegraphics[scale=0.65]{Chapters/chapter18/figures/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{figure}
 
 
 
 
 
-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
-new PRNG  has a strong  level of  security, which is  necessarily paid by  a speed
-reduction.
 
-\begin{figure}[htbp]
-\begin{center}
-  \includegraphics[scale=0.65]{Chapters/chapter18/figures/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{figure}
 
-All  these  experiments allow  us  to conclude  that  it  is possible  to
+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.
-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.
-
 
 
+\section{Summary}
 
 
+In this chapter,  a PRNG based on chaotic iterations is  presented.   It is proven to be
+chaotic according to Devaney.   Efficient implementations  on GPU
+using xor-like PRNGs  as input generators have shown that  a very large quantity
+of pseudorandom  numbers can be  generated per second (about  20Gsamples/s on a Tesla C1060), and
+that these proposed PRNGs succeed in passing the hardest battery in TestU01, namely, the {\it BigCrush}.
 
 
 
 
 
 
-\putbib[Chapters/chapter18/biblio]
+\putbib[Chapters/chapter18/biblio18]
 
