X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/47000eb06d8648eaafeadfdda00e1e4ce76e9ee6..228143ea0da3dc28561a873ee227024d21bb1e57:/prng_gpu.tex?ds=sidebyside

diff --git a/prng_gpu.tex b/prng_gpu.tex
index d2e5e05..84efafa 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -34,113 +34,183 @@
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
-\title{Efficient Generation of Pseudo-Random Numbers based on Chaotic Iterations
-on GPU}
+\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
 \begin{document}
 
-\author{Jacques M. Bahi, Rapha\"{e}l Couturier, and Christophe
-Guyeux, Pierre-Cyrille Heam\thanks{Authors in alphabetic order}}
-
+\author{Jacques M. Bahi, Rapha\"{e}l Couturier,  Christophe
+Guyeux, and Pierre-Cyrille Heam\thanks{Authors in alphabetic order}}
+   
 \maketitle
 
 \begin{abstract}
-In this paper we present a new pseudo-random numbers generator (PRNG) on
-graphics processing units  (GPU). This PRNG is based  on chaotic iterations.  it
-is proven  to be chaotic  in the Devanay's  formulation. We propose  an efficient
-implementation  for  GPU which  succeeds  to  the  {\it BigCrush},  the  hardest
-batteries of test of TestU01.  Experimentations show that this PRNG can generate
+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
+is firstly proven  to be chaotic according to the Devaney's  formulation. We thus propose  an efficient
+implementation  for  GPU that successfully passes the   {\it BigCrush} tests, deemed to be the  hardest
+battery of tests in TestU01.  Experiments show that this PRNG can generate
 about 20 billions of random numbers  per second on Tesla C1060 and NVidia GTX280
 cards.
+It is finally established that, under reasonable assumptions, the proposed PRNG can be cryptographically 
+secure.
 
 
 \end{abstract}
 
 \section{Introduction}
 
-Random  numbers are  used in  many scientific  applications and  simulations. On
-finite  state machines,  as computers,  it is  not possible  to  generate random
-numbers but only pseudo-random numbers. In practice, a good pseudo-random numbers
-generator (PRNG) needs  to verify some features to be used  by scientists. It is
-important  to  be  able  to  generate  pseudo-random  numbers  efficiently,  the
-generation  needs to  be reproducible  and a  PRNG needs  to satisfy  many usual
-statistical properties. Finally, from our point a view, it is essential to prove
-that  a PRNG  is  chaotic.  Concerning  the  statistical tests,  TestU01 is  the
-best-known public-domain statistical testing package.   So we use it for all our
-PRNGs, especially the {\it BigCrush}  which provides the largest serie of tests.
-Concerning  the  chaotic properties,  Devaney~\cite{Devaney}  proposed a  common
-mathematical formulation of chaotic dynamical systems.
-
-In a  previous work~\cite{bgw09:ip}  we have proposed  a new familly  of chaotic
-PRNG  based on  chaotic iterations. We  have proven  that these  PRNGs are
-chaotic in the Devaney's sense.  In this paper we propose a faster version which
-is also proven to be chaotic.
-
-Although graphics  processing units (GPU)  was initially designed  to accelerate
+Randomness is of importance in many fields as scientific simulations or cryptography. 
+``Random numbers'' can mainly be generated either by a deterministic and reproducible algorithm
+called a pseudorandom number generator (PRNG), or by a physical non-deterministic 
+process having all the characteristics of a random noise, called a truly random number
+generator (TRNG). 
+In this paper, we focus on reproducible generators, useful for instance in
+Monte-Carlo based simulators or in several cryptographic schemes.
+These domains need PRNGs that are statistically irreproachable. 
+On some fields 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 deflate 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 define \emph{secure} generators being 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. 
+Finally, a small part of the community working in this domain focus 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 words chaotic.
+Their 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.
+This distortion leads to a deflation of both chaotic properties and speed.
+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 authors' opinion is that topological properties of disorder, as they are
+properly defined in the mathematical theory of chaos, can reinforce the quality
+of a PRNG. But they are not substitutable for security or statistical perfection.
+Indeed, to the authors' point of view, such properties can be useful in the two following situations. On the
+one hand, a post-treatment based on a chaotic dynamical system can be applied
+to a PRNG statistically deflective, in order to improve its statistical 
+properties. Such an improvement can be found, for instance, in~\cite{bgw09:ip,bcgr11:ip}.
+On the other hand, chaos can be added to a fast, statistically perfect PRNG and/or a
+cryptographically secure one, in case where chaos can be of interest,
+\emph{only if these last properties are not lost during
+the proposed post-treatment}. Such an assumption is behind this research work.
+It leads to the attempts to define a 
+family of PRNGs that are chaotic while being fast and statistically perfect,
+or cryptographically secure.
+Let us finish this paragraph by noticing that, in this paper, 
+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 into 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}.
+
+
+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
+PRNGs. We have shown that proofs of Devaney's chaos can be established for this
+family, and that the sequence obtained after this post-treatment can pass the
+NIST~\cite{Nist10}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} batteries of tests, even if the inputted generators
+cannot.
+The proposition of this paper is to improve widely the speed of the formerly
+proposed generator, without any lack of chaos or statistical properties.
+In particular, a version of this PRNG on graphics processing units (GPU)
+is proposed.
+Although GPU was initially designed  to accelerate
 the manipulation of  images, they are nowadays commonly  used in many scientific
-applications. Therefore,  it is important  to be able to  generate pseudo-random
-numbers inside a GPU when a scientific application runs in a GPU. That is why we
-also provide  an efficient  PRNG for  GPU respecting based  on IC.  Such devices
-allows us to generated almost 20 billions of random numbers per second.
-
-In order  to establish  that our  PRNGs are chaotic  according to  the Devaney's
-formulation, we  extend what we  have proposed in~\cite{guyeux10}.
-
-The rest of this paper  is organised as follows. In Section~\ref{section:related
-  works} we  review some GPU implementions  of PRNG.  Section~\ref{section:BASIC
-  RECALLS} gives some basic recalls  on Devanay's formation of chaos and chaotic
-iterations. In  Section~\ref{sec:pseudo-random} the proof of chaos  of our PRNGs
-is   studied.    Section~\ref{sec:efficient    prng}   presents   an   efficient
-implementation of  our chaotic PRNG  on a CPU.   Section~\ref{sec:efficient prng
-  gpu}   describes   the  GPU   implementation   of   our   chaotic  PRNG.    In
-Section~\ref{sec:experiments}     some    experimentations     are    presented.
- Finally, we give a conclusion and some perspectives.
+applications. Therefore,  it is important  to be able to  generate pseudorandom
+numbers inside a GPU when a scientific application runs in it. This remark
+motivates our proposal of a chaotic and statistically perfect PRNG for GPU.  
+Such device
+allows us to generated almost 20 billions of pseudorandom numbers per second.
+Last, but not least, we show that the proposed post-treatment preserves the
+cryptographical security of the inputted PRNG, when this last has such a 
+property.
+
+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. 
+Proofs 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
+  gpu}   describes   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.
+Such a proof leads to the proposition of a cryptographically secure and
+chaotic generator on GPU based on the famous Blum Blum Shum
+in Section~\ref{sec:CSGPU}.
+This research work ends by a conclusion section, in which the contribution is
+summarized and intended future work is presented.
 
 
 
 
 \section{Related works on GPU based PRNGs}
 \label{section:related works}
-In the litterature many authors have work on defining GPU based PRNGs. We do not
-want to be exhaustive and we just give the most significant works from our point
-of view. When authors mention the  number of random numbers generated per second
-we mention  it. We  consider that  a million numbers  per second  corresponds to
-1MSample/s and than a billion numbers per second corresponds to 1GSample/s.
-
-In \cite{Pang:2008:cec},  the authors define  a PRNG based on  cellular automata
-which  does   not  require  high  precision  integer   arithmetics  nor  bitwise
-operations. There is no mention of statistical tests nor proof that this PRNG is
-chaotic.  Concerning   the  speed  of   generation,  they  can   generate  about
-3.2MSample/s on a GeForce 7800 GTX GPU (which is quite old now).
+
+Numerous research works on defining GPU based PRNGs have yet been proposed  in the
+literature, so that completeness is impossible.
+This is why authors of this document only give reference to the most significant attempts 
+in this domain, from their subjective point of view. 
+The  quantity of pseudorandom numbers generated per second is mentioned here 
+only when the information is given in the related work. 
+A million numbers  per second will be simply written as
+1MSample/s whereas a billion numbers per second is 1GSample/s.
+
+In \cite{Pang:2008:cec}  a PRNG based on  cellular automata is defined
+with no  requirement to an high  precision  integer   arithmetic  or to any bitwise
+operations. Authors can   generate  about
+3.2MSamples/s on a GeForce 7800 GTX GPU, which is quite an old card now.
+However, there is neither a mention of statistical tests nor any proof of
+chaos or cryptography in this document.
 
 In \cite{ZRKB10}, the authors propose  different versions of efficient GPU PRNGs
-based on  Lagged Fibonacci, Hybrid  Taus or Hybrid  Taus.  They have  used these
+based on  Lagged Fibonacci or Hybrid  Taus.  They have  used these
 PRNGs   for  Langevin   simulations   of  biomolecules   fully  implemented   on
 GPU. Performance of  the GPU versions are far better than  those obtained with a
-CPU and these PRNGs succeed to pass the {\it BigCrush} test of TestU01. There is
-no mention that their PRNGs have chaos mathematical properties.
+CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01. 
+However the evaluations of the proposed PRNGs are only statistical ones.
 
 
 Authors of~\cite{conf/fpga/ThomasHL09}  have studied the  implementation of some
-PRNGs on  diferrent computing architectures: CPU,  field-programmable gate array
-(FPGA), GPU and massively parallel  processor. This study is interesting because
-it  shows the  performance  of the  same  PRNGs on  different architeture.   For
-example,  the FPGA  is globally  the  fastest architecture  and it  is also  the
-efficient one because it provides the fastest number of generated random numbers
-per joule. Concerning the GPU,  authors can generate betweend 11 and 16GSample/s
-with a GTX 280  GPU. The drawback of this work is  that those PRNGs only succeed
-the {\it Crush} test which is easier than the {\it Big Crush} test.
-
-Cuda  has developped  a  library for  the  generation of  random numbers  called
-Curand~\cite{curand11}.        Several       PRNGs        are       implemented:
-Xorwow~\cite{Marsaglia2003} and  some variants of Sobol. Some  tests report that
-the  fastest version provides  15GSample/s on  the new  Fermi C2050  card. Their
-PRNGs fail to succeed the whole tests of TestU01 on only one test.
+PRNGs on  different computing architectures: CPU,  field-programmable gate array
+(FPGA), massively parallel  processors, and GPU. This study is of interest, because
+the  performance  of the  same  PRNGs on  different architectures are compared. 
+FPGA appears as  the  fastest  and the most
+efficient architecture, providing the fastest number of generated pseudorandom numbers
+per joule. 
+However, we can notice that authors can ``only'' generate between 11 and 16GSamples/s
+with a GTX 280  GPU, which should be compared with
+the results presented in this document.
+We can remark too that the PRNGs proposed in~\cite{conf/fpga/ThomasHL09} are only
+able to pass the {\it Crush} battery, which is very easy compared to the {\it Big Crush} one.
+
+Lastly, Cuda  has developed  a  library for  the  generation of  pseudorandom numbers  called
+Curand~\cite{curand11}.        Several       PRNGs        are       implemented, among
+other things 
+Xorwow~\cite{Marsaglia2003} and  some variants of Sobol. The  tests reported show that
+their  fastest version provides  15GSamples/s on  the new  Fermi C2050  card. 
+But their PRNGs cannot pass the whole TestU01 battery (only one test is failed).
 \newline
 \newline
-To the best of our knowledge no GPU implementation have been proven to have chaotic properties.
+We can finally remark that, to the best of our knowledge, no GPU implementation have been proven to be chaotic, and the cryptographically secure property is surprisingly never regarded.
 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
+
 This section is devoted to basic definitions and terminologies in the fields of
 topological chaos and chaotic iterations.
 \subsection{Devaney's Chaotic Dynamical Systems}
@@ -355,17 +425,18 @@ if and only if $\Gamma(f)$ is strongly connected.
 \end{theorem}
 
 This result of chaos has lead us to study the possibility to build a
-pseudo-random number generator (PRNG) based on the chaotic iterations. 
+pseudorandom number generator (PRNG) based on the chaotic iterations. 
 As $G_f$, defined on the domain   $\llbracket 1 ;  \mathsf{N} \rrbracket^{\mathds{N}} 
 \times \mathds{B}^\mathsf{N}$, is build from Boolean networks $f : \mathds{B}^\mathsf{N}
 \rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$
 during implementations (due to the discrete nature of $f$). It is as if
 $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ;  \mathsf{N}
-\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
+\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG).
+
+\section{Application to pseudorandomness}
+\label{sec:pseudorandom}
 
-\section{Application to Pseudo-Randomness}
-\label{sec:pseudo-random}
-\subsection{A First Pseudo-Random Number Generator}
+\subsection{A First pseudorandom Number Generator}
 
 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, 
@@ -410,7 +481,7 @@ return $y$\;
 
 
 This generator is synthesized in Algorithm~\ref{CI Algorithm}.
-It takes as input: a function $f$;
+It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:CaractÃ©risation   des   IC   chaotiques};
 an integer $b$, ensuring that the number of executed iterations is at least $b$
 and at most $2b+1$; and an initial configuration $x^0$.
 It returns the new generated configuration $x$.  Internally, it embeds two
@@ -435,7 +506,7 @@ We have proven in \cite{bcgr11:ip} that,
   if and only if $M$ is a double stochastic matrix.
 \end{theorem} 
 
-This former generator as successively passed various batteries of statistical tests, as the NIST tests~\cite{bcgr11:ip}.
+This former generator as successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07}.
 
 \subsection{Improving the Speed of the Former Generator}
 
@@ -490,7 +561,7 @@ the vectorial negation, leads to a speed improvement. However, proofs
 of chaos obtained in~\cite{bg10:ij} have been established
 only for chaotic iterations of the form presented in Definition 
 \ref{Def:chaotic iterations}. The question is now to determine whether the
-use of more general chaotic iterations to generate pseudo-random numbers 
+use of more general chaotic iterations to generate pseudorandom numbers 
 faster, does not deflate their topological chaos properties.
 
 \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
@@ -900,7 +971,7 @@ tab1, tab2: Arrays containing permutations of size permutation\_size\;}
 
 \KwOut{NewNb: array containing random numbers in global memory}
 \If{threadId is concerned} {
-  retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory\;
+  retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory and x\;
   offset = threadIdx\%permutation\_size\;
   o1 = threadIdx-offset+tab1[offset]\;
   o2 = threadIdx-offset+tab2[offset]\;
@@ -952,12 +1023,14 @@ Intel  Xeon E5530 cadenced  at 2.40  GHz for  our experiments  and we  have used
 another one  equipped with  a less performant  CPU and  a GeForce GTX  280. Both
 cards have 240 cores.
 
-In Figure~\ref{fig:time_gpu}  we compare the number of  random numbers generated
-per second. The xor-like prng  is a xor64 described in~\cite{Marsaglia2003}.  In
-order to obtain the optimal performance  we remove the storage of random numbers
-in the GPU memory. This step is time consumming and slows down the random number
-generation.  Moreover, if you are interested by applications that consome random
-numbers  directly   when  they  are  generated,  their   storage  is  completely
+In  Figure~\ref{fig:time_xorlike_gpu} we  compare the  number of  random numbers
+generated per second with the xor-like based PRNG. In this figure, the optimized
+version use the {\it xor64} described in~\cite{Marsaglia2003}. The naive version
+use  the three  xor-like  PRNGs described  in Listing~\ref{algo:seqCIprng}.   In
+order to obtain the optimal performance we removed the storage of random numbers
+in the GPU memory. This step is time consuming and slows down the random numbers
+generation.  Moreover, if one is  interested by applications that consume random
+numbers  directly   when  they  are  generated,  their   storage  are  completely
 useless. In this  figure we can see  that when the number of  threads is greater
 than approximately 30,000 upto 5 millions the number of random numbers generated
 per second  is almost constant.  With the  naive version, it is  between 2.5 and
@@ -968,10 +1041,10 @@ should be of better quality.
 
 \begin{figure}[htbp]
 \begin{center}
-  \includegraphics[scale=.7]{curve_time_gpu.pdf}
+  \includegraphics[scale=.7]{curve_time_xorlike_gpu.pdf}
 \end{center}
-\caption{Number of random numbers generated per second}
-\label{fig:time_gpu}
+\caption{Number of random numbers generated per second with the xorlike based PRNG}
+\label{fig:time_xorlike_gpu}
 \end{figure}
 
 
@@ -979,7 +1052,22 @@ In  comparison,   Listing~\ref{algo:seqCIprng}  allows  us   to  generate  about
 138MSample/s with only one core of the Xeon E5530.
 
 
+In Figure~\ref{fig:time_bbs_gpu}  we highlight the performance  of the optimized
+BBS based  PRNG on GPU. Performances are  less important. On the  Tesla C1060 we
+obtain approximately 1.8GSample/s and on the GTX 280 about 1.6GSample/s.
+
+\begin{figure}[htbp]
+\begin{center}
+  \includegraphics[scale=.7]{curve_time_bbs_gpu.pdf}
+\end{center}
+\caption{Number of random numbers generated per second with the BBS based PRNG}
+\label{fig:time_bbs_gpu}
+\end{figure}
 
+Both  these  experimentations allows  us  to conclude  that  it  is possible  to
+generate a  huge number of pseudorandom  numbers with the  xor-like version and
+about tens  times less with the BBS  based version. The former  version has only
+chaotic properties whereas the latter also has cryptographically properties.
 
 
 %% \section{Cryptanalysis of the Proposed PRNG}
@@ -1518,13 +1606,13 @@ In  comparison,   Listing~\ref{algo:seqCIprng}  allows  us   to  generate  about
 
 
 \section{Security Analysis}
-
+\label{sec:security analysis}
 
 
 
 In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.
-In a cryptographic context, a pseudo-random generator is a deterministic
+In a cryptographic context, a pseudorandom generator is a deterministic
 algorithm $G$ transforming strings  into strings and such that, for any
 seed $w$ of length $N$, $G(w)$ (the output of $G$ on the input $w$) has size
 $\ell_G(N)$ with $\ell_G(N)>N$.
@@ -1548,7 +1636,7 @@ quite easily possible to change the function $\ell$ into any polynomial
 function $\ell^\prime$ satisfying $\ell^\prime(N)>N)$~\cite[Chapter 3.3]{Goldreich}.
 
 The generation schema developed in (\ref{equation Oplus}) is based on a
-pseudo-random generator. Let $H$ be a cryptographic PRNG. We may assume,
+pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,
 without loss of generality, that for any string $S_0$ of size $N$, the size
 of $H(S_0)$ is $kN$, with $k>2$. It means that $\ell_H(N)=kN$. 
 Let $S_1,\ldots,S_k$ be the 
@@ -1629,20 +1717,45 @@ proving that $H$ is not secure, a contradiction.
 
 
 
+\section{A cryptographically secure prng for GPU}
+\label{sec:CSGPU}
+It is  possible to build a  cryptographically secure prng based  on the previous
+algorithm (algorithm~\ref{algo:gpu_kernel2}).   It simply consists  in replacing
+the  {\it  xor-like} algorithm  by  another  cryptographically  secure prng.  In
+practice, we suggest  to use the BBS algorithm~\cite{BBS}  which takes the form:
+$$x_{n+1}=x_n^2~ mod~ M$$  where $M$ is the product of  two prime numbers. Those
+prime numbers  need to be congruent  to 3 modulus  4. In practice, this  PRNG is
+known to  be slow and  not efficient for  the generation of random  numbers. For
+current  GPU   cards,  the  modulus   operation  is  the  most   time  consuming
+operation. So in  order to obtain quite reasonable  performances, it is required
+to use only modulus on 32  bits integer numbers. Consequently $x_n^2$ need to be
+less than  $2^{32}$ and the  number $M$  need to be  less than $2^{16}$.   So in
+pratice we can  choose prime numbers around 256 that are  congruent to 3 modulus
+4.  With  32 bits numbers,  only the  4 least significant  bits of $x_n$  can be
+chosen  (the   maximum  number  of   undistinguishing  is  less  or   equals  to
+$log_2(log_2(x_n))$). So  to generate a 32 bits  number, we need to  use 8 times
+the BBS algorithm, with different combinations of $M$ is required.
+
+Currently this PRNG does not succeed to pass all the tests of TestU01.
+
 
 \section{Conclusion}
 
 
 In  this  paper  we have  presented  a  new  class  of  PRNGs based  on  chaotic
-iterations. We have proven that these PRNGs are chaotic in the sense of Devenay. 
+iterations. We have proven that these PRNGs are chaotic in the sense of Devenay.
+We also propose a PRNG cryptographically secure and its implementation on GPU.
+
+An  efficient implementation  on  GPU based  on  a xor-like  PRNG  allows us  to
+generate   a  huge   number   of  pseudorandom   numbers   per  second   (about
+20Gsample/s). This PRNG succeeds to pass the hardest batteries of TestU01.
+
+In future  work we plan to  extend this work  for parallel PRNG for  clusters or
+grid computing. We also plan to improve  the BBS version in order to succeed all
+the tests of TestU01.
 
-An efficient implementation on GPU allows us to generate a huge number of pseudo
-random numbers  per second  (about 20Gsample/s). Our  PRNGs succeed to  pass the
-hardest batteries of test (TestU01).
 
-In future  work we plan  to extend our  work in order to  have cryptographically
-secure PRNGs because in some situations this property may be important.
 
-\bibliographystyle{plain}
+\bibliographystyle{plain} 
 \bibliography{mabase}
 \end{document}
