X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/baf2ff7c5bd7a44b68600f37eb61b76e67a8be17..HEAD:/prng_gpu.tex?ds=sidebyside

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 4880738..e90d0c2 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -1,6 +1,6 @@
-%\documentclass{article}
+\documentclass{article}
 %\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
-\documentclass[preprint,12pt]{elsarticle}
+%\documentclass[preprint,12pt]{elsarticle}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{fullpage}
@@ -18,6 +18,8 @@
 \usepackage{tabularx}
 \usepackage{multirow}
 
+%\usepackage{color}
+
 % Pour mathds : les ensembles IR, IN, etc.
 \usepackage{dsfont}
 
@@ -38,28 +40,30 @@
 \newcommand{\BN}{\mathds{B}^\mathsf{N}}
 \let\sur=\overline
 
-\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
+%\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
 \begin{document}
-\begin{frontmatter}
-\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
 
+\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
 
-\author{Jacques M. Bahi}
-\ead{jacques.bahi@univ-fcomte.fr}
-\author{ Rapha\"{e}l Couturier \corref{cor1}}
-\ead{raphael.couturier@univ-fcomte.fr}
-\cortext[cor1]{Corresponding author}
-\author{  Christophe Guyeux}
-\ead{christophe.guyeux@univ-fcomte.fr}
-\author{ Pierre-Cyrille HÃ©am }
-\ead{pierre-cyrille.heam@univ-fcomte.fr}
 
-\address{FEMTO-ST Institute, UMR  6174 CNRS,\\ University of Franche Comte, Belfort, France\\  Authors in alphabetic order}
+%% \author{Jacques M. Bahi}
+%% \ead{jacques.bahi@univ-fcomte.fr}
+%% \author{ Rapha\"{e}l Couturier \corref{cor1}}
+%% \ead{raphael.couturier@univ-fcomte.fr}
+%% \cortext[cor1]{Corresponding author}
+%% \author{  Christophe Guyeux}
+%% \ead{christophe.guyeux@univ-fcomte.fr}
+%% \author{ Pierre-Cyrille HÃ©am }
+%% \ead{pierre-cyrille.heam@univ-fcomte.fr}
 
+\author{Christophe Guyeux \and  Rapha\"{e}l Couturier \and    Pierre-Cyrille HÃ©am \and Jacques M. Bahi\\
+FEMTO-ST Institute, UMR  6174 CNRS,\\ University of Bourgogne Franche Comte, Belfort, France}
 
+\maketitle
 
 
+%\begin{frontmatter}
 %\IEEEcompsoctitleabstractindextext{
 \begin{abstract}
 In this paper we present a new pseudorandom number generator (PRNG) on
@@ -76,13 +80,11 @@ A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is fin
 
 \end{abstract}
 %}
-\begin{keyword}
-   pseudo random number\sep parallelization\sep GPU\sep cryptography\sep chaos
-
-\end{keyword}
-\end{frontmatter}
+%\begin{keyword}
+%   pseudo random number\sep parallelization\sep GPU\sep cryptography\sep chaos
+%\end{keyword}
+%\end{frontmatter}
 
-%\maketitle
 
 %\IEEEdisplaynotcompsoctitleabstractindextext
 %\IEEEpeerreviewmaketitle
@@ -104,8 +106,8 @@ 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 define \emph{secure} generators able to withstand malicious
+On the other hand, speed is not the main requirement in cryptography: the most
+important aspect 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
@@ -117,15 +119,15 @@ of the difficulty of this challenge when the size of the parameters of the PRNG
 
 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 
+The main idea is to take advantage from a chaotic dynamical system to obtain a
+generator that is unpredictable, disordered, sensible to its seed, or in other words chaotic.
+Their goal 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 
+However, the chaotic maps used as patterns 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
+are 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.
 
@@ -134,7 +136,7 @@ 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' mind, 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 
+to a statistically deflective PRNG, 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,
@@ -149,7 +151,7 @@ statistical perfection refers to the ability to pass the whole
 stringent statistical evaluation of a sequence claimed as random.
 This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}.
 More precisely, each time we performed a test on a PRNG, we ran it
-twice in order to observe if all $p-$values are inside [0.01, 0.99]. In
+twice in order to observe if all $p-$values were inside [0.01, 0.99]. In
 fact, we observed that few $p-$values (less than ten) are sometimes
 outside this interval but inside [0.001, 0.999], so that is why a
 second run allows us to confirm that the values outside are not for
@@ -169,15 +171,15 @@ 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
+Although GPUs were 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 pseudorandom
-numbers inside a GPU when a scientific application runs in it. This remark
+numbers inside a GPU when it is run by 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 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 
+cryptographical security of the inputted PRNG, when the latter has such a 
 property.
 Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric
 key encryption protocol by using the proposed method.
@@ -185,13 +187,16 @@ key encryption protocol by using the proposed method.
 
 {\bf Main contributions.} In this paper a new PRNG using chaotic iteration
 is defined. From a theoretical point of view, it is proven that it has fine
-topological chaotic properties and that it is cryptographically secured (when
-the initial PRNG is also cryptographically secured). From a practical point of
+topological chaotic properties and that it is cryptographically secure (when
+the initial PRNG is also cryptographically secure). From a practical point of
 view, experiments point out a very good statistical behavior. An optimized
 original implementation of this PRNG is also proposed and experimented.
 Pseudorandom numbers are generated at a rate of 20GSamples/s, which is faster
 than in~\cite{conf/fpga/ThomasHL09,Marsaglia2003} (and with a better
-statistical behavior). Experiments are also provided using BBS as the initial
+statistical behavior). Experiments are also provided using 
+ the well-known Blum-Blum-Shub
+(BBS) 
+as the initial
 random generator. The generation speed is significantly weaker.
 %Note also that an original qualitative comparison between topological chaotic
 %properties and statistical tests is also proposed.
@@ -216,7 +221,7 @@ our pseudorandom generators,  in particular with a  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 is cryptographically secure, then it is also the case of the
 generator provided by the post-treatment.
 A practical
 security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.
@@ -235,7 +240,7 @@ summarized and intended future work is presented.
 
 Numerous research works on defining GPU based PRNGs have already been proposed  in the
 literature, so that exhaustivity is impossible.
-This is why authors of this document only give reference to the most significant attempts 
+This is why the authors of this document only only refer 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. 
@@ -243,7 +248,7 @@ 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
+with no  requirement to a 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
@@ -259,12 +264,12 @@ 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  different computing architectures: CPU,  field-programmable gate array
-(FPGA), massively parallel  processors, and GPU. This study is of interest, because
+(FPGA), massively parallel  processors, and GPU. This study is interesting, 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 notice that authors can ``only'' generate between 11 and 16GSamples/s
+However, we notice that the 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
@@ -275,10 +280,10 @@ Curand~\cite{curand11}.        Several       PRNGs        are       implemented,
 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).
+But their PRNGs cannot pass the whole TestU01 battery (only one test has failed).
 \newline
 \newline
-We can finally remark that, to the best of our knowledge, no GPU implementation has been proven to be chaotic, and the cryptographically secure property has surprisingly never been considered.
+We can finally remark that, to the best of our knowledge, no GPU implementation has ever been proven to be chaotic, and the cryptographically secure property has surprisingly never been considered.
 
 \section{Basic Recalls}
 \label{section:BASIC RECALLS}
@@ -356,7 +361,7 @@ Let us consider  a \emph{system} with a finite  number $\mathsf{N} \in
 \mathds{N}^*$ of elements  (or \emph{cells}), so that each  cell has a
 Boolean  \emph{state}. Having $\mathsf{N}$ Boolean values for these
  cells  leads to the definition of a particular \emph{state  of the
-system}. A sequence which  elements belong to $\llbracket 1;\mathsf{N}
+system}. A sequence whose  elements belong to $\llbracket 1;\mathsf{N}
 \rrbracket $ is called a \emph{strategy}. The set of all strategies is
 denoted by $\llbracket 1, \mathsf{N} \rrbracket^\mathds{N}.$
 
@@ -685,8 +690,8 @@ inputted defective generators we have investigated.
 For instance, when considering the TestU01 battery with its 588 tests, we obtained 261 
 failures for a PRNG based on the logistic map alone, and 
 this number of failures falls below 138 in the Old CI(Logistic,Logistic) generator.
-In the XORshift case (146 failures when considering it alone), the results are more amazing,
-as the chaotic iterations post-treatment makes it fails only 8 tests. 
+In the XORshift case (146 failures when considering it alone), the results are more impressive,
+as the chaotic iterations post-treatment  fails with only 8 tests of the TestU01 battery. 
 Further investigations have been systematically realized in \cite{bfg12a:ip}
 using a large set of inputted defective PRNGs, the three most used batteries of
 tests (DieHARD, NIST, and TestU01), and for all the versions of generators we have proposed.
@@ -718,6 +723,9 @@ 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 
 component of this state (a binary digit) changes if and only if the $k-$th 
 digit in the binary decomposition of $S^n$ is 1.
+Obviously, when $S$ is periodic of period $p$, then $x$ is periodic too of
+period either $p$ or $2p$, depending on the fact that, after $p$ iterations,
+the state of the system may or not be the same as before these iterations.
 
 The single basic component presented in Eq.~\ref{equation Oplus} is of 
 ordinary use as a good elementary brick in various PRNGs. It corresponds
@@ -852,7 +860,7 @@ The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
 
 \begin{proof}
  $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
-too, thus $d$, as being the sum of two distances, will also be a distance.
+too, thus $d$, being the sum of two distances, will also be a distance.
  \begin{itemize}
 \item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then 
 $d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then 
@@ -971,7 +979,7 @@ where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
 claimed in the lemma.
 \end{proof}
 
-We can now prove the Theorem~\ref{t:chaos des general}.
+We can now prove Theorem~\ref{t:chaos des general}.
 
 \begin{proof}[Theorem~\ref{t:chaos des general}]
 Firstly, strong transitivity implies transitivity.
@@ -1420,7 +1428,7 @@ In order to  take benefits 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,  ...),  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
@@ -1434,14 +1442,14 @@ called {\it kernels}.
 \subsection{Naive Version for GPU}
 
  
-It is possible to deduce from the CPU version a quite similar version adapted to GPU.
+It is possible to deduce from the CPU version a fairly similar version adapted to GPU.
 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 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 so, the  ISAAC  PRNG~\cite{Jenkins96} is used to  set  all  the
 parameters embedded into each thread.   
 
 The implementation of  the three
@@ -1483,6 +1491,12 @@ then   the  memory   required   to  store all of the  internals   variables  of
 PRNGs\footnote{we multiply this number by $2$ in order to count 32-bits 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.
+Remark that the only requirement regarding the seed regarding the security of our PRNG is
+that it must be randomly picked. Indeed, the asymptotic security of BBS guarantees
+that, as the seed length increases, no polynomial time statistical test can 
+distinguish the pseudorandom sequences from truly random sequences with non-negligible probability,
+see, \emph{e.g.},~\cite{Sidorenko:2005:CSB:2179218.2179250}.
+
 
 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 
@@ -1498,7 +1512,7 @@ for all the different nodes involved in the computation.
 
 \subsection{Improved Version for GPU}
 
-As GPU cards using CUDA have shared memory between threads of the same block, it
+As GPU cards using CUDA have a 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
@@ -1526,20 +1540,20 @@ NumThreads: Number of threads\;
 array\_comb1, array\_comb2: Arrays containing combinations of size combination\_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 and x\;
+\If{threadIdx is concerned} {
+  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]\;
   \For{i=1 to n} {
     t=xor-like()\;
     t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\;
-    shared\_mem[threadId]=t\;
+    shared\_mem[threadIdx]=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
@@ -1559,7 +1573,7 @@ To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
 we must guarantee that this dynamical system iterates on the space 
 $\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
 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 
+To prevent  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 $\llbracket 1, \mathsf{N} \rrbracket$. 
 
@@ -1600,7 +1614,7 @@ per second  is almost constant.  With the  naive version, this value ranges from
 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
 practice,  the Tesla C1060  has more  memory than  the GTX  280, and  this memory
-should be of better quality.
+is 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.
 
@@ -1636,7 +1650,7 @@ generate a very large quantity of pseudorandom  numbers statistically perfect wi
 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.
+as  shown in the next sections.
 
 
 
@@ -1855,7 +1869,7 @@ integer.
 
 
 A direct numerical application shows that this attacker 
-cannot achieve its $(10^{12},0.2)$ distinguishing
+cannot achieve his/her $(10^{12},0.2)$ distinguishing
 attack in that context.
 
 
@@ -1878,8 +1892,8 @@ very slow and only usable for cryptographic applications.
 The modulus operation is the most time consuming operation for current
 GPU cards.  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 lesser than $2^{32}$,  and thus the number $M$ must be
-lesser than $2^{16}$.  So in practice we can choose prime numbers around
+$x_n^2$ need  to be inferior than $2^{32}$,  and thus the number $M$ must be
+inferior than $2^{16}$.  So in practice 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
 indistinguishable    bits    is    lesser    than   or    equals    to
@@ -1929,8 +1943,8 @@ array\_shift[4]=\{0,1,3,7\}\;
 }
 
 \KwOut{NewNb: array containing random numbers in global memory}
-\If{threadId is concerned} {
-  retrieve data from InternalVarBBSArray[threadId] in local variables including shared memory and x\;
+\If{threadIdx is concerned} {
+  retrieve data from InternalVarBBSArray[threadIdx] in local variables including shared memory and x\;
   we consider that bbs1 ... bbs8 represent the internal states of the 8 BBS numbers\;
   offset = threadIdx\%combination\_size\;
   o1 = threadIdx-offset+array\_comb[bbs1\&7][offset]\;
@@ -1949,12 +1963,12 @@ array\_shift[4]=\{0,1,3,7\}\;
     t$<<$=shift\;
     t|=BBS2(bbs2)\&array\_shift[shift]\;
     t=t\textasciicircum  shmem[o1]\textasciicircum     shmem[o2]\;
-    shared\_mem[threadId]=t\;
+    shared\_mem[threadIdx]=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] using a rotation\;
+  store internal variables in InternalVarXorLikeArray[threadIdx] using a rotation\;
 }
 \end{small}
 \caption{main kernel for the BBS based PRNG GPU}
@@ -2011,7 +2025,7 @@ proposed parameters, or if it is only a very fast
 and statistically perfect generator on GPU, its
 $(T,\varepsilon)-$security must be determined, and
 a formulation similar to Eq.\eqref{mesureConcrete}
-must be established. Authors
+must be established. The authors
 hope to achieve this difficult task in a future
 work.
 
@@ -2097,17 +2111,24 @@ have shown that a very large quantity of pseudorandom numbers can be generated p
 namely the BigCrush.
 Furthermore, we have shown that when the inputted generator is cryptographically
 secure, then it is the case too for the PRNG we propose, thus leading to
-the possibility to develop fast and secure PRNGs using the GPU architecture.
+the possibility of developping fast and secure PRNGs using the GPU architecture.
 An improvement of the Blum-Goldwasser cryptosystem, making it 
 behave chaotically, has finally been proposed. 
 
 In future  work we plan to extend this research, building a parallel PRNG for  clusters or
 grid computing. Topological properties of the various proposed generators will be investigated,
 and the use of other categories of PRNGs as input will be studied too. The improvement
-of Blum-Goldwasser will be deepened. Finally, we
+of Blum-Goldwasser will be deepened.
+Another aspect to consider might be different accelerator-based systems like 
+Intel Xeon Phi cards and speed measurements using such cards: as heterogeneity of
+supercomputers tends to increase using other accelerators than GPGPUs,
+a Xeon Phi solution might be interesting to investigate.
+ Finally, we
 will try to enlarge the quantity of pseudorandom numbers generated per second either
 in a simulation context or in a cryptographic one.
 
+\section*{Acknowledgment}
+This work is  partially funded by the Labex ACTION program (contract ANR-11-LABX-01-01).
 
 
 \bibliographystyle{plain} 
