petites corrections

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 678217540467750ffd9322330118fe8e6593fa47..4feac7cc64b8730163298a05ba2edca48d894eeb 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -7,7 +7,7 @@
 \usepackage{amscd}
 \usepackage{moreverb}
 \usepackage{commath}
 \usepackage{amscd}
 \usepackage{moreverb}
 \usepackage{commath}

-\usepackage{algorithm2e}
+\usepackage[ruled,vlined]{algorithm2e}

 \usepackage{listings}
 \usepackage[standard]{ntheorem}
 
 \usepackage{listings}
 \usepackage[standard]{ntheorem}
 


@@ -50,8 +50,9 @@ implementation  for  GPU that successfully passes the   {\it BigCrush} tests, de
 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.
 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 
+It is then established that, under reasonable assumptions, the proposed PRNG can be cryptographically 

 secure.
 secure.

+A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is finally proposed.

 
 
 \end{abstract}
 
 
 \end{abstract}


@@ -80,7 +81,7 @@ 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
 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.
+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 
 and unassailable due to chaos.
 However, the chaotic maps used as a pattern are defined in the real line 
 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 


@@ -131,19 +132,21 @@ 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.
 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
+Furthermore, we show that the proposed post-treatment preserves the

 cryptographical security of the inputted PRNG, when this last has such a 
 property.
 cryptographical security of the inputted PRNG, when this last has such a 
 property.

+Last, but not least, we propose a rewritten 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
   works} we  review some GPU implementations  of PRNGs.  Section~\ref{section:BASIC
   RECALLS} gives some basic recalls  on the well-known Devaney's formulation of chaos, 
   and on an iteration process called ``chaotic
 iterations'' on which the post-treatment is based. 
 
 The 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}.
+The proposed PRNG and its proof of chaos are given in  Section~\ref{sec:pseudorandom}.

 Section~\ref{sec:efficient    PRNG}   presents   an   efficient
 implementation of  this chaotic PRNG  on a CPU, whereas   Section~\ref{sec:efficient PRNG
 Section~\ref{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. 
+  gpu}   describes and evaluates theoretically  the  GPU   implementation. 

 Such generators are experimented in 
 Section~\ref{sec:experiments}.
 We show in Section~\ref{sec:security analysis} that, if the inputted
 Such generators are experimented in 
 Section~\ref{sec:experiments}.
 We show in Section~\ref{sec:security analysis} that, if the inputted


@@ -151,7 +154,8 @@ 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
 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}.
+in Section~\ref{sec:CSGPU}, and to an improvement of the
+Blum-Goldwasser protocol in Sect.~\ref{Blum-Goldwasser}.

 This research work ends by a conclusion section, in which the contribution is
 summarized and intended future work is presented.
 
 This research work ends by a conclusion section, in which the contribution is
 summarized and intended future work is presented.
 


@@ -192,7 +196,7 @@ 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. 
 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
+However, we 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
 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


@@ -414,8 +418,7 @@ The relation between $\Gamma(f)$ and $G_f$ is clear: there exists a
 path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
 strategy $s$ such that the parallel iteration of $G_f$ from the
 initial point $(s,x)$ reaches the point $x'$.
 path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
 strategy $s$ such that the parallel iteration of $G_f$ from the
 initial point $(s,x)$ reaches the point $x'$.

-
-We have finally proven in \cite{bcgr11:ip} that,
+We have then proven in \cite{bcgr11:ip} that,

 
 
 \begin{theorem}
 
 
 \begin{theorem}


@@ -424,14 +427,33 @@ Let $f:\mathds{B}^\mathsf{N}\to\mathds{B}^\mathsf{N}$. $G_f$ is chaotic  (accord
 if and only if $\Gamma(f)$ is strongly connected.
 \end{theorem}
 
 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
+Finally, we have established in \cite{bcgr11:ip} that,
+\begin{theorem}
+  Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
+  iteration graph, $\check{M}$ its adjacency
+  matrix and $M$
+  a $n\times n$ matrix defined by 
+  $
+  M_{ij} = \frac{1}{n}\check{M}_{ij}$ %\textrm{ 
+  if $i \neq j$ and  
+  $M_{ii} = 1 - \frac{1}{n} \sum\limits_{j=1, j\neq i}^n \check{M}_{ij}$ otherwise.
+  
+  If $\Gamma(f)$ is strongly connected, then 
+  the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows 
+  a law that tends to the uniform distribution 
+  if and only if $M$ is a double stochastic matrix.
+\end{theorem} 
+
+
+These results of chaos and uniform distribution have lead us to study the possibility to build a

 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 $G_f$
 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
+during implementations (due to the discrete nature of $f$). Indeed, 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, in PRNG, or a physical noise in TRNG).
 $\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ;  \mathsf{N}
 \rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG).

+Let us finally remark that the vectorial negation satisfies the hypotheses of the two theorems above.

 
 \section{Application to Pseudorandomness}
 \label{sec:pseudorandom}
 
 \section{Application to Pseudorandomness}
 \label{sec:pseudorandom}


@@ -494,19 +516,7 @@ with a bit shifted version of it. This PRNG, which has a period of
 $2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used
 in our PRNG to compute the strategy length and the strategy elements.
 
 $2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used
 in our PRNG to compute the strategy length and the strategy elements.
 

-
-We have proven in \cite{bcgr11:ip} that,
-\begin{theorem}
-  Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
-  iteration graph, $\check{M}$ its adjacency
-  matrix and $M$ a $n\times n$ matrix defined as in the previous lemma.
-  If $\Gamma(f)$ is strongly connected, then 
-  the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows 
-  a law that tends to the uniform distribution 
-  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~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07}.
+This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones.

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


@@ -846,7 +856,9 @@ $$
 
 
 
 
 
 

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

 \begin{lstlisting}
 unsigned int CIPRNG() {
   static unsigned int x = 123123123;
 \begin{lstlisting}
 unsigned int CIPRNG() {
   static unsigned int x = 123123123;


@@ -866,19 +878,18 @@ 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
+\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
+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}.

 
 

-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  \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 32 least significant bits of a given integer, and the code
-\texttt{(unsigned int)(t3$>>$32)}  in order to obtain the 32 most significant  bits of \texttt{t}.   
-
-So 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
+So 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}.
 
 \section{Efficient PRNGs based on Chaotic Iterations on GPU}
 stringent BigCrush battery of tests~\cite{LEcuyerS07}.
 
 \section{Efficient PRNGs based on Chaotic Iterations on GPU}


@@ -890,12 +901,12 @@ 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.
 Obviously, having these requirements in  mind, it is possible to build
 more local  memory is  used, and the  less branching  instructions are
 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   Algorithm
+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}
 environment,    threads    have     a    local    identifier    called
 \texttt{ThreadIdx},  which   is  relative  to   the  block  containing
 \ref{algo:seqCIPRNG}, which computes  pseudorandom numbers on GPU.  To
 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. With  CUDA parts of  the code which  are executed by the  GPU are
+them. Furthermore, in  CUDA, parts of  the code that are executed by the  GPU are

 called {\it kernels}.
 
 
 called {\it kernels}.
 
 


@@ -972,12 +983,14 @@ 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. 
 
 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
+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}
 \texttt{combination\_size}.   Then we  can compute  \texttt{o1}  and \texttt{o2}

-representing the indexes of the  other threads whose results are used
-by the  current one. In  this algorithm, we  consider that a  64-bits xor-like
-PRNG has been chosen, and so its two 32-bits parts are used.
+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
+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).

 
 This version also can pass the whole {\it BigCrush} battery of tests.
 
 
 This version also can pass the whole {\it BigCrush} battery of tests.
 


@@ -986,28 +999,28 @@ This version also can pass the whole {\it BigCrush} battery of tests.
 \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\;

-tab1, tab2: Arrays containing combinations of size combination\_size\;}
+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\;
   offset = threadIdx\%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\;
   offset = threadIdx\%combination\_size\;

-  o1 = threadIdx-offset+tab1[offset]\;
-  o2 = threadIdx-offset+tab2[offset]\;
+  o1 = threadIdx-offset+array\_comb1[offset]\;
+  o2 = threadIdx-offset+array\_comb2[offset]\;

   \For{i=1 to n} {
     t=xor-like()\;
   \For{i=1 to n} {
     t=xor-like()\;

-    t=t $\hat{ }$ shmem[o1] $\hat{ }$ shmem[o2]\;
+    t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\;

     shared\_mem[threadId]=t\;
     shared\_mem[threadId]=t\;

-    x = x $\hat{ }$ t\;
+    x = x\textasciicircum t\;

 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
   store internal variables in InternalVarXorLikeArray[threadId]\;
 }
 
 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
   store internal variables in InternalVarXorLikeArray[threadId]\;
 }
 

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

 \end{algorithm}
 
 \subsection{Theoretical Evaluation of the Improved Version}
 \end{algorithm}
 
 \subsection{Theoretical Evaluation of the Improved Version}


@@ -1032,8 +1045,7 @@ 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, 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,

-(this can be stated by an immediate mathematical
-induction), and thus the next $x$ is finally uniformly distributed.
+(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 
 
 Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
 chaotic iterations presented previously, and for this reason, it satisfies the 


@@ -1051,7 +1063,7 @@ All the
 cards have 240 cores.
 
 In  Figure~\ref{fig:time_xorlike_gpu} we  compare the  quantity of  pseudorandom numbers
 cards have 240 cores.
 
 In  Figure~\ref{fig:time_xorlike_gpu} we  compare the  quantity of  pseudorandom numbers

-generated per second with various xor-like based PRNG. In this figure, the optimized
+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
 embed  the three  xor-like  PRNGs described  in Listing~\ref{algo:seqCIPRNG}.   In
 order to obtain the optimal performances, the storage of pseudorandom numbers
 versions use the {\it 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


@@ -1230,84 +1242,87 @@ algorithm (Algorithm~\ref{algo:gpu_kernel2}).   Due to Proposition~\ref{cryptopr
 it simply consists  in replacing
 the  {\it  xor-like} PRNG  by  a  cryptographically  secure one.  
 We have chosen the Blum Blum Shum generator~\cite{BBS} (usually denoted by BBS) having the form:
 it simply consists  in replacing
 the  {\it  xor-like} PRNG  by  a  cryptographically  secure one.  
 We have chosen the Blum Blum Shum generator~\cite{BBS} (usually denoted by BBS) having the form:

-$$x_{n+1}=x_n^2~ mod~ M$$  where $M$ is the product of  two prime numbers. These
-prime numbers  need to be congruent  to 3 modulus  4. BBS is
+$$x_{n+1}=x_n^2~ mod~ M$$  where $M$ is the product of  two prime numbers (these
+prime numbers  need to be congruent  to 3 modulus  4). BBS is known to be

 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
 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 less than $2^{32}$  and the number $M$  need to be
-less 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
+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
+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
 4 least significant bits of $x_n$ can be chosen (the maximum number of
 indistinguishable    bits    is    lesser    than   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  $M$. This
-approach is  not sufficient to pass  all the tests  of TestU01 because
-the fact  of having chosen  small values of  $M$ for the BBS  leads to
-have a  small period. So, in  order to add randomness  we proceed with
+$log_2(log_2(M))$). 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,
+as small values of  $M$ for the BBS  lead to
+  small periods. So, in  order to add randomness  we proceed with

 the followings  modifications. 
 \begin{itemize}
 \item
 the followings  modifications. 
 \begin{itemize}
 \item

-First we  define 16 arrangement arrays  instead of 2  (as described in
-algorithm \ref{algo:gpu_kernel2}) but only 2  are used at each call of
-the  PRNG kernels. In  practice, the  selection of  which combinations
-arrays will be used is different for all the threads and is determined
+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
+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.
 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  \& performs the  AND bitwise.  So using  \&7 with  a number
-gives the last 3 bits, so it provides a number between 0 and 7.
+%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.

 \item
 \item

-Second, after the  generation of the 8 BBS numbers  for each thread we
-have a 32 bits number for which the period is possibly quite small. So
-to add randomness,  we generate 4 more BBS numbers  which allows us to
-shift  the 32 bits  numbers and  add upto  6 new  bits.  This  part is
-described  in algorithm~\ref{algo:bbs_gpu}.  In  practice, if  we call
-{\it strategy}, the number representing  the strategy, the last 2 bits
-of the first new BBS number are  used to make a left shift of at least
+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
+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

 3 bits. The  last 3 bits of the  second new BBS number are  add 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.
 \item
 3 bits. The  last 3 bits of the  second new BBS number are  add 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.
 \item

-Finally, as  we use 8 BBS numbers  for each thread, the  store of these
+Finally, as  we use 8 BBS numbers  for each thread, the  storage of these

 numbers at the end of the  kernel is performed using a rotation. So,
 internal  variable for  BBS number  1 is  stored in  place  2, internal
 numbers at the end of the  kernel is performed using a rotation. So,
 internal  variable for  BBS number  1 is  stored in  place  2, internal

-variable  for BBS  number 2  is  store ind  place 3,  ... and  internal
+variable  for BBS  number 2  is  stored in  place 3,  ..., and finally, internal

 variable for BBS number 8 is stored in place 1.
 \end{itemize}
 
 variable for BBS number 8 is stored in place 1.
 \end{itemize}
 

-

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

-tab: 2D Arrays containing 16 combinations (in first dimension)  of size combination\_size (in second dimension)\;}
+array\_comb: 2D Arrays containing 16 combinations (in first dimension)  of size combination\_size (in second dimension)\;
+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\;
   we consider that bbs1 ... bbs8 represent the internal states of the 8 BBS numbers\;
   offset = threadIdx\%combination\_size\;
 
 \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\;
   we consider that bbs1 ... bbs8 represent the internal states of the 8 BBS numbers\;
   offset = threadIdx\%combination\_size\;

-  o1 = threadIdx-offset+tab[bbs1\&7][offset]\;
-  o2 = threadIdx-offset+tab[8+bbs2\&7][offset]\;
+  o1 = threadIdx-offset+array\_comb[bbs1\&7][offset]\;
+  o2 = threadIdx-offset+array\_comb[8+bbs2\&7][offset]\;

   \For{i=1 to n} {
   \For{i=1 to n} {

-    t<<=4\;
+    t$<<$=4\;

     t|=BBS1(bbs1)\&15\;
     ...\;
     t|=BBS1(bbs1)\&15\;
     ...\;

-    t<<=4\;
+    t$<<$=4\;

     t|=BBS8(bbs8)\&15\;
     t|=BBS8(bbs8)\&15\;

-    //two new shifts\;
-    t<<=BBS3(bbs3)\&3\;
-    t|=BBS1(bbs1)\&7\;
-     t<<=BBS7(bbs7)\&3\;
-    t|=BBS2(bbs2)\&7\;
-    t=t $\hat{ }$ shmem[o1] $\hat{ }$ shmem[o2]\;
+    \tcp{two new shifts}
+    shift=BBS3(bbs3)\&3\;
+    t$<<$=shift\;
+    t|=BBS1(bbs1)\&array\_shift[shift]\;
+    shift=BBS7(bbs7)\&3\;
+    t$<<$=shift\;
+    t|=BBS2(bbs2)\&array\_shift[shift]\;
+    t=t\textasciicircum  shmem[o1]\textasciicircum     shmem[o2]\;

     shared\_mem[threadId]=t\;
     shared\_mem[threadId]=t\;

-    x = x $\hat{ }$ t\;
+    x = x\textasciicircum   t\;

 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }


@@ -1318,16 +1333,31 @@ tab: 2D Arrays containing 16 combinations (in first dimension)  of size combinat
 \label{algo:bbs_gpu}
 \end{algorithm}
 
 \label{algo:bbs_gpu}
 \end{algorithm}
 

-In algorithm~\ref{algo:bbs_gpu}, t<<=4 performs a left shift of 4 bits
-on the variable  t and stores the result  in t. BBS1(bbs1)\&15 selects
-the last  four bits of the result  of BBS1. It should  be noticed that
-for the two new shifts, we use arbitrarily 4 BBSs that have previously
-been used.
+In Algorithm~\ref{algo:bbs_gpu}, $n$ is for  the quantity of random numbers that
+a thread has to  generate.  The operation t<<=4 performs a left  shift of 4 bits
+on the variable  $t$ and stores the result in  $t$, and $BBS1(bbs1)\&15$ selects
+the last  four bits  of the  result of $BBS1$.   Thus an  operation of  the form
+$t<<=4; t|=BBS1(bbs1)\&15\;$  realizes in $t$ a  left shift of 4  bits, and then
+puts the 4 last bits of $BBS1(bbs1)$  in the four last positions of $t$.  Let us
+remark that to initialize $t$ is not a necessity as we fill it 4 bits by 4 bits,
+until having obtained 32-bits.  The two last new shifts are realized in order to
+enlarge  the  small periods  of  the  BBS used  here,  to  introduce  a kind  of
+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 $t$.
+
+It should  be noticed that this generator has another time the form $x^{n+1} = x^n \oplus S^n$,
+where $S^n$ is referred in this algorithm as $t$: each iteration of this
+PRNG ends with $x = x \wedge t$. This $S^n$ is only constituted
+by secure bits produced by the BBS generator, and thus, due to
+Proposition~\ref{cryptopreuve}, the resulted PRNG is cryptographically
+secure

 
 
 
 \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
 
 
 
 \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}

-
+\label{Blum-Goldwasser}

 We finish this research work by giving some thoughts about the use of
 the proposed PRNG in an asymmetric cryptosystem.
 This first approach will be further investigated in a future work.
 We finish this research work by giving some thoughts about the use of
 the proposed PRNG in an asymmetric cryptosystem.
 This first approach will be further investigated in a future work.


@@ -1355,7 +1385,7 @@ Suppose Bob wishes to send a string $m=(m_0, \dots, m_{L-1})$ of $L$ bits to Ali
 \item $i=0$.
 \item While $i \leqslant L-1$:
 \begin{itemize}
 \item $i=0$.
 \item While $i \leqslant L-1$:
 \begin{itemize}

-\item Set $b_i$ equal to the least-significant\footnote{BBS can securely output up to $\mathsf{N} = \lfloor log(log(N)) \rfloor$ of the least-significant bits of $x_i$ during each round.} bit of $x_i$,
+\item Set $b_i$ equal to the least-significant\footnote{As signaled previously, BBS can securely output up to $\mathsf{N} = \lfloor log(log(N)) \rfloor$ of the least-significant bits of $x_i$ during each round.} bit of $x_i$,

 \item $i=i+1$,
 \item $x_i = (x_{i-1})^2~mod~N.$
 \end{itemize}
 \item $i=i+1$,
 \item $x_i = (x_{i-1})^2~mod~N.$
 \end{itemize}


@@ -1396,16 +1426,25 @@ the inheritance of all the properties presented in this paper.
 \section{Conclusion}
 
 
 \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 Devaney.
-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
-20Gsamples/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.
+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
+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
+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.
+Thoughts about an improvement of the Blum-Goldwasser cryptosystem, using the 
+proposed method, has been finally proposed.
+
+In future  work we plan to extend these researches, 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
+will try to enlarge the quantity of pseudorandom numbers generated per second either
+in a simulation context or in a cryptographic one.