X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/26aa4e0cf2939580eae07e8bbad140d527f47234..ec3fcbcdbf09ed6be33553cbb41c938cf74d3a74:/prng_gpu.tex?ds=inline

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 0279f03..f21f19e 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -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.
-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.
+A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is finally proposed.
 
 
 \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
-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 
@@ -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.
-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.
+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. 
-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
-  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
@@ -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
-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.
 
@@ -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. 
-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
@@ -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 $x'$.
-
-We have finally proven in \cite{bcgr11:ip} that,
+We have then proven in \cite{bcgr11:ip} that,
 
 
 \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}
 
-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 $G_f$
-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).
+Let us finally remark that the vectorial negation satisfies the hypotheses of the two theorems above.
 
 \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.
 
-
-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}
 
@@ -884,16 +894,19 @@ stringent BigCrush battery of tests~\cite{LEcuyerS07}.
 \section{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 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.  
-Obviously, having these requirements in mind, it is possible to  build a program similar to 
-the one presented in Algorithm  \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.
+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.
+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}
+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
+called {\it kernels}.
 
 
 \subsection{Naive Version for GPU}
@@ -965,13 +978,13 @@ 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
 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 permutation array that
-contains  the indexes  of  all threads  and  for which  a  permutation has  been
+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 permutations arrays are used.
+In Algorithm~\ref{algo:gpu_kernel2}, two combination arrays are used.
 The    variable   \texttt{offset}    is    computed   using    the   value    of
-\texttt{permutation\_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.
@@ -983,19 +996,19 @@ 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\;
-tab1, tab2: Arrays containing permutations of size permutation\_size\;}
+tab1, tab2: 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\%permutation\_size\;
+  offset = threadIdx\%combination\_size\;
   o1 = threadIdx-offset+tab1[offset]\;
   o2 = threadIdx-offset+tab2[offset]\;
   \For{i=1 to n} {
     t=xor-like()\;
-    t=t$\oplus$shmem[o1]$\oplus$shmem[o2]\;
+    t=t $\wedge$ shmem[o1] $\wedge$ shmem[o2]\;
     shared\_mem[threadId]=t\;
-    x = x $\oplus$ t\;
+    x = x $\wedge$ t\;
 
     store the new PRNG in NewNb[NumThreads*threadId+i]\;
   }
@@ -1029,8 +1042,7 @@ 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,
-(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 
@@ -1048,7 +1060,7 @@ All the
 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
@@ -1079,7 +1091,7 @@ As a  comparison,   Listing~\ref{algo:seqCIPRNG}  leads   to the  generation of
 
 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 1.8GSample/s and on the GTX 280 about 1.6GSample/s, which is
+obtain approximately 700MSample/s and on the GTX 280 about 670MSample/s, which is
 obviously slower than the xorlike-based PRNG on GPU. However, we will show in the 
 next sections that 
 this new PRNG has a strong level of security, which is necessary paid by a speed
@@ -1115,15 +1127,15 @@ In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.
 In a cryptographic context, a pseudorandom generator is a deterministic
 algorithm $G$ transforming strings  into strings and such that, for any
-seed $w$ of length $N$, $G(w)$ (the output of $G$ on the input $w$) has size
-$\ell_G(N)$ with $\ell_G(N)>N$.
+seed $k$ of length $k$, $G(k)$ (the output of $G$ on the input $k$) has size
+$\ell_G(k)$ with $\ell_G(k)>k$.
 The notion of {\it secure} PRNGs can now be defined as follows. 
 
 \begin{definition}
 A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time
 algorithm $D$, for any positive polynomial $p$, and for all sufficiently
 large $k$'s,
-$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(N)},$$
+$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(k)},$$
 where $U_r$ is the uniform distribution over $\{0,1\}^r$ and the
 probabilities are taken over $U_N$, $U_{\ell_G(N)}$ as well as over the
 internal coin tosses of $D$. 
@@ -1227,44 +1239,210 @@ 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:
-$$x_{n+1}=x_n^2~ mod~ M$$  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 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  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 $M$.
+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
+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
+$log_2(log_2(M))$). 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
+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.
+%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
+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
+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
+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}
+
+\begin{algorithm}
 
-Currently this PRNG does not succeed to pass all the tests of TestU01.
+\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)\;}
+
+\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]\;
+  \For{i=1 to n} {
+    t<<=4\;
+    t|=BBS1(bbs1)\&15\;
+    ...\;
+    t<<=4\;
+    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 \^{ } shmem[o1] \^{ }    shmem[o2]\;
+    shared\_mem[threadId]=t\;
+    x = x \^{ }    t\;
+
+    store the new PRNG in NewNb[NumThreads*threadId+i]\;
+  }
+  store internal variables in InternalVarXorLikeArray[threadId] using a rotation\;
+}
+
+\caption{main kernel for the BBS based PRNG GPU}
+\label{algo:bbs_gpu}
+\end{algorithm}
 
+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 and we put \emph{exactly} the 3 last bits from a BBS into 
+the 3 last bits of $t$, leading possibly to a loss of a few 
+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}
+\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.
+
+\subsubsection{Recalls of the Blum-Goldwasser Probabilistic Cryptosystem}
+
+The Blum-Goldwasser cryptosystem is a cryptographically secure asymmetric key encryption algorithm 
+proposed in 1984~\cite{Blum:1985:EPP:19478.19501}.  The encryption algorithm 
+implements a XOR-based stream cipher using the BBS PRNG, in order to generate 
+the keystream. Decryption is done by obtaining the initial seed thanks to
+the final state of the BBS generator and the secret key, thus leading to the
+ reconstruction of the keystream.
+
+The key generation consists in generating two prime numbers $(p,q)$, 
+randomly and independently of each other, that are
+ congruent to 3 mod 4, and to compute the modulus $N=pq$.
+The public key is $N$, whereas the secret key is the factorization $(p,q)$.
+
+
+Suppose Bob wishes to send a string $m=(m_0, \dots, m_{L-1})$ of $L$ bits to Alice:
+\begin{enumerate}
+\item Bob picks an integer $r$ randomly in the interval $\llbracket 1,N\rrbracket$ and computes $x_0 = r^2~mod~N$.
+\item He uses the BBS to generate the keystream of $L$ pseudorandom bits $(b_0, \dots, b_{L-1})$, as follows. For $i=0$ to $L-1$,
+\begin{itemize}
+\item $i=0$.
+\item While $i \leqslant L-1$:
+\begin{itemize}
+\item Set $b_i$ equal to the least-significant\footnote{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}
+\end{itemize}
+\item The ciphertext is computed by XORing the plaintext bits $m$ with the keystream: $ c = (c_0, \dots, c_{L-1}) = m \oplus  b$. This ciphertext is $[c, y]$, where $y=x_{0}^{2^{L}}~mod~N.$
+\end{enumerate}
 
-\subsection{A Secure Asymetric Cryptosystem}
 
+When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$ as follows:
+\begin{enumerate}
+\item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$.
+\item The initial seed can be obtained using the following procedure: $x_0=q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q~{mod}~N$.
+\item She recomputes the bit-vector $b$ by using BBS and $x_0$.
+\item Alice computes finally the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus  b$.
+\end{enumerate}
 
 
+\subsubsection{Proposal of a new Asymmetric Cryptosystem Adapted from Blum-Goldwasser}
 
-\section{Conclusion}
+We propose to adapt the Blum-Goldwasser protocol as follows. 
+Let $\mathsf{N} = \lfloor log(log(N)) \rfloor$ be the number of bits that can
+be obtained securely with the BBS generator using the public key $N$ of Alice.
+Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too, and
+her new public key will be $(S^0, N)$.
+
+To encrypt his message, Bob will compute
+\begin{equation}
+c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)
+\end{equation}
+instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$. 
 
+The same decryption stage as in Blum-Goldwasser leads to the sequence 
+$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$.
+Thus, with a simple use of $S^0$, Alice can obtained the plaintext.
+By doing so, the proposed generator is used in place of BBS, leading to
+the inheritance of all the properties presented in this paper.
 
-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.
+\section{Conclusion}
 
-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. We also plan to improve  the BBS version in order to succeed all
-the tests of TestU01.
+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.
 
 
 
