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
+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}.
\section{Efficient PRNG based on Chaotic Iterations}
-\label{sec:efficient prng}
+\label{sec:efficient PRNG}
Based on the proof presented in the previous section, it is now possible to
improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}.
-\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 iterations},label=algo:seqCIPRNG}
\begin{lstlisting}
-unsigned int CIprng() {
+unsigned int CIPRNG() {
static unsigned int x = 123123123;
unsigned long t1 = xorshift();
unsigned long t2 = xor128();
-In Listing~\ref{algo:seqCIprng} a sequential version of the proposed PRNG based on chaotic iterations
+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
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.
+\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. With CUDA parts of the code which are executed by the GPU are
+called {\it kernels}.
\subsection{Naive Version for GPU}
\If{threadIdx is concerned by the computation} {
retrieve data from InternalVarXorLikeArray[threadIdx] in local variables\;
\For{i=1 to n} {
- compute a new PRNG as in Listing\ref{algo:seqCIprng}\;
+ compute a new PRNG as in Listing\ref{algo:seqCIPRNG}\;
store the new PRNG in NewNb[NumThreads*threadIdx+i]\;
}
store internal variables in InternalVarXorLikeArray[threadIdx]\;
As GPU cards using CUDA have shared memory between threads of the same block, it
is possible to use this feature in order to simplify the previous algorithm,
-i.e., using less than 3 xor-like PRNGs. The solution consists in computing only
-one xor-like PRNG by thread, saving it into shared memory and using the results
+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 which
-contains the indexes of all threads and for which a permutation has been
-performed. In Algorithm~\ref{algo:gpu_kernel2}, 2 permutations arrays are used.
+thread uses the result of which other one, we can use a combination array that
+contains the indexes of all threads and for which a combination has been
+performed.
+
+In Algorithm~\ref{algo:gpu_kernel2}, two combination arrays are used.
The variable \texttt{offset} is computed using the value of
-\texttt{permutation\_size}. Then we can compute \texttt{o1} and \texttt{o2}
-which represent the indexes of the other threads for which the results are used
-by the current thread. In the algorithm, we consider that a 64-bits xor-like
-PRNG is used, that is why both 32-bits parts are used.
+\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.
-This version also succeeds to the {\it BigCrush} batteries of tests.
+This version also can pass the whole {\it BigCrush} battery of tests.
\begin{algorithm}
\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]\;
}
\subsection{Theoretical Evaluation of the Improved Version}
-A run of Algorithm~\ref{algo:gpu_kernel2} consists in three operations having
+A run of Algorithm~\ref{algo:gpu_kernel2} consists in an operation ($x=x\oplus t$) having
the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
-system of Eq.~\ref{eq:generalIC}. That is, three iterations of the general chaotic
-iterations are realized between two stored values of the PRNG.
+system of Eq.~\ref{eq:generalIC}. That is, an iteration of the general chaotic
+iterations is realized between the last stored value $x$ of the thread and a strategy $t$
+(obtained by a bitwise exclusive or between a value provided by a xor-like() call
+and two values previously obtained by two other threads).
To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
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 into $\mathds{B}^ \mathsf{N}$.
-To prevent from any flaws of chaotic properties, we must check that each right
-term, corresponding to terms of the strategies, can possibly be equal to any
+To prevent from any flaws of chaotic properties, we must check that the right
+term (the last $t$), corresponding to the strategies, can possibly be equal to any
integer of $\llbracket 1, \mathsf{N} \rrbracket$.
-Such a result is obvious for the two first lines, as for the xor-like(), all the
-integers belonging into its interval of definition can occur at each iteration.
-It can be easily stated for the two last lines by an immediate mathematical
-induction.
+Such a result is obvious, as for the xor-like(), all the
+integers belonging into its interval of definition can occur at each iteration, and thus the
+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.
Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
chaotic iterations presented previously, and for this reason, it satisfies the
\label{sec:experiments}
Different experiments have been performed in order to measure the generation
-speed. We have used a computer equiped with Tesla C1060 NVidia GPU card and an
-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
+speed. We have used a first computer equipped with a Tesla C1060 NVidia GPU card
+and an
+Intel Xeon E5530 cadenced at 2.40 GHz, and
+a second computer equipped with a smaller CPU and a GeForce GTX 280.
+All the
cards have 240 cores.
-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
-3GSample/s. With the optimized version, it is approximately equals to
-20GSample/s. Finally we can remark that both GPU cards are quite similar. In
-practice, the Tesla C1060 has more memory than the GTX 280 and this memory
+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
+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
+into the GPU memory has been removed. This step is time consuming and slows down the numbers
+generation. Moreover this storage is completely
+useless, in case of applications that consume the pseudorandom
+numbers directly after generation. We can see that when the number of threads is greater
+than approximately 30,000 and lower than 5 millions, the number of pseudorandom numbers generated
+per second is almost constant. With the naive version, this value ranges from 2.5 to
+3GSamples/s. With the optimized version, it is approximately equal to
+20GSamples/s. Finally we can remark that both GPU cards are quite similar, but in
+practice, the Tesla C1060 has more memory than the GTX 280, and this memory
should be of better quality.
+As a comparison, Listing~\ref{algo:seqCIPRNG} leads to the generation of about
+138MSample/s when using one core of the Xeon E5530.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.7]{curve_time_xorlike_gpu.pdf}
\end{center}
-\caption{Number of random numbers generated per second with the xorlike based PRNG}
+\caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG}
\label{fig:time_xorlike_gpu}
\end{figure}
-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.
+
+In Figure~\ref{fig:time_bbs_gpu} we highlight the performances of the optimized
+BBS-based PRNG on GPU. On the Tesla C1060 we
+obtain approximately 700MSample/s and on the GTX 280 about 670MSample/s, which is
+obviously slower than the xorlike-based PRNG on GPU. However, we will show in the
+next sections that
+this new PRNG has a strong level of security, which is necessary paid by a speed
+reduction.
\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}
+\caption{Quantity of pseudorandom numbers generated per second using 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}
-
-
-%% Mettre ici la preuve de PCH
-
-%\section{The relativity of disorder}
-%\label{sec:de la relativité du désordre}
-
-%In the next two sections, we investigate the impact of the choices that have
-%lead to the definitions of measures in Sections \ref{sec:chaotic iterations} and \ref{deuxième def}.
-
-%\subsection{Impact of the topology's finenesse}
-
-%Let us firstly introduce the following notations.
-
-%\begin{notation}
-%$\mathcal{X}_\tau$ will denote the topological space
-%$\left(\mathcal{X},\tau\right)$, whereas $\mathcal{V}_\tau (x)$ will be the set
-%of all the neighborhoods of $x$ when considering the topology $\tau$ (or simply
-%$\mathcal{V} (x)$, if there is no ambiguity).
-%\end{notation}
-
-
-
-%\begin{theorem}
-%\label{Th:chaos et finesse}
-%Let $\mathcal{X}$ a set and $\tau, \tau'$ two topologies on $\mathcal{X}$ s.t.
-%$\tau'$ is finer than $\tau$. Let $f:\mathcal{X} \to \mathcal{X}$, continuous
-%both for $\tau$ and $\tau'$.
-
-%If $(\mathcal{X}_{\tau'},f)$ is chaotic according to Devaney, then
-%$(\mathcal{X}_\tau,f)$ is chaotic too.
-%\end{theorem}
-
-%\begin{proof}
-%Let us firstly establish the transitivity of $(\mathcal{X}_\tau,f)$.
-
-%Let $\omega_1, \omega_2$ two open sets of $\tau$. Then $\omega_1, \omega_2 \in
-%\tau'$, becaus $\tau'$ is finer than $\tau$. As $f$ is $\tau'-$transitive, we
-%can deduce that $\exists n \in \mathds{N}, \omega_1 \cap f^{(n)}(\omega_2) =
-%\varnothing$. Consequently, $f$ is $\tau-$transitive.
-
-%Let us now consider the regularity of $(\mathcal{X}_\tau,f)$, \emph{i.e.}, for
-%all $x \in \mathcal{X}$, and for all $\tau-$neighborhood $V$ of $x$, there is a
-%periodic point for $f$ into $V$.
-
-%Let $x \in \mathcal{X}$ and $V \in \mathcal{V}_\tau (x)$ a $\tau-$neighborhood
-%of $x$. By definition, $\exists \omega \in \tau, x \in \omega \subset V$.
-
-%But $\tau \subset \tau'$, so $\omega \in \tau'$, and then $V \in
-%\mathcal{V}_{\tau'} (x)$. As $(\mathcal{X}_{\tau'},f)$ is regular, there is a
-%periodic point for $f$ into $V$, and the regularity of $(\mathcal{X}_\tau,f)$ is
-%proven.
-%\end{proof}
-
-%\subsection{A given system can always be claimed as chaotic}
-
-%Let $f$ an iteration function on $\mathcal{X}$ having at least a fixed point.
-%Then this function is chaotic (in a certain way):
-
-%\begin{theorem}
-%Let $\mathcal{X}$ a nonempty set and $f: \mathcal{X} \to \X$ a function having
-%at least a fixed point.
-%Then $f$ is $\tau_0-$chaotic, where $\tau_0$ is the trivial (indiscrete)
-%topology on $\X$.
-%\end{theorem}
-
-
-%\begin{proof}
-%$f$ is transitive when $\forall \omega, \omega' \in \tau_0 \setminus
-%\{\varnothing\}, \exists n \in \mathds{N}, f^{(n)}(\omega) \cap \omega' \neq
-%\varnothing$.
-%As $\tau_0 = \left\{ \varnothing, \X \right\}$, this is equivalent to look for
-%an integer $n$ s.t. $f^{(n)}\left( \X \right) \cap \X \neq \varnothing$. For
-%instance, $n=0$ is appropriate.
-
-%Let us now consider $x \in \X$ and $V \in \mathcal{V}_{\tau_0} (x)$. Then $V =
-%\mathcal{X}$, so $V$ has at least a fixed point for $f$. Consequently $f$ is
-%regular, and the result is established.
-%\end{proof}
-
-
-
-
-%\subsection{A given system can always be claimed as non-chaotic}
-
-%\begin{theorem}
-%Let $\mathcal{X}$ be a set and $f: \mathcal{X} \to \X$.
-%If $\X$ is infinite, then $\left( \X_{\tau_\infty}, f\right)$ is not chaotic
-%(for the Devaney's formulation), where $\tau_\infty$ is the discrete topology.
-%\end{theorem}
-
-%\begin{proof}
-%Let us prove it by contradiction, assuming that $\left(\X_{\tau_\infty},
-%f\right)$ is both transitive and regular.
-
-%Let $x \in \X$ and $\{x\}$ one of its neighborhood. This neighborhood must
-%contain a periodic point for $f$, if we want that $\left(\X_{\tau_\infty},
-%f\right)$ is regular. Then $x$ must be a periodic point of $f$.
-
-%Let $I_x = \left\{ f^{(n)}(x), n \in \mathds{N}\right\}$. This set is finite
-%because $x$ is periodic, and $\mathcal{X}$ is infinite, then $\exists y \in
-%\mathcal{X}, y \notin I_x$.
-
-%As $\left(\X_{\tau_\infty}, f\right)$ must be transitive, for all open nonempty
-%sets $A$ and $B$, an integer $n$ must satisfy $f^{(n)}(A) \cap B \neq
-%\varnothing$. However $\{x\}$ and $\{y\}$ are open sets and $y \notin I_x
-%\Rightarrow \forall n, f^{(n)}\left( \{x\} \right) \cap \{y\} = \varnothing$.
-%\end{proof}
-
-
-
-
-
-
-%\section{Chaos on the order topology}
-%\label{sec: chaos order topology}
-%\subsection{The phase space is an interval of the real line}
-
-%\subsubsection{Toward a topological semiconjugacy}
-
-%In what follows, our intention is to establish, by using a topological
-%semiconjugacy, that chaotic iterations over $\mathcal{X}$ can be described as
-%iterations on a real interval. To do so, we must firstly introduce some
-%notations and terminologies.
-
-%Let $\mathcal{S}_\mathsf{N}$ be the set of sequences belonging into $\llbracket
-%1; \mathsf{N}\rrbracket$ and $\mathcal{X}_{\mathsf{N}} = \mathcal{S}_\mathsf{N}
-%\times \B^\mathsf{N}$.
-
-
-%\begin{definition}
-%The function $\varphi: \mathcal{S}_{10} \times\mathds{B}^{10} \rightarrow \big[
-%0, 2^{10} \big[$ is defined by:
-%\begin{equation}
-% \begin{array}{cccl}
-%\varphi: & \mathcal{X}_{10} = \mathcal{S}_{10} \times\mathds{B}^{10}&
-%\longrightarrow & \big[ 0, 2^{10} \big[ \\
-% & (S,E) = \left((S^0, S^1, \hdots ); (E_0, \hdots, E_9)\right) & \longmapsto &
-%\varphi \left((S,E)\right)
-%\end{array}
-%\end{equation}
-%where $\varphi\left((S,E)\right)$ is the real number:
-%\begin{itemize}
-%\item whose integral part $e$ is $\displaystyle{\sum_{k=0}^9 2^{9-k} E_k}$, that
-%is, the binary digits of $e$ are $E_0 ~ E_1 ~ \hdots ~ E_9$.
-%\item whose decimal part $s$ is equal to $s = 0,S^0~ S^1~ S^2~ \hdots =
-%\sum_{k=1}^{+\infty} 10^{-k} S^{k-1}.$
-%\end{itemize}
-%\end{definition}
-
-
-
-%$\varphi$ realizes the association between a point of $\mathcal{X}_{10}$ and a
-%real number into $\big[ 0, 2^{10} \big[$. We must now translate the chaotic
-%iterations $\Go$ on this real interval. To do so, two intermediate functions
-%over $\big[ 0, 2^{10} \big[$ must be introduced:
-
-
-%\begin{definition}
-%\label{def:e et s}
-%Let $x \in \big[ 0, 2^{10} \big[$ and:
-%\begin{itemize}
-%\item $e_0, \hdots, e_9$ the binary digits of the integral part of $x$:
-%$\displaystyle{\lfloor x \rfloor = \sum_{k=0}^{9} 2^{9-k} e_k}$.
-%\item $(s^k)_{k\in \mathds{N}}$ the digits of $x$, where the chosen decimal
-%decomposition of $x$ is the one that does not have an infinite number of 9:
-%$\displaystyle{x = \lfloor x \rfloor + \sum_{k=0}^{+\infty} s^k 10^{-k-1}}$.
-%\end{itemize}
-%$e$ and $s$ are thus defined as follows:
-%\begin{equation}
-%\begin{array}{cccl}
-%e: & \big[ 0, 2^{10} \big[ & \longrightarrow & \mathds{B}^{10} \\
-% & x & \longmapsto & (e_0, \hdots, e_9)
-%\end{array}
-%\end{equation}
-%and
-%\begin{equation}
-% \begin{array}{cccc}
-%s: & \big[ 0, 2^{10} \big[ & \longrightarrow & \llbracket 0, 9
-%\rrbracket^{\mathds{N}} \\
-% & x & \longmapsto & (s^k)_{k \in \mathds{N}}
-%\end{array}
-%\end{equation}
-%\end{definition}
-
-%We are now able to define the function $g$, whose goal is to translate the
-%chaotic iterations $\Go$ on an interval of $\mathds{R}$.
-
-%\begin{definition}
-%$g:\big[ 0, 2^{10} \big[ \longrightarrow \big[ 0, 2^{10} \big[$ is defined by:
-%\begin{equation}
-%\begin{array}{cccc}
-%g: & \big[ 0, 2^{10} \big[ & \longrightarrow & \big[ 0, 2^{10} \big[ \\
-% & x & \longmapsto & g(x)
-%\end{array}
-%\end{equation}
-%where g(x) is the real number of $\big[ 0, 2^{10} \big[$ defined bellow:
-%\begin{itemize}
-%\item its integral part has a binary decomposition equal to $e_0', \hdots,
-%e_9'$, with:
-% \begin{equation}
-%e_i' = \left\{
-%\begin{array}{ll}
-%e(x)_i & \textrm{ if } i \neq s^0\\
-%e(x)_i + 1 \textrm{ (mod 2)} & \textrm{ if } i = s^0\\
-%\end{array}
-%\right.
-%\end{equation}
-%\item whose decimal part is $s(x)^1, s(x)^2, \hdots$
-%\end{itemize}
-%\end{definition}
-
-%\bigskip
-
-
-%In other words, if $x = \displaystyle{\sum_{k=0}^{9} 2^{9-k} e_k +
-%\sum_{k=0}^{+\infty} s^{k} ~10^{-k-1}}$, then:
-%\begin{equation}
-%g(x) =
-%\displaystyle{\sum_{k=0}^{9} 2^{9-k} (e_k + \delta(k,s^0) \textrm{ (mod 2)}) +
-%\sum_{k=0}^{+\infty} s^{k+1} 10^{-k-1}}.
-%\end{equation}
-
-
-%\subsubsection{Defining a metric on $\big[ 0, 2^{10} \big[$}
-
-%Numerous metrics can be defined on the set $\big[ 0, 2^{10} \big[$, the most
-%usual one being the Euclidian distance recalled bellow:
-
-%\begin{notation}
-%\index{distance!euclidienne}
-%$\Delta$ is the Euclidian distance on $\big[ 0, 2^{10} \big[$, that is,
-%$\Delta(x,y) = |y-x|^2$.
-%\end{notation}
-
-%\medskip
-
-%This Euclidian distance does not reproduce exactly the notion of proximity
-%induced by our first distance $d$ on $\X$. Indeed $d$ is finer than $\Delta$.
-%This is the reason why we have to introduce the following metric:
-
-
-
-%\begin{definition}
-%Let $x,y \in \big[ 0, 2^{10} \big[$.
-%$D$ denotes the function from $\big[ 0, 2^{10} \big[^2$ to $\mathds{R}^+$
-%defined by: $D(x,y) = D_e\left(e(x),e(y)\right) + D_s\left(s(x),s(y)\right)$,
-%where:
-%\begin{center}
-%$\displaystyle{D_e(E,\check{E}) = \sum_{k=0}^\mathsf{9} \delta (E_k,
-%\check{E}_k)}$, ~~and~ $\displaystyle{D_s(S,\check{S}) = \sum_{k = 1}^\infty
-%\dfrac{|S^k-\check{S}^k|}{10^k}}$.
-%\end{center}
-%\end{definition}
-
-%\begin{proposition}
-%$D$ is a distance on $\big[ 0, 2^{10} \big[$.
-%\end{proposition}
-
-%\begin{proof}
-%The three axioms defining a distance must be checked.
-%\begin{itemize}
-%\item $D \geqslant 0$, because everything is positive in its definition. If
-%$D(x,y)=0$, then $D_e(x,y)=0$, so the integral parts of $x$ and $y$ are equal
-%(they have the same binary decomposition). Additionally, $D_s(x,y) = 0$, then
-%$\forall k \in \mathds{N}^*, s(x)^k = s(y)^k$. In other words, $x$ and $y$ have
-%the same $k-$th decimal digit, $\forall k \in \mathds{N}^*$. And so $x=y$.
-%\item $D(x,y)=D(y,x)$.
-%\item Finally, the triangular inequality is obtained due to the fact that both
-%$\delta$ and $\Delta(x,y)=|x-y|$ satisfy it.
-%\end{itemize}
-%\end{proof}
-
-
-%The convergence of sequences according to $D$ is not the same than the usual
-%convergence related to the Euclidian metric. For instance, if $x^n \to x$
-%according to $D$, then necessarily the integral part of each $x^n$ is equal to
-%the integral part of $x$ (at least after a given threshold), and the decimal
-%part of $x^n$ corresponds to the one of $x$ ``as far as required''.
-%To illustrate this fact, a comparison between $D$ and the Euclidian distance is
-%given Figure \ref{fig:comparaison de distances}. These illustrations show that
-%$D$ is richer and more refined than the Euclidian distance, and thus is more
-%precise.
-
-
-%\begin{figure}[t]
-%\begin{center}
-% \subfigure[Function $x \to dist(x;1,234) $ on the interval
-%$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien.pdf}}\quad
-% \subfigure[Function $x \to dist(x;3) $ on the interval
-%$(0;5)$.]{\includegraphics[scale=.35]{DvsEuclidien2.pdf}}
-%\end{center}
-%\caption{Comparison between $D$ (in blue) and the Euclidian distane (in green).}
-%\label{fig:comparaison de distances}
-%\end{figure}
-
-
-
-
-%\subsubsection{The semiconjugacy}
-
-%It is now possible to define a topological semiconjugacy between $\mathcal{X}$
-%and an interval of $\mathds{R}$:
-
-%\begin{theorem}
-%Chaotic iterations on the phase space $\mathcal{X}$ are simple iterations on
-%$\mathds{R}$, which is illustrated by the semiconjugacy of the diagram bellow:
-%\begin{equation*}
-%\begin{CD}
-%\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right) @>G_{f_0}>>
-%\left(~\mathcal{S}_{10} \times\mathds{B}^{10}, d~\right)\\
-% @V{\varphi}VV @VV{\varphi}V\\
-%\left( ~\big[ 0, 2^{10} \big[, D~\right) @>>g> \left(~\big[ 0, 2^{10} \big[,
-%D~\right)
-%\end{CD}
-%\end{equation*}
-%\end{theorem}
-
-%\begin{proof}
-%$\varphi$ has been constructed in order to be continuous and onto.
-%\end{proof}
-
-%In other words, $\mathcal{X}$ is approximately equal to $\big[ 0, 2^\mathsf{N}
-%\big[$.
-
-
-
-
-
-
-%\subsection{Study of the chaotic iterations described as a real function}
-
-
-%\begin{figure}[t]
-%\begin{center}
-% \subfigure[ICs on the interval
-%$(0,9;1)$.]{\includegraphics[scale=.35]{ICs09a1.pdf}}\quad
-% \subfigure[ICs on the interval
-%$(0,7;1)$.]{\includegraphics[scale=.35]{ICs07a95.pdf}}\\
-% \subfigure[ICs on the interval
-%$(0,5;1)$.]{\includegraphics[scale=.35]{ICs05a1.pdf}}\quad
-% \subfigure[ICs on the interval
-%$(0;1)$]{\includegraphics[scale=.35]{ICs0a1.pdf}}
-%\end{center}
-%\caption{Representation of the chaotic iterations.}
-%\label{fig:ICs}
-%\end{figure}
-
-
-
-
-%\begin{figure}[t]
-%\begin{center}
-% \subfigure[ICs on the interval
-%$(510;514)$.]{\includegraphics[scale=.35]{ICs510a514.pdf}}\quad
-% \subfigure[ICs on the interval
-%$(1000;1008)$]{\includegraphics[scale=.35]{ICs1000a1008.pdf}}
-%\end{center}
-%\caption{ICs on small intervals.}
-%\label{fig:ICs2}
-%\end{figure}
-
-%\begin{figure}[t]
-%\begin{center}
-% \subfigure[ICs on the interval
-%$(0;16)$.]{\includegraphics[scale=.3]{ICs0a16.pdf}}\quad
-% \subfigure[ICs on the interval
-%$(40;70)$.]{\includegraphics[scale=.45]{ICs40a70.pdf}}\quad
-%\end{center}
-%\caption{General aspect of the chaotic iterations.}
-%\label{fig:ICs3}
-%\end{figure}
-
-
-%We have written a Python program to represent the chaotic iterations with the
-%vectorial negation on the real line $\mathds{R}$. Various representations of
-%these CIs are given in Figures \ref{fig:ICs}, \ref{fig:ICs2} and \ref{fig:ICs3}.
-%It can be remarked that the function $g$ is a piecewise linear function: it is
-%linear on each interval having the form $\left[ \dfrac{n}{10},
-%\dfrac{n+1}{10}\right[$, $n \in \llbracket 0;2^{10}\times 10 \rrbracket$ and its
-%slope is equal to 10. Let us justify these claims:
-
-%\begin{proposition}
-%\label{Prop:derivabilite des ICs}
-%Chaotic iterations $g$ defined on $\mathds{R}$ have derivatives of all orders on
-%$\big[ 0, 2^{10} \big[$, except on the 10241 points in $I$ defined by $\left\{
-%\dfrac{n}{10} ~\big/~ n \in \llbracket 0;2^{10}\times 10\rrbracket \right\}$.
-
-%Furthermore, on each interval of the form $\left[ \dfrac{n}{10},
-%\dfrac{n+1}{10}\right[$, with $n \in \llbracket 0;2^{10}\times 10 \rrbracket$,
-%$g$ is a linear function, having a slope equal to 10: $\forall x \notin I,
-%g'(x)=10$.
-%\end{proposition}
-
-
-%\begin{proof}
-%Let $I_n = \left[ \dfrac{n}{10}, \dfrac{n+1}{10}\right[$, with $n \in \llbracket
-%0;2^{10}\times 10 \rrbracket$. All the points of $I_n$ have the same integral
-%prat $e$ and the same decimal part $s^0$: on the set $I_n$, functions $e(x)$
-%and $x \mapsto s(x)^0$ of Definition \ref{def:e et s} only depend on $n$. So all
-%the images $g(x)$ of these points $x$:
-%\begin{itemize}
-%\item Have the same integral part, which is $e$, except probably the bit number
-%$s^0$. In other words, this integer has approximately the same binary
-%decomposition than $e$, the sole exception being the digit $s^0$ (this number is
-%then either $e+2^{10-s^0}$ or $e-2^{10-s^0}$, depending on the parity of $s^0$,
-%\emph{i.e.}, it is equal to $e+(-1)^{s^0}\times 2^{10-s^0}$).
-%\item A shift to the left has been applied to the decimal part $y$, losing by
-%doing so the common first digit $s^0$. In other words, $y$ has been mapped into
-%$10\times y - s^0$.
-%\end{itemize}
-%To sum up, the action of $g$ on the points of $I$ is as follows: first, make a
-%multiplication by 10, and second, add the same constant to each term, which is
-%$\dfrac{1}{10}\left(e+(-1)^{s^0}\times 2^{10-s^0}\right)-s^0$.
-%\end{proof}
-
-%\begin{remark}
-%Finally, chaotic iterations are elements of the large family of functions that
-%are both chaotic and piecewise linear (like the tent map).
-%\end{remark}
-
-
-
-%\subsection{Comparison of the two metrics on $\big[ 0, 2^\mathsf{N} \big[$}
-
-%The two propositions bellow allow to compare our two distances on $\big[ 0,
-%2^\mathsf{N} \big[$:
-
-%\begin{proposition}
-%Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,\Delta~\right) \to \left(~\big[ 0,
-%2^\mathsf{N} \big[, D~\right)$ is not continuous.
-%\end{proposition}
-
-%\begin{proof}
-%The sequence $x^n = 1,999\hdots 999$ constituted by $n$ 9 as decimal part, is
-%such that:
-%\begin{itemize}
-%\item $\Delta (x^n,2) \to 0.$
-%\item But $D(x^n,2) \geqslant 1$, then $D(x^n,2)$ does not converge to 0.
-%\end{itemize}
-
-%The sequential characterization of the continuity concludes the demonstration.
-%\end{proof}
-
-
-
-%A contrario:
-
-%\begin{proposition}
-%Id: $\left(~\big[ 0, 2^\mathsf{N} \big[,D~\right) \to \left(~\big[ 0,
-%2^\mathsf{N} \big[, \Delta ~\right)$ is a continuous fonction.
-%\end{proposition}
-
-%\begin{proof}
-%If $D(x^n,x) \to 0$, then $D_e(x^n,x) = 0$ at least for $n$ larger than a given
-%threshold, because $D_e$ only returns integers. So, after this threshold, the
-%integral parts of all the $x^n$ are equal to the integral part of $x$.
-
-%Additionally, $D_s(x^n, x) \to 0$, then $\forall k \in \mathds{N}^*, \exists N_k
-%\in \mathds{N}, n \geqslant N_k \Rightarrow D_s(x^n,x) \leqslant 10^{-k}$. This
-%means that for all $k$, an index $N_k$ can be found such that, $\forall n
-%\geqslant N_k$, all the $x^n$ have the same $k$ firsts digits, which are the
-%digits of $x$. We can deduce the convergence $\Delta(x^n,x) \to 0$, and thus the
-%result.
-%\end{proof}
-
-%The conclusion of these propositions is that the proposed metric is more precise
-%than the Euclidian distance, that is:
-
-%\begin{corollary}
-%$D$ is finer than the Euclidian distance $\Delta$.
-%\end{corollary}
-
-%This corollary can be reformulated as follows:
-
-%\begin{itemize}
-%\item The topology produced by $\Delta$ is a subset of the topology produced by
-%$D$.
-%\item $D$ has more open sets than $\Delta$.
-%\item It is harder to converge for the topology $\tau_D$ inherited by $D$, than
-%to converge with the one inherited by $\Delta$, which is denoted here by
-%$\tau_\Delta$.
-%\end{itemize}
-
-
-%\subsection{Chaos of the chaotic iterations on $\mathds{R}$}
-%\label{chpt:Chaos des itérations chaotiques sur R}
-
-
-
-%\subsubsection{Chaos according to Devaney}
-
-%We have recalled previously that the chaotic iterations $\left(\Go,
-%\mathcal{X}_d\right)$ are chaotic according to the formulation of Devaney. We
-%can deduce that they are chaotic on $\mathds{R}$ too, when considering the order
-%topology, because:
-%\begin{itemize}
-%\item $\left(\Go, \mathcal{X}_d\right)$ and $\left(g, \big[ 0, 2^{10}
-%\big[_D\right)$ are semiconjugate by $\varphi$,
-%\item Then $\left(g, \big[ 0, 2^{10} \big[_D\right)$ is a system chaotic
-%according to Devaney, because the semiconjugacy preserve this character.
-%\item But the topology generated by $D$ is finer than the topology generated by
-%the Euclidian distance $\Delta$ -- which is the order topology.
-%\item According to Theorem \ref{Th:chaos et finesse}, we can deduce that the
-%chaotic iterations $g$ are indeed chaotic, as defined by Devaney, for the order
-%topology on $\mathds{R}$.
-%\end{itemize}
+All these experiments allow us to conclude that it is possible to
+generate a very large quantity of pseudorandom numbers statistically perfect with the xor-like version.
+In a certain extend, it is the case too 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.
-%This result can be formulated as follows.
-
-%\begin{theorem}
-%\label{th:IC et topologie de l'ordre}
-%The chaotic iterations $g$ on $\mathds{R}$ are chaotic according to the
-%Devaney's formulation, when $\mathds{R}$ has his usual topology, which is the
-%order topology.
-%\end{theorem}
-
-%Indeed this result is weaker than the theorem establishing the chaos for the
-%finer topology $d$. However the Theorem \ref{th:IC et topologie de l'ordre}
-%still remains important. Indeed, we have studied in our previous works a set
-%different from the usual set of study ($\mathcal{X}$ instead of $\mathds{R}$),
-%in order to be as close as possible from the computer: the properties of
-%disorder proved theoretically will then be preserved when computing. However, we
-%could wonder whether this change does not lead to a disorder of a lower quality.
-%In other words, have we replaced a situation of a good disorder lost when
-%computing, to another situation of a disorder preserved but of bad quality.
-%Theorem \ref{th:IC et topologie de l'ordre} prove exactly the contrary.
-%
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 $D$.
then the new one is secure too.
\begin{proposition}
+\label{cryptopreuve}
If $H$ is a secure cryptographic PRNG, then $X$ is a secure cryptographic
PRNG too.
\end{proposition}
\end{proof}
+\section{Cryptographical Applications}
-
-\section{A Cryptographically Secure PRNG for GPU}
+\subsection{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.
+It is possible to build a cryptographically secure PRNG based on the previous
+algorithm (Algorithm~\ref{algo:gpu_kernel2}). Due to Proposition~\ref{cryptopreuve},
+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 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 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}
+
+\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 $\wedge$ shmem[o1] $\wedge$ shmem[o2]\;
+ shared\_mem[threadId]=t\;
+ x = x $\wedge$ 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 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}
+
+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{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}
+
+
+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}
+
+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.
\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 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
-20Gsample/s). This PRNG succeeds to pass the hardest batteries of TestU01.
+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.
+grid computing.
