$log_2(log_2(x_n))$). So to generate a 32 bits number, we need to use 8 times
the BBS algorithm, with different combinations of $M$ is required.
+Currently this PRNG does not succeed to pass all the tests of TestU01.
+
\section{Experiments}
\label{sec:experiments}
another one equipped with a less performant CPU and a GeForce GTX 280. Both
cards have 240 cores.
-In Figure~\ref{fig:time_gpu} we compare the number of random numbers generated
-per second. The xor-like prng is a xor64 described in~\cite{Marsaglia2003}. In
-order to obtain the optimal performance we remove the storage of random numbers
-in the GPU memory. This step is time consuming and slows down the random number
-generation. Moreover, if you are interested by applications that consume random
-numbers directly when they are generated, their storage is completely
+In Figure~\ref{fig:time_xorlike_gpu} we compare the number of random numbers
+generated per second with the xor-like based PRNG. In this figure, the optimized
+version use the {\it xor64} described in~\cite{Marsaglia2003}. The naive version
+use the three xor-like PRNGs described in Listing~\ref{algo:seqCIprng}. In
+order to obtain the optimal performance we removed the storage of random numbers
+in the GPU memory. This step is time consuming and slows down the random numbers
+generation. Moreover, if one is interested by applications that consume random
+numbers directly when they are generated, their storage are completely
useless. In this figure we can see that when the number of threads is greater
than approximately 30,000 upto 5 millions the number of random numbers generated
per second is almost constant. With the naive version, it is between 2.5 and
\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{Number of random numbers generated per second with the xorlike based PRNG}
\label{fig:time_xorlike_gpu}
\end{figure}
138MSample/s with only one core of the Xeon E5530.
-
+In Figure~\ref{fig:time_bbs_gpu} we highlight the performance of the optimized
+BBS based PRNG on GPU. Performances are less important. On the Tesla C1060 we
+obtain approximately 1.8GSample/s and on the GTX 280 about 1.6GSample/s.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=.7]{curve_time_bbs_gpu.pdf}
\end{center}
-\caption{Number of random numbers generated per second with the bbs based prng}
+\caption{Number of random numbers generated per second with the BBS based PRNG}
\label{fig:time_bbs_gpu}
\end{figure}
+Both these experimentations allows us to conclude that it is possible to
+generate a huge number of pseudo-random 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}
In this paper we have presented a new class of PRNGs based on chaotic
-iterations. We have proven that these PRNGs are chaotic in the sense of Devenay.
+iterations. We have proven that these PRNGs are chaotic in the sense of Devenay.
+We also propose a PRNG cryptographically secure and its implementation on GPU.
+
+An efficient implementation on GPU based on a xor-like PRNG allows us to
+generate a huge number of pseudo-random numbers per second (about
+20Gsample/s). This PRNG succeeds to pass the hardest batteries of TestU01.
+
+In future work we plan to extend this work for parallel PRNG for clusters or
+grid computing. We also plan to improve the BBS version in order to succeed all
+the tests of TestU01.
-An efficient implementation on GPU allows us to generate a huge number of pseudo
-random numbers per second (about 20Gsample/s). Our PRNGs succeed to pass the
-hardest batteries of test (TestU01).
-In future work we plan to extend our work in order to have cryptographically
-secure PRNGs because in some situations this property may be important.
-\bibliographystyle{plain}
+\bibliographystyle{plain}
\bibliography{mabase}
\end{document}
