In \cite{Pang:2008:cec} a PRNG based on cellular automata is defined
with no requirement to an high precision integer arithmetic or to any bitwise
operations. Authors can generate about
-3.2MSample/s on a GeForce 7800 GTX GPU, which is quite an old card now.
+3.2MSamples/s on a GeForce 7800 GTX GPU, which is quite an old card now.
However, there is neither a mention of statistical tests nor any proof of
chaos or cryptography in this document.
Authors of~\cite{conf/fpga/ThomasHL09} have studied the implementation of some
-PRNGs on diferrent computing architectures: CPU, field-programmable gate array
-(FPGA), GPU and massively parallel processor. This study is interesting because
-it shows the performance of the same PRNGs on different architectures. For
-example, the FPGA is globally the fastest architecture and it is also the
-efficient one because it provides the fastest number of generated random numbers
-per joule. Concerning the GPU, authors can generate betweend 11 and 16GSample/s
-with a GTX 280 GPU. The drawback of this work is that those PRNGs only succeed
-the {\it Crush} test which is easier than the {\it Big Crush} test.
-
-Cuda has developped a library for the generation of random numbers called
-Curand~\cite{curand11}. Several PRNGs are implemented:
-Xorwow~\cite{Marsaglia2003} and some variants of Sobol. Some tests report that
-the fastest version provides 15GSample/s on the new Fermi C2050 card. Their
-PRNGs fail to succeed the whole tests of TestU01 on only one test.
+PRNGs on different computing architectures: CPU, field-programmable gate array
+(FPGA), massively parallel processors, and GPU. This study is of interest, because
+the performance of the same PRNGs on different architectures are compared.
+FPGA appears as the fastest and the most
+efficient architecture, providing the fastest number of generated pseudorandom numbers
+per joule.
+However, we can 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
+able to pass the {\it Crush} battery, which is very easy compared to the {\it Big Crush} one.
+
+Lastly, Cuda has developed a library for the generation of pseudorandom numbers called
+Curand~\cite{curand11}. Several PRNGs are implemented, among
+other things
+Xorwow~\cite{Marsaglia2003} and some variants of Sobol. The tests reported show that
+their fastest version provides 15GSamples/s on the new Fermi C2050 card.
+But their PRNGs cannot pass the whole TestU01 battery (only one test is failed).
\newline
\newline
-To the best of our knowledge no GPU implementation have been proven to have chaotic properties.
+We can finally remark that, to the best of our knowledge, no GPU implementation have been proven to be chaotic, and the cryptographically secure property is surprisingly never regarded.
\section{Basic Recalls}
\label{section:BASIC RECALLS}
+
This section is devoted to basic definitions and terminologies in the fields of
topological chaos and chaotic iterations.
\subsection{Devaney's Chaotic Dynamical Systems}
