statistical perfection refers to the ability to pass the whole
{\it BigCrush} battery of tests, which is widely considered as the most
stringent statistical evaluation of a sequence claimed as random.
-This battery can be found into the well-known TestU01 package.
+This battery can be found into the well-known TestU01 package~\cite{LEcuyerS07}.
Chaos, for its part, refers to the well-established definition of a
chaotic dynamical system proposed by Devaney~\cite{Devaney}.
as a chaotic dynamical system. Such a post-treatment leads to a new category of
PRNGs. We have shown that proofs of Devaney's chaos can be established for this
family, and that the sequence obtained after this post-treatment can pass the
-NIST, DieHARD, and TestU01 batteries of tests, even if the inputted generators
+NIST~\cite{Nist10}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} batteries of tests, even if the inputted generators
cannot.
The proposition of this paper is to improve widely the speed of the formerly
proposed generator, without any lack of chaos or statistical properties.
\section{Related works on GPU based PRNGs}
\label{section:related works}
-In the litterature many authors have work on defining GPU based PRNGs. We do not
-want to be exhaustive and we just give the most significant works from our point
-of view. When authors mention the number of random numbers generated per second
-we mention it. We consider that a million numbers per second corresponds to
-1MSample/s and than a billion numbers per second corresponds to 1GSample/s.
-
-In \cite{Pang:2008:cec}, the authors define a PRNG based on cellular automata
-which does not require high precision integer arithmetics nor bitwise
-operations. There is no mention of statistical tests nor proof that this PRNG is
-chaotic. Concerning the speed of generation, they can generate about
-3.2MSample/s on a GeForce 7800 GTX GPU (which is quite old now).
+
+Numerous research works on defining GPU based PRNGs have yet been proposed in the
+literature, so that completeness is impossible.
+This is why authors of this document only give reference to the most significant attempts
+in this domain, from their subjective point of view.
+The quantity of pseudorandom numbers generated per second is mentioned here
+only when the information is given in the related work.
+A million numbers per second will be simply written as
+1MSample/s whereas a billion numbers per second is 1GSample/s.
+
+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
+3.2MSamples/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.
In \cite{ZRKB10}, the authors propose different versions of efficient GPU PRNGs
-based on Lagged Fibonacci, Hybrid Taus or Hybrid Taus. They have used these
+based on Lagged Fibonacci or Hybrid Taus. They have used these
PRNGs for Langevin simulations of biomolecules fully implemented on
GPU. Performance of the GPU versions are far better than those obtained with a
-CPU and these PRNGs succeed to pass the {\it BigCrush} test of TestU01. There is
-no mention that their PRNGs have chaos mathematical properties.
+CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01.
+However the evaluations of the proposed PRNGs are only statistical ones.
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 architeture. 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}
\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
$\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ; \mathsf{N}
-\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance).
+\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG).
\section{Application to pseudorandomness}
\label{sec:pseudorandom}
+
\subsection{A First pseudorandom Number Generator}
We have proposed in~\cite{bgw09:ip} a new family of generators that receives
This generator is synthesized in Algorithm~\ref{CI Algorithm}.
-It takes as input: a function $f$;
+It takes as input: a Boolean function $f$ satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques};
an integer $b$, ensuring that the number of executed iterations is at least $b$
and at most $2b+1$; and an initial configuration $x^0$.
It returns the new generated configuration $x$. Internally, it embeds two
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 tests~\cite{bcgr11:ip}.
+This former generator as successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07}.
\subsection{Improving the Speed of the Former Generator}
