+Randomness is of importance in many fields such as scientific simulations or cryptography.
+``Random numbers'' can mainly be generated either by a deterministic and reproducible algorithm
+called a pseudorandom number generator (PRNG), or by a physical non-deterministic
+process having all the characteristics of a random noise, called a truly random number
+generator (TRNG).
+In this paper, we focus on reproducible generators, useful for instance in
+Monte-Carlo based simulators or in several cryptographic schemes.
+These domains need PRNGs that are statistically irreproachable.
+In some fields such as in numerical simulations, speed is a strong requirement
+that is usually attained by using parallel architectures. In that case,
+a recurrent problem is that a deflation of the statistical qualities is often
+reported, when the parallelization of a good PRNG is realized.
+This is why ad-hoc PRNGs for each possible architecture must be found to
+achieve both speed and randomness.
+On the other side, speed is not the main requirement in cryptography: the great
+need is to define \emph{secure} generators able to withstand malicious
+attacks. Roughly speaking, an attacker should not be able in practice to make
+the distinction between numbers obtained with the secure generator and a true random
+sequence.
+Finally, a small part of the community working in this domain focuses 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 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
+whereas computers deal with finite precision numbers.
+This distortion leads to a deflation of both chaotic properties and speed.
+Furthermore, authors of such chaotic generators often claim their PRNG
+as secure due to their chaos properties, but there is no obvious relation
+between chaos and security as it is understood in cryptography.
+This is why the use of chaos for PRNG still remains marginal and disputable.
+
+The authors' opinion is that topological properties of disorder, as they are
+properly defined in the mathematical theory of chaos, can reinforce the quality
+of a PRNG. But they are not substitutable for security or statistical perfection.
+Indeed, to the authors' mind, such properties can be useful in the two following situations. On the
+one hand, a post-treatment based on a chaotic dynamical system can be applied
+to a PRNG statistically deflective, in order to improve its statistical
+properties. Such an improvement can be found, for instance, in~\cite{bgw09:ip,bcgr11:ip}.
+On the other hand, chaos can be added to a fast, statistically perfect PRNG and/or a
+cryptographically secure one, in case where chaos can be of interest,
+\emph{only if these last properties are not lost during
+the proposed post-treatment}. Such an assumption is behind this research work.
+It leads to the attempts to define a
+family of PRNGs that are chaotic while being fast and statistically perfect,
+or cryptographically secure.
+Let us finish this paragraph by noticing that, in this paper,
+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 in 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}.
+
+
+In a previous work~\cite{bgw09:ip,guyeux10} we have proposed a post-treatment on PRNGs making them behave
+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~\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.
+In particular, a version of this PRNG on graphics processing units (GPU)
+is proposed.
+Although GPU was initially designed to accelerate
+the manipulation of images, they are nowadays commonly used in many scientific
+applications. Therefore, it is important to be able to generate pseudorandom
+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 generate almost 20 billion of pseudorandom numbers per second.
+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 rewriting 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.
+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 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
+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 Shub
+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.
+
+
+
+
+\section{Related works on GPU based PRNGs}
+\label{section:related works}
+
+Numerous research works on defining GPU based PRNGs have already been proposed in the
+literature, so that exhaustivity 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 or Hybrid Taus. They have used these
+PRNGs for Langevin simulations of biomolecules fully implemented on
+GPU. Performances of the GPU versions are far better than those obtained with a
+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 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 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 far easier than 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
+We can finally remark that, to the best of our knowledge, no GPU implementation has been proven to be chaotic, and the cryptographically secure property has surprisingly never been considered.
+
+\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. We assume the reader is familiar
+with basic notions on topology (see for instance~\cite{Devaney}).
+
+
+\subsection{Devaney's Chaotic Dynamical Systems}
+
+In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
+denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$
+is for the $k^{th}$ composition of a function $f$. Finally, the following
+notation is used: $\llbracket1;N\rrbracket=\{1,2,\hdots,N\}$.
+
+
+Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f :
+\mathcal{X} \rightarrow \mathcal{X}$.
+
+\begin{definition}
+The function $f$ is said to be \emph{topologically transitive} if, for any pair of open sets
+$U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq
+\varnothing$.
+\end{definition}
+
+\begin{definition}
+An element $x$ is a \emph{periodic point} for $f$ of period $n\in \mathds{N}^*$
+if $f^{n}(x)=x$.% The set of periodic points of $f$ is denoted $Per(f).$
+\end{definition}
