\usepackage[standard]{ntheorem}
\usepackage{algorithmic}
\usepackage{slashbox}
+\usepackage{ctable}
+\usepackage{tabularx}
+\usepackage{multirow}
% Pour mathds : les ensembles IR, IN, etc.
\usepackage{dsfont}
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.
+sequence. \begin{color}{red} Or, in an equivalent formulation, he or she should not be
+able (in practice) to predict the next bit of the generator, having the knowledge of all the
+binary digits that have been already released. ``Being able in practice'' refers here
+to the possibility to achieve this attack in polynomial time, and to the exponential growth
+of the difficulty of this challenge when the size of the parameters of the PRNG increases.
+\end{color}
+
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
{\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}.
+\begin{color}{red}
+More precisely, each time we performed a test on a PRNG, we ran it
+twice in order to observe if all $p-$values are inside [0.01, 0.99]. In
+fact, we observed that few $p-$values (less than ten) are sometimes
+outside this interval but inside [0.001, 0.999], so that is why a
+second run allows us to confirm that the values outside are not for
+the same test. With this approach all our PRNGs pass the {\it
+ BigCrush} successfully and all $p-$values are at least once inside
+[0.01, 0.99].
+\end{color}
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
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
+\begin{color}{red}
+Section~\ref{The generation of pseudorandom sequence} illustrates the statistical
+improvement related to the chaotic iteration based post-treatment, for
+our previously released PRNGs and a new efficient
+implementation on CPU.
+\end{color}
+ Section~\ref{sec:efficient PRNG
gpu} describes and evaluates theoretically the GPU implementation.
Such generators are experimented in
Section~\ref{sec:experiments}.
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
+in Section~\ref{sec:CSGPU}, \begin{color}{red} to a practical
+security evaluation in Section~\ref{sec:Practicak evaluation}, \end{color} 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}
+\section{Related work on GPU based PRNGs}
\label{section:related works}
Numerous research works on defining GPU based PRNGs have already been proposed in the
\subsection{Devaney's Chaotic Dynamical Systems}
-
+\label{subsec:Devaney}
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
two PRNGs as inputs. These two generators are mixed with chaotic iterations,
leading thus to a new PRNG that
\begin{color}{red}
-should improves the statistical properties of each
+should improve the statistical properties of each
generator taken alone.
Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input
present.
First of all, some chaotic iterations have to be done to generate a sequence
$\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^{32}\right)^\mathds{N}$
of Boolean vectors, which are the successive states of the iterated system.
-Some of these vectors will be randomly extracted and our pseudo-random bit
+Some of these vectors will be randomly extracted and our pseudorandom bit
flow will be constituted by their components. Such chaotic iterations are
realized as follows. Initial state $x^0 \in \mathds{B}^{32}$ is a Boolean
vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in
Such a procedure is equivalent to achieve chaotic iterations with
the Boolean vectorial negation $f_0$ and some well-chosen strategies.
Finally, some $x^n$ are selected
-by a sequence $m^n$ as the pseudo-random bit sequence of our generator.
+by a sequence $m^n$ as the pseudorandom bit sequence of our generator.
$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers.
The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}.
The internal state is $x$, the output state is $r$. $a$ and $b$ are those computed by the two input
PRNGs. Lastly, the value $g(a)$ is an integer defined as in Eq.~\ref{Formula}.
-This function is required to make the outputs uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$
-(the reader is referred to~\cite{bg10:ip} for more information).
+This function must be chosen such that the outputs of the resulted PRNG are uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$. Function of \eqref{Formula} achieves this
+goal (other candidates and more information can be found in ~\cite{bg10:ip}).
\begin{equation}
\label{Formula}
}
\ENDFOR
\STATE$a\leftarrow{PRNG_1()}$\;
-\STATE$m\leftarrow{g(a)}$\;
-\STATE$k\leftarrow{m}$\;
+\STATE$k\leftarrow{g(a)}$\;
\WHILE{$i=0,\dots,k$}
\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^n,
\end{array}
\right.
-\label{equation Oplus0}
+\label{equation Oplus}
\end{equation}
where $\oplus$ is for the bitwise exclusive or between two integers.
This rewriting can be understood as follows. The $n-$th term $S^n$ of the
component of this state (a binary digit) changes if and only if the $k-$th
digit in the binary decomposition of $S^n$ is 1.
-The single basic component presented in Eq.~\ref{equation Oplus0} is of
+The single basic component presented in Eq.~\ref{equation Oplus} is of
ordinary use as a good elementary brick in various PRNGs. It corresponds
to the following discrete dynamical system in chaotic iterations:
Obviously, replacing the previous CI PRNG Algorithms by
-Equation~\ref{equation Oplus0}, which is possible when the iteration function is
-the vectorial negation, leads to a speed improvement. However, proofs
+Equation~\ref{equation Oplus}, which is possible when the iteration function is
+the vectorial negation, leads to a speed improvement
+(the resulting generator will be referred as ``Xor CI PRNG''
+in what follows).
+However, proofs
of chaos obtained in~\cite{bg10:ij} have been established
only for chaotic iterations of the form presented in Definition
\ref{Def:chaotic iterations}. The question is now to determine whether the
In conclusion,
%%RAPH : ici j'ai rajouté une ligne
+%%TOF : ici j'ai rajouté un commentaire
+%%TOF : ici aussi
$
\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
,$ $\forall n\geqslant N_{0},$
\begin{color}{red}
\section{Statistical Improvements Using Chaotic Iterations}
-\label{The generation of pseudo-random sequence}
+\label{The generation of pseudorandom sequence}
Let us now explain why we are reasonable grounds to believe that chaos
can improve statistical properties.
-We will show in this section that, when mixing defective PRNGs with
-chaotic iterations, the result presents better statistical properties
-(this section summarizes the work of~\cite{bfg12a:ip}).
+We will show in this section that chaotic properties as defined in the
+mathematical theory of chaos are related to some statistical tests that can be found
+in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with
+chaotic iterations, the new generator presents better statistical properties
+(this section summarizes and extends the work of~\cite{bfg12a:ip}).
+
+
+
+\subsection{Qualitative relations between topological properties and statistical tests}
+
+
+There are various relations between topological properties that describe an unpredictable behavior for a discrete
+dynamical system on the one
+hand, and statistical tests to check the randomness of a numerical sequence
+on the other hand. These two mathematical disciplines follow a similar
+objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a
+recurrent sequence), with two different but complementary approaches.
+It is true that the following illustrative links give only qualitative arguments,
+and proofs should be provided later to make such arguments irrefutable. However
+they give a first understanding of the reason why we think that chaotic properties should tend
+to improve the statistical quality of PRNGs.
+%
+Let us now list some of these relations between topological properties defined in the mathematical
+theory of chaos and tests embedded into the NIST battery. %Such relations need to be further
+%investigated, but they presently give a first illustration of a trend to search similar properties in the
+%two following fields: mathematical chaos and statistics.
+
+
+\begin{itemize}
+ \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney}, a chaotic dynamical system must
+have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of
+a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity
+is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a
+knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in
+the two following NIST tests~\cite{Nist10}:
+ \begin{itemize}
+ \item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern.
+ \item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are near each other) in the tested sequence that would indicate a deviation from the assumption of randomness.
+ \end{itemize}
+
+\item \textbf{Transitivity}. This topological property introduced previously states that the dynamical system is intrinsically complicated: it cannot be simplified into
+two subsystems that do not interact, as we can find in any neighborhood of any point another point whose orbit visits the whole phase space.
+This focus on the places visited by orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory
+of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention
+is brought on states visited during a random walk in the two tests below~\cite{Nist10}:
+ \begin{itemize}
+ \item \textbf{Random Excursions Variant Test}. Detect deviations from the expected number of visits to various states in the random walk.
+ \item \textbf{Random Excursions Test}. Determine if the number of visits to a particular state within a cycle deviates from what one would expect for a random sequence.
+ \end{itemize}
+
+\item \textbf{Chaos according to Li and Yorke}. Two points of the phase space $(x,y)$ define a couple of Li-Yorke when $\limsup_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))>0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillates as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according
+to Li-Yorke~\cite{Li75,Ruette2001}. A similar property is regarded in the following NIST test~\cite{Nist10}.
+ \begin{itemize}
+ \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow.
+ \end{itemize}
+ \item \textbf{Topological entropy}. The desire to formulate an equivalency of the thermodynamics entropy
+has emerged both in the topological and statistical fields. Another time, a similar objective has led to two different
+rewritten of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach,
+whereas topological entropy is defined as follows.
+$x,y \in \mathcal{X}$ are $\varepsilon-$\emph{separated in time $n$} if there exists $k \leqslant n$ such that $d\left(f^{(k)}(x),f^{(k)}(y)\right)>\varepsilon$. Then $(n,\varepsilon)-$separated sets are sets of points that are all $\varepsilon-$separated in time $n$, which
+leads to the definition of $s_n(\varepsilon,Y)$, being the maximal cardinality of all $(n,\varepsilon)-$separated sets. Using these notations,
+the topological entropy is defined as follows: $$h_{top}(\mathcal{X},f) = \displaystyle{\lim_{\varepsilon \rightarrow 0} \Big[ \limsup_{n \rightarrow +\infty} \dfrac{1}{n} \log s_n(\varepsilon,\mathcal{X})\Big]}.$$
+This value measures the average exponential growth of the number of distinguishable orbit segments.
+In this sense, it measures complexity of the topological dynamical system, whereas
+the Shannon approach is in mind when defining the following test~\cite{Nist10}:
+ \begin{itemize}
+\item \textbf{Approximate Entropy Test}. Compare the frequency of overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence.
+ \end{itemize}
+
+ \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are
+not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}.
+ \begin{itemize}
+\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence.
+\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random.
+ \end{itemize}
+\end{itemize}
+
+
+We have proven in our previous works~\cite{} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques} are, among other
+things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke,
+and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$,
+where $\mathsf{N}$ is the size of the iterated vector.
+These topological properties make that we are ground to believe that a generator based on chaotic
+iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like
+the NIST one. The following subsections, in which we prove that defective generators have their
+statistical properties improved by chaotic iterations, show that such an assumption is true.
\subsection{Details of some Existing Generators}
The list of defective PRNGs we will use
as inputs for the statistical tests to come is introduced here.
-Firstly, the simple linear congruency generator (LCGs) will be used.
-It is defined by the following recurrence:
+Firstly, the simple linear congruency generators (LCGs) will be used.
+They are defined by the following recurrence:
\begin{equation}
-x^n = (ax^{n-1} + c)~mod~m
+x^n = (ax^{n-1} + c)~mod~m,
\label{LCG}
\end{equation}
where $a$, $c$, and $x^0$ must be, among other things, non-negative and less than
$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer as two (resp. three)
combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
-Secondly, the multiple recursive generators (MRGs) will be used too, which
+Secondly, the multiple recursive generators (MRGs) will be used, which
are based on a linear recurrence of order
$k$, modulo $m$~\cite{LEcuyerS07}:
\begin{equation}
-x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m
+x^n = (a^1x^{n-1}+~...~+a^kx^{n-k})~mod~m .
\label{MRG}
\end{equation}
-Combination of two MRGs (referred as 2MRGs) is also used in these experimentations.
+Combination of two MRGs (referred as 2MRGs) is also used in these experiments.
Generators based on linear recurrences with carry will be regarded too.
This family of generators includes the add-with-carry (AWC) generator, based on the recurrence:
Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
\begin{equation}
-x^n = x^{n-r} \oplus x^{n-k}
+x^n = x^{n-r} \oplus x^{n-k} .
\label{GFSR}
\end{equation}
-Finally, the nonlinear inversive generator~\cite{LEcuyerS07} has been regarded too, which is:
+Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:
\begin{equation}
\label{INV}
+\begin{table}
+\renewcommand{\arraystretch}{1.3}
+\caption{TestU01 Statistical Test}
+\label{TestU011}
+\centering
+ \begin{tabular}{lccccc}
+ \toprule
+Test name &Tests& Logistic & XORshift & ISAAC\\
+Rabbit & 38 &21 &14 &0 \\
+Alphabit & 17 &16 &9 &0 \\
+Pseudo DieHARD &126 &0 &2 &0 \\
+FIPS\_140\_2 &16 &0 &0 &0 \\
+SmallCrush &15 &4 &5 &0 \\
+Crush &144 &95 &57 &0 \\
+Big Crush &160 &125 &55 &0 \\ \hline
+Failures & &261 &146 &0 \\
+\bottomrule
+ \end{tabular}
+\end{table}
+
+
+
+\begin{table}
+\renewcommand{\arraystretch}{1.3}
+\caption{TestU01 Statistical Test for Old CI algorithms ($\mathsf{N}=4$)}
+\label{TestU01 for Old CI}
+\centering
+ \begin{tabular}{lcccc}
+ \toprule
+\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
+&Logistic& XORshift& ISAAC&ISAAC \\
+&+& +& + & + \\
+&Logistic& XORshift& XORshift&ISAAC \\ \cmidrule(r){2-5}
+Rabbit &7 &2 &0 &0 \\
+Alphabit & 3 &0 &0 &0 \\
+DieHARD &0 &0 &0 &0 \\
+FIPS\_140\_2 &0 &0 &0 &0 \\
+SmallCrush &2 &0 &0 &0 \\
+Crush &47 &4 &0 &0 \\
+Big Crush &79 &3 &0 &0 \\ \hline
+Failures &138 &9 &0 &0 \\
+\bottomrule
+ \end{tabular}
+\end{table}
+
+
+
\subsection{Statistical tests}
cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot
fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
iterations can solve this issue.
-More precisely, to
-illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
-\begin{enumerate}
- \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
- \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
- \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
-\begin{equation}
-\begin{array}{l}
-\left\{
-\begin{array}{l}
-x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
-\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
-\end{equation}
-$m$ is called the \emph{functional power}.
-\end{enumerate}
-
-
-We have performed statistical analysis of each of the aforementioned CIPRNGs.
-The results are reproduced in Tables~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} and \ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
-The scores written in boldface indicate that all the tests have been passed successfully, whereas an asterisk ``*'' means that the considered passing rate has been improved.
+%More precisely, to
+%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
+%\begin{enumerate}
+% \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
+% \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
+% \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
+%\begin{equation}
+%\begin{array}{l}
+%\left\{
+%\begin{array}{l}
+%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
+%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
+%\end{equation}
+%$m$ is called the \emph{functional power}.
+%\end{enumerate}
+%
+The obtained results are reproduced in Table
+\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
+The scores written in boldface indicate that all the tests have been passed successfully, whereas an
+asterisk ``*'' means that the considered passing rate has been improved.
+The improvements are obvious for both the ``Old CI'' and ``New CI'' generators.
+Concerning the ``Xor CI PRNG'', the score is less spectacular: a large speed improvement makes that statistics
+ are not as good as for the two other versions of these CIPRNGs.
+However 8 tests have been improved (with no deflation for the other results).
-\subsubsection{Tests based on the Single CIPRNG}
\begin{table*}
\renewcommand{\arraystretch}{1.3}
\end{tabular}
\end{table*}
-The statistical tests results of the PRNGs using the single CIPRNG method are given in Table~\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
-We can observe that, except for the Xor CIPRNG, all of the CIPRNGs have passed the 15 tests of the NIST battery and the 18 tests of the DieHARD one.
-Moreover, considering these scores, we can deduce that both the single Old CIPRNG and the single New CIPRNG are relatively steadier than the single Xor CIPRNG approach, when applying them to different PRNGs.
-However, the Xor CIPRNG is obviously the fastest approach to generate a CI random sequence, and it still improves the statistical properties relative to each generator taken alone, although the test values are not as good as desired.
-
-Therefore, all of these three ways are interesting, for different reasons, in the production of pseudorandom numbers and,
-on the whole, the single CIPRNG method can be considered to adapt to or improve all kinds of PRNGs.
-
-To have a realization of the Xor CIPRNG that can pass all the tests embedded into the NIST battery, the Xor CIPRNG with multiple functional powers are investigated in Section~\ref{Tests based on Multiple CIPRNG}.
-
-
-\subsubsection{Tests based on the Mixed CIPRNG}
-
-To compare the previous approach with the CIPRNG design that uses a Mixed CIPRNG, we have taken into account the same inputted generators than in the previous section.
-These inputted couples $(PRNG_1,PRNG_2)$ of PRNGs are used in the Mixed approach as follows:
-\begin{equation}
-\left\{
-\begin{array}{l}
-x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\
-\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus PRNG_1\oplus PRNG_2,
-\end{array}
-\right.
-\label{equation Oplus}
-\end{equation}
-
-With this Mixed CIPRNG approach, both the Old CIPRNG and New CIPRNG continue to pass all the NIST and DieHARD suites.
-In addition, we can see that the PRNGs using a Xor CIPRNG approach can pass more tests than previously.
-The main reason of this success is that the Mixed Xor CIPRNG has a longer period.
-Indeed, let $n_{P}$ be the period of a PRNG $P$, then the period deduced from the single Xor CIPRNG approach is obviously equal to:
-\begin{equation}
-n_{SXORCI}=
-\left\{
-\begin{array}{ll}
-n_{P}&\text{if~}x^0=x^{n_{P}}\\
-2n_{P}&\text{if~}x^0\neq x^{n_{P}}.\\
-\end{array}
-\right.
-\label{equation Oplus}
-\end{equation}
-
-Let us now denote by $n_{P1}$ and $n_{P2}$ the periods of respectively the $PRNG_1$ and $PRNG_2$ generators, then the period of the Mixed Xor CIPRNG will be:
-\begin{equation}
-n_{XXORCI}=
-\left\{
-\begin{array}{ll}
-LCM(n_{P1},n_{P2})&\text{if~}x^0=x^{LCM(n_{P1},n_{P2})}\\
-2LCM(n_{P1},n_{P2})&\text{if~}x^0\neq x^{LCM(n_{P1},n_{P2})}.\\
-\end{array}
-\right.
-\label{equation Oplus}
-\end{equation}
-
-In Table~\ref{DieHARD fail mixex CIPRNG}, we only show the results for the Mixed CIPRNGs that cannot pass all DieHARD suites (the NIST tests are all passed). It demonstrates that Mixed Xor CIPRNG involving LCG, MRG, LCG2, LCG3, MRG2, or INV cannot pass the two following tests, namely the ``Matrix Rank 32x32'' and the ``COUNT-THE-1's'' tests contained into the DieHARD battery. Let us recall their definitions:
-
-\begin{itemize}
- \item \textbf{Matrix Rank 32x32.} A random 32x32 binary matrix is formed, each row having a 32-bit random vector. Its rank is an integer that ranges from 0 to 32. Ranks less than 29 must be rare, and their occurences must be pooled with those of rank 29. To achieve the test, ranks of 40,000 such random matrices are obtained, and a chisquare test is performed on counts for ranks 32,31,30 and for ranks $\leq29$.
-
- \item \textbf{COUNT-THE-1's TEST} Consider the file under test as a stream of bytes (four per 2 bit integer). Each byte can contain from 0 to 8 1's, with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let the stream of bytes provide a string of overlapping 5-letter words, each ``letter'' taking values A,B,C,D,E. The letters are determined by the number of 1's in a byte: 0,1, or 2 yield A, 3 yields B, 4 yields C, 5 yields D and 6,7, or 8 yield E. Thus we have a monkey at a typewriter hitting five keys with various probabilities (37,56,70,56,37 over 256). There are $5^5$ possible 5-letter words, and from a string of 256,000 (over-lapping) 5-letter words, counts are made on the frequencies for each word. The quadratic form in the weak inverse of the covariance matrix of the cell counts provides a chisquare test: Q5-Q4, the difference of the naive Pearson sums of $(OBS-EXP)^2/EXP$ on counts for 5- and 4-letter cell counts.
-\end{itemize}
-
-The reason of these fails is that the output of LCG, LCG2, LCG3, MRG, and MRG2 under the experiments are in 31-bit. Compare with the Single CIPRNG, using different PRNGs to build CIPRNG seems more efficient in improving random number quality (mixed Xor CI can 100\% pass NIST, but single cannot).
-
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{Scores of mixed Xor CIPRNGs when considering the DieHARD battery}
-\label{DieHARD fail mixex CIPRNG}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|}
- \hline
-\backslashbox{\textbf{$PRNG_1$}} {\textbf{$PRNG_0$}} & LCG & MRG & INV & LCG2 & LCG3 & MRG2 \\ \hline\hline
-LCG &\backslashbox{} {} &16/18&16/18 &16/18 &16/18 &16/18\\ \hline
-MRG &16/18 &\backslashbox{} {} &16/18&16/18 &16/18 &16/18\\ \hline
-INV &16/18 &16/18&\backslashbox{} {} &16/18 &16/18&16/18 \\ \hline
-LCG2 &16/18 &16/18 &16/18 &\backslashbox{} {} &16/18&16/18\\ \hline
-LCG3 &16/18 &16/18 &16/18&16/18&\backslashbox{} {} &16/18\\ \hline
-MRG2 &16/18 &16/18 &16/18&16/18 &16/18 &\backslashbox{} {} \\ \hline
-\end{tabular}
-\end{table*}
-
-\subsubsection{Tests based on the Multiple CIPRNG}
-\label{Tests based on Multiple CIPRNG}
-
-Until now, the combination of at most two input PRNGs has been investigated.
-We now regard the possibility to use a larger number of generators to improve the statistics of the generated pseudorandom numbers, leading to the multiple functional power approach.
-For the CIPRNGs which have already pass both the NIST and DieHARD suites with 2 inputted PRNGs (all the Old and New CIPRNGs, and some of the Xor CIPRNGs), it is not meaningful to consider their adaption of this multiple CIPRNG method, hence only the Multiple Xor CIPRNGs, having the following form, will be investigated.
-\begin{equation}
-\left\{
-\begin{array}{l}
-x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\
-\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^{nm}\oplus S^{nm+1}\ldots \oplus S^{nm+m-1} ,
-\end{array}
-\right.
-\label{equation Oplus}
-\end{equation}
-
-The question is now to determine the value of the threshold $m$ (the functional power) making the multiple CIPRNG being able to pass the whole NIST battery.
-Such a question is answered in Table~\ref{threshold}.
+We have then investigate in~\cite{bfg12a:ip} if it is possible to improve
+the statistical behavior of the Xor CI version by combining more than one
+$\oplus$ operation. Results are summarized in Table~\ref{threshold}, illustrating
+the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in.
+Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained
+using chaotic iterations on defective generators.
\begin{table*}
\renewcommand{\arraystretch}{1.3}
-\caption{Functional power $m$ making it possible to pass the whole NIST battery}
+\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
\label{threshold}
\centering
\begin{tabular}{|l||c|c|c|c|c|c|c|c|}
\end{tabular}
\end{table*}
-\subsubsection{Results Summary}
+Finally, the TestU01 battery has been launched on three well-known generators
+(a logistic map, a simple XORshift, and the cryptographically secure ISAAC,
+see Table~\ref{TestU011}). These results can be compared with
+Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
+Old CI PRNG that has received these generators.
+The obvious improvement speaks for itself, and together with the other
+results recalled in this section, it reinforces the opinion that a strong
+correlation between topological properties and statistical behavior exists.
-We can summarize the obtained results as follows.
-\begin{enumerate}
-\item The CIPRNG method is able to improve the statistical properties of a large variety of PRNGs.
-\item Using different PRNGs in the CIPRNG approach is better than considering several instances of one unique PRNG.
-\item The statistical quality of the outputs increases with the functional power $m$.
-\end{enumerate}
+Next subsection will now give a concrete original implementation of the Xor CI PRNG, the
+fastest generator in the chaotic iteration based family. In the remainder,
+this generator will be simply referred as CIPRNG, or ``the proposed PRNG'', if this statement does not
+raise ambiguity.
\end{color}
-\section{Efficient PRNG based on Chaotic Iterations}
+\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
\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}.
-The first idea is to consider
-that the provided strategy is a pseudorandom Boolean vector obtained by a
-given PRNG.
-An iteration of the system is simply the bitwise exclusive or between
-the last computed state and the current strategy.
-Topological properties of disorder exhibited by chaotic
-iterations can be inherited by the inputted generator, we hope by doing so to
-obtain some statistical improvements while preserving speed.
-
+%
+%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}.
+%The first idea is to consider
+%that the provided strategy is a pseudorandom Boolean vector obtained by a
+%given PRNG.
+%An iteration of the system is simply the bitwise exclusive or between
+%the last computed state and the current strategy.
+%Topological properties of disorder exhibited by chaotic
+%iterations can be inherited by the inputted generator, we hope by doing so to
+%obtain some statistical improvements while preserving speed.
+%
%%RAPH : j'ai viré tout ca
%% Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
%% are
-\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{small}
\begin{lstlisting}
Thus producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers
that are provided by 3 64-bits PRNGs. This version successfully passes the
-stringent BigCrush battery of tests~\cite{LEcuyerS07}.
+stringent BigCrush battery of tests~\cite{LEcuyerS07}.
+\begin{color}{red}At this point, we thus
+have defined an efficient and statistically unbiased generator. Its speed is
+directly related to the use of linear operations, but for the same reason,
+this fast generator cannot be proven as secure.
+\end{color}
+
\section{Efficient PRNGs based on Chaotic Iterations on GPU}
\label{sec:efficient PRNG gpu}
\begin{color}{red}
\subsection{Practical Security Evaluation}
+\label{sec:Practicak evaluation}
Suppose now that the PRNG will work during
$M=100$ time units, and that during this period,
