\usepackage[standard]{ntheorem}
\usepackage{algorithmic}
\usepackage{slashbox}
-
+\usepackage{ctable}
+\usepackage{cite}
+\usepackage{tabularx}
+\usepackage{multirow}
% Pour mathds : les ensembles IR, IN, etc.
\usepackage{dsfont}
\usepackage{graphicx}
% Pour faire des sous-figures dans les figures
\usepackage{subfigure}
+\usepackage{xr-hyper}
+\usepackage{hyperref}
+\externaldocument[A-]{supplementary}
+
-\usepackage{color}
\newtheorem{notation}{Notation}
\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
+
+
\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
\begin{document}
\IEEEcompsoctitleabstractindextext{
\begin{abstract}
In this paper we present a new pseudorandom number generator (PRNG) on
-graphics processing units (GPU). This PRNG is based on the so-called chaotic iterations. It
-is firstly proven to be chaotic according to the Devaney's formulation. We thus propose an efficient
+graphics processing units (GPU). This PRNG is based on the so-called chaotic iterations and
+it is thus chaotic according to the Devaney's formulation. We propose an efficient
implementation for GPU that successfully passes the {\it BigCrush} tests, deemed to be the hardest
battery of tests in TestU01. Experiments show that this PRNG can generate
about 20 billion of random numbers per second on Tesla C1060 and NVidia GTX280
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. However, 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.
+
+
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}.
+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].
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
Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric
key encryption protocol by using the proposed method.
+
+{\bf Main contributions.} In this paper a new PRNG using chaotic iteration
+is defined. From a theoretical point of view, it is proven that it has fine
+topological chaotic properties and that it is cryptographically secured (when
+the initial PRNG is also cryptographically secured). From a practical point of
+view, experiments point out a very good statistical behavior. An optimized
+original implementation of this PRNG is also proposed and experimented.
+Pseudorandom numbers are generated at a rate of 20GSamples/s, which is faster
+than in~\cite{conf/fpga/ThomasHL09,Marsaglia2003} (and with a better
+statistical behavior). Experiments are also provided using BBS as the initial
+random generator. The generation speed is significantly weaker.
+Note also that an original qualitative comparison between topological chaotic
+properties and statistical test is also proposed.
+
+
+
+
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
+Section~\ref{sec:efficient PRNG} %{The generation of pseudorandom sequence} %illustrates the statistical
+%improvement related to the chaotic iteration based post-treatment, for
+%our previously released PRNGs and
+ contains a new efficient
+implementation on CPU.
+ 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.
+%A practical
+%security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.
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} 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
\end{proposition}
The chaotic property of $G_f$ has been firstly established for the vectorial
-Boolean negation $f(x_1,\hdots, x_\mathsf{N}) = (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly
+Boolean negation $f_0(x_1,\hdots, x_\mathsf{N}) = (\overline{x_1},\hdots, \overline{x_\mathsf{N}})$ \cite{guyeux10}. To obtain a characterization, we have secondly
introduced the notion of asynchronous iteration graph recalled bellow.
Let $f$ be a map from $\mathds{B}^\mathsf{N}$ to itself. The
We have proposed in~\cite{bgw09:ip} a new family of generators that receives
two PRNGs as inputs. These two generators are mixed with chaotic iterations,
-leading thus to a new PRNG that improves the statistical properties of each
-generator taken alone. Furthermore, our generator
-possesses various chaos properties that none of the generators used as input
-present.
+leading thus to a new PRNG that
+should improve the statistical properties of each
+generator taken alone.
+Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as present input.
+
\begin{algorithm}[h!]
($n$ bits)}
\KwOut{a configuration $x$ ($n$ bits)}
$x\leftarrow x^0$\;
-$k\leftarrow b + \textit{XORshift}(b)$\;
+$k\leftarrow b + PRNG_1(b)$\;
\For{$i=0,\dots,k$}
{
-$s\leftarrow{\textit{XORshift}(n)}$\;
+$s\leftarrow{PRNG_2(n)}$\;
$x\leftarrow{F_f(s,x)}$\;
}
return $x$\;
\end{small}
-\caption{PRNG with chaotic functions}
+\caption{An arbitrary round of $Old~ CI~ PRNG_f(PRNG_1,PRNG_2)$}
\label{CI Algorithm}
\end{algorithm}
+This generator is synthesized in Algorithm~\ref{CI Algorithm}.
+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
+between two outputs 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
+inputted generators $PRNG_i(k), i=1,2$,
+ which must return integers
+uniformly distributed
+into $\llbracket 1 ; k \rrbracket$.
+For instance, these PRNGs can be the \textit{XORshift}~\cite{Marsaglia2003},
+being a category of very fast PRNGs designed by George Marsaglia
+that repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number
+with a bit shifted version of it. Such a PRNG, which has a period of
+$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}.
+This XORshift, or any other reasonable PRNG, is used
+in our own generator to compute both the number of iterations between two
+outputs (provided by $PRNG_1$) and the strategy elements ($PRNG_2$).
+
+%This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones.
+
+
\begin{algorithm}[h!]
\begin{small}
\KwIn{the internal configuration $z$ (a 32-bit word)}
\end{algorithm}
-
-
-
-This generator is synthesized in Algorithm~\ref{CI Algorithm}.
-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
-\textit{XORshift}$(k)$ PRNGs~\cite{Marsaglia2003} that return integers
-uniformly distributed
-into $\llbracket 1 ; k \rrbracket$.
-\textit{XORshift} is a category of very fast PRNGs designed by George Marsaglia,
-which repeatedly uses the transform of exclusive or (XOR, $\oplus$) on a number
-with a bit shifted version of it. This PRNG, which has a period of
-$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used
-in our PRNG to compute the strategy length and the strategy elements.
-
-This former generator has successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07} ones.
-
-\subsection{Improving the Speed of the Former Generator}
-
-Instead of updating only one cell at each iteration, we can try to choose a
-subset of components and to update them together. Such an attempt leads
-to a kind of merger of the two sequences used in Algorithm
-\ref{CI Algorithm}. When the updating function is the vectorial negation,
-this algorithm can be rewritten 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 S^n,
-\end{array}
-\right.
-\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
-sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents
-the list of cells to update in the state $x^n$ of the system (represented
-as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th
-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 Oplus} is of
-ordinary use as a good elementary brick in various PRNGs. It corresponds
-to the following discrete dynamical system in chaotic iterations:
-
-\begin{equation}
-\forall n\in \mathds{N}^{\ast }, \forall i\in
-\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
-\begin{array}{ll}
- x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\
- \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
-\end{array}\right.
-\label{eq:generalIC}
-\end{equation}
-where $f$ is the vectorial negation and $\forall n \in \mathds{N}$,
-$\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that
-$k \in \mathcal{S}^n$ if and only if the $k-$th digit in the binary
-decomposition of $S^n$ is 1. Such chaotic iterations are more general
-than the ones presented in Definition \ref{Def:chaotic iterations} because, instead of updating only one term at each iteration,
-we select a subset of components to change.
-
-
-Obviously, replacing Algorithm~\ref{CI Algorithm} by
-Equation~\ref{equation Oplus}, which is possible when the iteration function is
-the vectorial negation, leads to a speed improvement. 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
-use of more general chaotic iterations to generate pseudorandom numbers
-faster, does not deflate their topological chaos properties.
-
-\subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
-\label{deuxième def}
-Let us consider the discrete dynamical systems in chaotic iterations having
-the general form: $\forall n\in \mathds{N}^{\ast }$, $ \forall i\in
-\llbracket1;\mathsf{N}\rrbracket $,
-
-\begin{equation}
- x_i^n=\left\{
-\begin{array}{ll}
- x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\
- \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
-\end{array}\right.
-\label{general CIs}
-\end{equation}
-
-In other words, at the $n^{th}$ iteration, only the cells whose id is
-contained into the set $S^{n}$ are iterated.
-
-Let us now rewrite these general chaotic iterations as usual discrete dynamical
-system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation
-is required in order to study the topological behavior of the system.
-
-Let us introduce the following function:
-\begin{equation}
-\begin{array}{cccc}
- \chi: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
- & (i,X) & \longmapsto & \left\{ \begin{array}{ll} 0 & \textrm{if }i \notin X, \\ 1 & \textrm{if }i \in X, \end{array}\right.
-\end{array}
-\end{equation}
-where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.
-
-Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
-$F_{f}: \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}}
-\longrightarrow \mathds{B}^{\mathsf{N}}$
-\begin{equation*}
-\begin{array}{rll}
- (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%
-\end{array}%
-\end{equation*}%
-where + and . are the Boolean addition and product operations, and $\overline{x}$
-is the negation of the Boolean $x$.
-Consider the phase space:
-\begin{equation}
-\mathcal{X} = \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N} \times
-\mathds{B}^\mathsf{N},
-\end{equation}
-\noindent and the map defined on $\mathcal{X}$:
-\begin{equation}
-G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), %\label{Gf} %%RAPH, j'ai viré ce label qui existe déjà avant...
-\end{equation}
-\noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
-(S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in
-\mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}$ and $i$ is the \emph{initial function}
-$i:(S^{n})_{n\in \mathds{N}} \in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow S^{0}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)$.
-Then the general chaotic iterations defined in Equation \ref{general CIs} can
-be described by the following discrete dynamical system:
-\begin{equation}
-\left\{
-\begin{array}{l}
-X^0 \in \mathcal{X} \\
-X^{k+1}=G_{f}(X^k).%
-\end{array}%
-\right.
-\end{equation}%
-
-Once more, a shift function appears as a component of these general chaotic
-iterations.
-
-To study the Devaney's chaos property, a distance between two points
-$X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
-Let us introduce:
-\begin{equation}
-d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
-\label{nouveau d}
-\end{equation}
-\noindent where $ \displaystyle{d_{e}(E,\check{E})} = \displaystyle{\sum_{k=1}^{\mathsf{N}%
- }\delta (E_{k},\check{E}_{k})}$ is once more the Hamming distance, and
-$ \displaystyle{d_{s}(S,\check{S})} = \displaystyle{\dfrac{9}{\mathsf{N}}%
- \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}$,
-%%RAPH : ici, j'ai supprimé tous les sauts à la ligne
-%% \begin{equation}
-%% \left\{
-%% \begin{array}{lll}
-%% \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
-%% }\delta (E_{k},\check{E}_{k})} \textrm{ is once more the Hamming distance}, \\
-%% \displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
-%% \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.%
-%% \end{array}%
-%% \right.
-%% \end{equation}
-where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
-$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
-
-
-\begin{proposition}
-The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
-\end{proposition}
-
-\begin{proof}
- $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
-too, thus $d$, as being the sum of two distances, will also be a distance.
- \begin{itemize}
-\item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then
-$d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then
-$\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$.
- \item $d_s$ is symmetric
-($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property
-of the symmetric difference.
-\item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$,
-and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$,
-we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle
-inequality is obtained.
- \end{itemize}
-\end{proof}
-
-
-Before being able to study the topological behavior of the general
-chaotic iterations, we must first establish that:
-
-\begin{proposition}
- For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on
-$\left( \mathcal{X},d\right)$.
-\end{proposition}
-
-
-\begin{proof}
-We use the sequential continuity.
-Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
-\mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
-G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
-G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy,
-thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of
-sequences).\newline
-As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
-to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
-d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline
-In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
-cell will change its state:
-$\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$
-
-In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
-\mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
-n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
-first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$
-
-Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
-identical and strategies $S^n$ and $S$ start with the same first term.\newline
-Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
-so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline
-\noindent We now prove that the distance between $\left(
-G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to
-0. Let $\varepsilon >0$. \medskip
-\begin{itemize}
-\item If $\varepsilon \geqslant 1$, we see that the distance
-between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
-strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
-\medskip
-\item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
-\varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
-\begin{equation*}
-\exists n_{2}\in \mathds{N},\forall n\geqslant
-n_{2},d_{s}(S^n,S)<10^{-(k+2)},
-\end{equation*}%
-thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
-\end{itemize}
-\noindent As a consequence, the $k+1$ first entries of the strategies of $%
-G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
-the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
-10^{-(k+1)}\leqslant \varepsilon $.
-
-In conclusion,
-%%RAPH : ici j'ai rajouté une ligne
-$
-\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
-,$ $\forall n\geqslant N_{0},$
-$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
-\leqslant \varepsilon .
-$
-$G_{f}$ is consequently continuous.
-\end{proof}
-
-
-It is now possible to study the topological behavior of the general chaotic
-iterations. We will prove that,
-
-\begin{theorem}
-\label{t:chaos des general}
- The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
-the Devaney's property of chaos.
-\end{theorem}
-
-Let us firstly prove the following lemma.
-
-\begin{lemma}[Strong transitivity]
-\label{strongTrans}
- For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can
-find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
-\end{lemma}
-
-\begin{proof}
- Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$.
-Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$,
-are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define
-$\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
-We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
-that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
-the form $(S',E')$ where $E'=E$ and $S'$ starts with
-$(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
-\begin{itemize}
- \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
- \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
-\end{itemize}
-Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$,
-where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
-claimed in the lemma.
-\end{proof}
-
-We can now prove the Theorem~\ref{t:chaos des general}.
-
-\begin{proof}[Theorem~\ref{t:chaos des general}]
-Firstly, strong transitivity implies transitivity.
-
-Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
-prove that $G_f$ is regular, it is sufficient to prove that
-there exists a strategy $\tilde S$ such that the distance between
-$(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
-$(\tilde S,E)$ is a periodic point.
-
-Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
-configuration that we obtain from $(S,E)$ after $t_1$ iterations of
-$G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$
-and $t_2\in\mathds{N}$ such
-that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
-
-Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
-of $S$ and the first $t_2$ terms of $S'$:
-%%RAPH : j'ai coupé la ligne en 2
-$$\tilde
-S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,$$$$\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
-is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
-$t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
-point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
-have $d((S,E),(\tilde S,E))<\epsilon$.
-\end{proof}
-
-
-\begin{color}{red}
-\section{Improving Statistical Properties Using Chaotic Iterations}
-
-
-\subsection{The CIPRNG family}
-
-Three categories of PRNGs have been derived from chaotic iterations. They are
-recalled in what follows.
-
-\subsubsection{Old CIPRNG}
-
-Let $\mathsf{N} = 4$. Some chaotic iterations are fulfilled to generate a sequence $\left(x^n\right)_{n\in\mathds{N}} \in \left(\mathds{B}^4\right)^\mathds{N}$ of Boolean vectors: the successive states of the iterated system. Some of these vectors are randomly extracted and their components constitute our pseudorandom bit flow~\cite{bgw09:ip}.
-Chaotic iterations are realized as follows. Initial state $x^0 \in \mathds{B}^4$ is a Boolean vector taken as a seed and chaotic strategy $\left(S^n\right)_{n\in\mathds{N}}\in \llbracket 1, 4 \rrbracket^\mathds{N}$ is constructed with $PRNG_2$. Lastly, iterate function $f$ is the vectorial Boolean negation.
-At each iteration, only the $S^n$-th component of state $x^n$ is updated. Finally, some $x^n$ are selected by a sequence $m^n$, provided by a second generator $PRNG_1$, as the pseudorandom bit sequence of our generator.
-
-The basic design procedure of the Old CI generator is summed up in Algorithm~\ref{Chaotic iteration}.
-The internal state is $x$, the output array is $r$. $a$ and $b$ are those computed by $PRNG_1$ and $PRNG_2$.
-
-
-\begin{algorithm}
-\textbf{Input:} the internal state $x$ (an array of 4-bit words)\\
-\textbf{Output:} an array $r$ of 4-bit words
-\begin{algorithmic}[1]
-
-\STATE$a\leftarrow{PRNG_1()}$;
-\STATE$m\leftarrow{a~mod~2+13}$;
-\WHILE{$i=0,\dots,m$}
-\STATE$b\leftarrow{PRNG_2()}$;
-\STATE$S\leftarrow{b~mod~4}$;
-\STATE$x_S\leftarrow{ \overline{x_S}}$;
-\ENDWHILE
-\STATE$r\leftarrow{x}$;
-\STATE return $r$;
-\medskip
-\caption{An arbitrary round of the old CI generator}
-\label{Chaotic iteration}
-\end{algorithmic}
-\end{algorithm}
-
-\subsubsection{New CIPRNG}
-
-The New CI generator is designed by the following process~\cite{bg10:ip}. 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 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 $\left(S^n\right)_{n\in\mathds{N}}\in \llbracket 1, 32 \rrbracket^\mathds{N}$ is
-an \emph{irregular decimation} of $PRNG_2$ sequence, as described in Algorithm~\ref{Chaotic iteration1}.
-
-Another time, at each iteration, only the $S^n$-th component of state $x^n$ is updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$.
+\subsection{A ``New CI PRNG''}
+
+In order to make the Old CI PRNG usable in practice, we have proposed
+an adapted version of the chaotic iteration based generator in~\cite{bg10:ip}.
+In this ``New CI PRNG'', we prevent a given bit from changing twice between two outputs.
+This new generator is designed by the following process.
+
+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 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 $\left(S^n\right)_{n\in\mathds{N}}\in
+\llbracket 1, 32 \rrbracket^\mathds{N}$ is
+an \emph{irregular decimation} of $PRNG_2$ sequence, as described in
+Algorithm~\ref{Chaotic iteration1}.
+
+Then, at each iteration, only the $S^n$-th component of state $x^n$ is
+updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$.
+Such a procedure is equivalent to achieving 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 $g_1(a)$ is an integer defined as in Eq.~\ref{Formula}.
+PRNGs. Lastly, the value $g(a)$ is an integer defined as in Eq.~\ref{Formula}.
+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}
-m^n = g_1(y^n)=
+m^n = g(y^n)=
\left\{
\begin{array}{l}
0 \text{ if }0 \leqslant{y^n}<{C^0_{32}},\\
}
\ENDFOR
\STATE$a\leftarrow{PRNG_1()}$\;
-\STATE$m\leftarrow{f(a)}$\;
-\STATE$k\leftarrow{m}$\;
+\STATE$k\leftarrow{g(a)}$\;
\WHILE{$i=0,\dots,k$}
\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
\end{algorithmic}
\end{algorithm}
+\subsection{Improving the Speed of the Former Generator}
-\subsubsection{Xor CIPRNG}
-
-Instead of updating only one cell at each iteration as Old CI and New CI, we can try to choose a
-subset of components and to update them together. Such an attempt leads
-to a kind of merger of the two random sequences. When the updating function is the vectorial negation,
-this algorithm can be rewritten as follows~\cite{arxivRCCGPCH}:
+Instead of updating only one cell at each iteration, we now propose to choose a
+subset of components and to update them together, for speed improvement. Such a proposition leads
+to a kind of merger of the two sequences used in Algorithms
+\ref{CI Algorithm} and \ref{Chaotic iteration1}. When the updating function is the vectorial negation,
+this algorithm can be rewritten as follows:
\begin{equation}
\left\{
\right.
\label{equation Oplus}
\end{equation}
-%This rewriting can be understood as follows. The $n-$th term $S^n$ of the
-%sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents
-%the list of cells to update in the state $x^n$ of the system (represented
-%as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th
-%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 Oplus} is of
-ordinary use as a good elementary brick in various PRNGs. It corresponds
-to the discrete dynamical system in chaotic iterations.
-
-\subsection{About some Well-known PRNGs}
-\label{The generation of pseudo-random sequence}
-
-
-
-
-Let us now give illustration on the fact that chaos appears to improve statistical properties.
-
-\subsection{Details of some Existing Generators}
-
-Here are the modules of PRNGs we have chosen to experiment.
-
-\subsubsection{LCG}
-This PRNG implements either the simple or the combined linear congruency generator (LCGs). The simple LCG is defined by the recurrence:
-\begin{equation}
-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{testU01}. In what follows, 2LCGs and 3LCGs refer as two (resp. three) combinations of such LCGs.
-For further details, see~\cite{combined_lcg}.
-
-\subsubsection{MRG}
-This module implements multiple recursive generators (MRGs), based on a linear recurrence of order $k$, modulo $m$~\cite{testU01}:
-\begin{equation}
-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 be used in this paper.
-
-\subsubsection{UCARRY}
-Generators based on linear recurrences with carry are implemented in this module. This includes the add-with-carry (AWC) generator, based on the recurrence:
-\begin{equation}
-\label{AWC}
-\begin{array}{l}
-x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
-c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
-the SWB generator, having the recurrence:
-\begin{equation}
-\label{SWB}
-\begin{array}{l}
-x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
-c^n=\left\{
-\begin{array}{l}
-1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
-0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
-and the SWC generator designed by R. Couture, which is based on the following recurrence:
-\begin{equation}
-\label{SWC}
-\begin{array}{l}
-x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
-c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}
-
-\subsubsection{GFSR}
-This module implements the generalized feedback shift register (GFSR) generator, that is:
-\begin{equation}
-x^n = x^{n-r} \oplus x^{n-k}
-\label{GFSR}
-\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
+sequence $S$, which is an integer of $\mathsf{N}$ binary digits, presents
+the list of cells to update in the state $x^n$ of the system (represented
+as an integer having $\mathsf{N}$ bits too). More precisely, the $k-$th
+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.
-\subsubsection{INV}
-Finally, this module implements the nonlinear inversive generator, as defined in~\cite{testU01}, which is:
+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:
\begin{equation}
-\label{INV}
-\begin{array}{l}
-x^n=\left\{
+\forall n\in \mathds{N}^{\ast }, \forall i\in
+\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
\begin{array}{ll}
-(a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\
-a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
-
-
-
-
-
-\subsection{Statistical tests}
-\label{Security analysis}
-
-%A theoretical proof for the randomness of a generator is impossible to give, therefore statistical inference based on observed sample sequences produced by the generator seems to be the best option.
-Considering the properties of binary random sequences, various statistical tests can be designed to evaluate the assertion that the sequence is generated by a perfectly random source. We have performed some statistical tests for the CIPRNGs proposed here. These tests include NIST suite~\cite{ANDREW2008} and DieHARD battery of tests~\cite{DieHARD}. For completeness and for reference, we give in the following subsection a brief description of each of the aforementioned tests.
-
-
-
-\subsubsection{NIST statistical tests suite}
+ x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\
+ \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
+\end{array}\right.
+\label{eq:generalIC}
+\end{equation}
+where $f$ is the vectorial negation and $\forall n \in \mathds{N}$,
+$\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ is such that
+$k \in \mathcal{S}^n$ if and only if the $k-$th digit in the binary
+decomposition of $S^n$ is 1. Such chaotic iterations are more general
+than the ones presented in Definition \ref{Def:chaotic iterations} because, instead of updating only one term at each iteration,
+we select a subset of components to change.
-Among the numerous standard tests for pseudo-randomness, a convincing way to show the randomness of the produced sequences is to confront them to the NIST (National Institute of Standards and Technology) statistical tests, being an up-to-date tests suite proposed by the Information Technology Laboratory (ITL). A new version of the Statistical tests suite has been released in August 11, 2010.
-The NIST tests suite SP 800-22 is a statistical package consisting of 15 tests. They were developed to test the randomness of binary sequences produced by hardware or software based cryptographic pseudorandom number generators. These tests focus on a variety of different types of non-randomness that could exist in a sequence.
+Obviously, replacing the previous CI PRNG Algorithms by
+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 to determine whether the
+use of more general chaotic iterations to generate pseudorandom numbers
+faster, does not deflate their topological chaos properties, has been
+investigated in Annex~\ref{A-deuxième def}, leading to the following result.
-For each statistical test, a set of $P-values$ (corresponding to the set of sequences) is produced.
-The interpretation of empirical results can be conducted in various ways.
-In this paper, the examination of the distribution of P-values to check for uniformity ($ P-value_{T}$) is used.
-The distribution of $P-values$ is examined to ensure uniformity.
-If $P-value_{T} \geqslant 0.0001$, then the sequences can be considered to be uniformly distributed.
+ \begin{theorem}
+ \label{t:chaos des general}
+ The general chaotic iterations defined in Equation~\ref{eq:generalIC}
+satisfy
+ the Devaney's property of chaos.
+ \end{theorem}
-In our experiments, 100 sequences (s = 100), each with 1,000,000-bit long, are generated and tested. If the $P-value_{T}$ of any test is smaller than 0.0001, the sequences are considered to be not good enough and the generating algorithm is not suitable for usage.
+%%RAF proof en supplementary, j'ai mis le theorem.
+% A vérifier
+% \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations}
+%\label{deuxième def}
+%The proof is given in Section~\ref{A-deuxième def} of the annex document.
+%% \label{deuxième def}
+%% Let us consider the discrete dynamical systems in chaotic iterations having
+%% the general form: $\forall n\in \mathds{N}^{\ast }$, $ \forall i\in
+%% \llbracket1;\mathsf{N}\rrbracket $,
+%% \begin{equation}
+%% x_i^n=\left\{
+%% \begin{array}{ll}
+%% x_i^{n-1} & \text{ if } i \notin \mathcal{S}^n \\
+%% \left(f(x^{n-1})\right)_{S^n} & \text{ if }i \in \mathcal{S}^n.
+%% \end{array}\right.
+%% \label{general CIs}
+%% \end{equation}
+%% In other words, at the $n^{th}$ iteration, only the cells whose id is
+%% contained into the set $S^{n}$ are iterated.
-\subsubsection{DieHARD battery of tests}
-The DieHARD battery of tests has been the most sophisticated standard for over a decade. Because of the stringent requirements in the DieHARD tests suite, a generator passing this battery of
-tests can be considered good as a rule of thumb.
+%% Let us now rewrite these general chaotic iterations as usual discrete dynamical
+%% system of the form $X^{n+1}=f(X^n)$ on an ad hoc metric space. Such a formulation
+%% is required in order to study the topological behavior of the system.
-The DieHARD battery of tests consists of 18 different independent statistical tests. This collection
- of tests is based on assessing the randomness of bits comprising 32-bit integers obtained from
-a random number generator. Each test requires $2^{23}$ 32-bit integers in order to run the full set
-of tests. Most of the tests in DieHARD return a $P-value$, which should be uniform on $[0,1)$ if the input file
-contains truly independent random bits. These $P-values$ are obtained by
-$P=F(X)$, where $F$ is the assumed distribution of the sample random variable $X$ (often normal).
-But that assumed $F$ is just an asymptotic approximation, for which the fit will be worst
-in the tails. Thus occasional $P-values$ near 0 or 1, such as 0.0012 or 0.9983, can occur.
-An individual test is considered to be failed if the $P-value$ approaches 1 closely, for example $P>0.9999$.
+%% Let us introduce the following function:
+%% \begin{equation}
+%% \begin{array}{cccc}
+%% \chi: & \llbracket 1; \mathsf{N} \rrbracket \times \mathcal{P}\left(\llbracket 1; \mathsf{N} \rrbracket\right) & \longrightarrow & \mathds{B}\\
+%% & (i,X) & \longmapsto & \left\{ \begin{array}{ll} 0 & \textrm{if }i \notin X, \\ 1 & \textrm{if }i \in X, \end{array}\right.
+%% \end{array}
+%% \end{equation}
+%% where $\mathcal{P}\left(X\right)$ is for the powerset of the set $X$, that is, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$.
+
+%% Given a function $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, define the function:
+%% $F_{f}: \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}}
+%% \longrightarrow \mathds{B}^{\mathsf{N}}$
+%% \begin{equation*}
+%% \begin{array}{rll}
+%% (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%
+%% \end{array}%
+%% \end{equation*}%
+%% where + and . are the Boolean addition and product operations, and $\overline{x}$
+%% is the negation of the Boolean $x$.
+%% Consider the phase space:
+%% \begin{equation}
+%% \mathcal{X} = \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N} \times
+%% \mathds{B}^\mathsf{N},
+%% \end{equation}
+%% \noindent and the map defined on $\mathcal{X}$:
+%% \begin{equation}
+%% G_f\left(S,E\right) = \left(\sigma(S), F_f(i(S),E)\right), %\label{Gf} %%RAPH, j'ai viré ce label qui existe déjà avant...
+%% \end{equation}
+%% \noindent where $\sigma$ is the \emph{shift} function defined by $\sigma
+%% (S^{n})_{n\in \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow (S^{n+1})_{n\in
+%% \mathds{N}}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}$ and $i$ is the \emph{initial function}
+%% $i:(S^{n})_{n\in \mathds{N}} \in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)^\mathds{N}\longrightarrow S^{0}\in \mathcal{P}\left(\llbracket 1 ; \mathsf{N} \rrbracket\right)$.
+%% Then the general chaotic iterations defined in Equation \ref{general CIs} can
+%% be described by the following discrete dynamical system:
+%% \begin{equation}
+%% \left\{
+%% \begin{array}{l}
+%% X^0 \in \mathcal{X} \\
+%% X^{k+1}=G_{f}(X^k).%
+%% \end{array}%
+%% \right.
+%% \end{equation}%
+%% Once more, a shift function appears as a component of these general chaotic
+%% iterations.
-\subsection{Results and discussion}
-\label{Results and discussion}
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
-\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
- \hline\hline
-Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
-\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
-NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15} & 14/15 & 14/15 & 14/15 & 14/15& 14/15& 14/15 \\ \hline
-DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
-\end{tabular}
-\end{table*}
+%% To study the Devaney's chaos property, a distance between two points
+%% $X = (S,E), Y = (\check{S},\check{E})$ of $\mathcal{X}$ must be defined.
+%% Let us introduce:
+%% \begin{equation}
+%% d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
+%% \label{nouveau d}
+%% \end{equation}
+%% \noindent where $ \displaystyle{d_{e}(E,\check{E})} = \displaystyle{\sum_{k=1}^{\mathsf{N}%
+%% }\delta (E_{k},\check{E}_{k})}$ is once more the Hamming distance, and
+%% $ \displaystyle{d_{s}(S,\check{S})} = \displaystyle{\dfrac{9}{\mathsf{N}}%
+%% \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}$,
+%% %%RAPH : ici, j'ai supprimé tous les sauts à la ligne
+%% %% \begin{equation}
+%% %% \left\{
+%% %% \begin{array}{lll}
+%% %% \displaystyle{d_{e}(E,\check{E})} & = & \displaystyle{\sum_{k=1}^{\mathsf{N}%
+%% %% }\delta (E_{k},\check{E}_{k})} \textrm{ is once more the Hamming distance}, \\
+%% %% \displaystyle{d_{s}(S,\check{S})} & = & \displaystyle{\dfrac{9}{\mathsf{N}}%
+%% %% \sum_{k=1}^{\infty }\dfrac{|S^k\Delta {S}^k|}{10^{k}}}.%
+%% %% \end{array}%
+%% %% \right.
+%% %% \end{equation}
+%% where $|X|$ is the cardinality of a set $X$ and $A\Delta B$ is for the symmetric difference, defined for sets A, B as
+%% $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$.
+
+
+%% \begin{proposition}
+%% The function $d$ defined in Eq.~\ref{nouveau d} is a metric on $\mathcal{X}$.
+%% \end{proposition}
+
+%% \begin{proof}
+%% $d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
+%% too, thus $d$, as being the sum of two distances, will also be a distance.
+%% \begin{itemize}
+%% \item Obviously, $d_s(S,\check{S})\geqslant 0$, and if $S=\check{S}$, then
+%% $d_s(S,\check{S})=0$. Conversely, if $d_s(S,\check{S})=0$, then
+%% $\forall k \in \mathds{N}, |S^k\Delta {S}^k|=0$, and so $\forall k, S^k=\check{S}^k$.
+%% \item $d_s$ is symmetric
+%% ($d_s(S,\check{S})=d_s(\check{S},S)$) due to the commutative property
+%% of the symmetric difference.
+%% \item Finally, $|S \Delta S''| = |(S \Delta \varnothing) \Delta S''|= |S \Delta (S'\Delta S') \Delta S''|= |(S \Delta S') \Delta (S' \Delta S'')|\leqslant |S \Delta S'| + |S' \Delta S''|$,
+%% and so for all subsets $S,S',$ and $S''$ of $\llbracket 1, \mathsf{N} \rrbracket$,
+%% we have $d_s(S,S'') \leqslant d_e(S,S')+d_s(S',S'')$, and the triangle
+%% inequality is obtained.
+%% \end{itemize}
+%% \end{proof}
+
+
+%% Before being able to study the topological behavior of the general
+%% chaotic iterations, we must first establish that:
+
+%% \begin{proposition}
+%% For all $f:\mathds{B}^\mathsf{N} \longrightarrow \mathds{B}^\mathsf{N} $, the function $G_f$ is continuous on
+%% $\left( \mathcal{X},d\right)$.
+%% \end{proposition}
+
+
+%% \begin{proof}
+%% We use the sequential continuity.
+%% Let $(S^n,E^n)_{n\in \mathds{N}}$ be a sequence of the phase space $%
+%% \mathcal{X}$, which converges to $(S,E)$. We will prove that $\left(
+%% G_{f}(S^n,E^n)\right) _{n\in \mathds{N}}$ converges to $\left(
+%% G_{f}(S,E)\right) $. Let us remark that for all $n$, $S^n$ is a strategy,
+%% thus, we consider a sequence of strategies (\emph{i.e.}, a sequence of
+%% sequences).\newline
+%% As $d((S^n,E^n);(S,E))$ converges to 0, each distance $d_{e}(E^n,E)$ and $d_{s}(S^n,S)$ converges
+%% to 0. But $d_{e}(E^n,E)$ is an integer, so $\exists n_{0}\in \mathds{N},$ $%
+%% d_{e}(E^n,E)=0$ for any $n\geqslant n_{0}$.\newline
+%% In other words, there exists a threshold $n_{0}\in \mathds{N}$ after which no
+%% cell will change its state:
+%% $\exists n_{0}\in \mathds{N},n\geqslant n_{0}\Rightarrow E^n = E.$
+
+%% In addition, $d_{s}(S^n,S)\longrightarrow 0,$ so $\exists n_{1}\in %
+%% \mathds{N},d_{s}(S^n,S)<10^{-1}$ for all indexes greater than or equal to $%
+%% n_{1}$. This means that for $n\geqslant n_{1}$, all the $S^n$ have the same
+%% first term, which is $S^0$: $\forall n\geqslant n_{1},S_0^n=S_0.$
+
+%% Thus, after the $max(n_{0},n_{1})^{th}$ term, states of $E^n$ and $E$ are
+%% identical and strategies $S^n$ and $S$ start with the same first term.\newline
+%% Consequently, states of $G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are equal,
+%% so, after the $max(n_0, n_1)^{th}$ term, the distance $d$ between these two points is strictly less than 1.\newline
+%% \noindent We now prove that the distance between $\left(
+%% G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is convergent to
+%% 0. Let $\varepsilon >0$. \medskip
+%% \begin{itemize}
+%% \item If $\varepsilon \geqslant 1$, we see that the distance
+%% between $\left( G_{f}(S^n,E^n)\right) $ and $\left( G_{f}(S,E)\right) $ is
+%% strictly less than 1 after the $max(n_{0},n_{1})^{th}$ term (same state).
+%% \medskip
+%% \item If $\varepsilon <1$, then $\exists k\in \mathds{N},10^{-k}\geqslant
+%% \varepsilon > 10^{-(k+1)}$. But $d_{s}(S^n,S)$ converges to 0, so
+%% \begin{equation*}
+%% \exists n_{2}\in \mathds{N},\forall n\geqslant
+%% n_{2},d_{s}(S^n,S)<10^{-(k+2)},
+%% \end{equation*}%
+%% thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal.
+%% \end{itemize}
+%% \noindent As a consequence, the $k+1$ first entries of the strategies of $%
+%% G_{f}(S^n,E^n)$ and $G_{f}(S,E)$ are the same ($G_{f}$ is a shift of strategies) and due to the definition of $d_{s}$, the floating part of
+%% the distance between $(S^n,E^n)$ and $(S,E)$ is strictly less than $%
+%% 10^{-(k+1)}\leqslant \varepsilon $.
+
+%% In conclusion,
+%% %%RAPH : ici j'ai rajouté une ligne
+%% %%TOF : ici j'ai rajouté un commentaire
+%% %%TOF : ici aussi
+%% $
+%% \forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
+%% ,$ $\forall n\geqslant N_{0},$
+%% $ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
+%% \leqslant \varepsilon .
+%% $
+%% $G_{f}$ is consequently continuous.
+%% \end{proof}
+
+
+%% It is now possible to study the topological behavior of the general chaotic
+%% iterations. We will prove that,
+
+%% \begin{theorem}
+%% \label{t:chaos des general}
+%% The general chaotic iterations defined on Equation~\ref{general CIs} satisfy
+%% the Devaney's property of chaos.
+%% \end{theorem}
+
+%% Let us firstly prove the following lemma.
+
+%% \begin{lemma}[Strong transitivity]
+%% \label{strongTrans}
+%% For all couples $X,Y \in \mathcal{X}$ and any neighborhood $V$ of $X$, we can
+%% find $n \in \mathds{N}^*$ and $X' \in V$ such that $G^n(X')=Y$.
+%% \end{lemma}
+
+%% \begin{proof}
+%% Let $X=(S,E)$, $\varepsilon>0$, and $k_0 = \lfloor log_{10}(\varepsilon)+1 \rfloor$.
+%% Any point $X'=(S',E')$ such that $E'=E$ and $\forall k \leqslant k_0, S'^k=S^k$,
+%% are in the open ball $\mathcal{B}\left(X,\varepsilon\right)$. Let us define
+%% $\check{X} = \left(\check{S},\check{E}\right)$, where $\check{X}= G^{k_0}(X)$.
+%% We denote by $s\subset \llbracket 1; \mathsf{N} \rrbracket$ the set of coordinates
+%% that are different between $\check{E}$ and the state of $Y$. Thus each point $X'$ of
+%% the form $(S',E')$ where $E'=E$ and $S'$ starts with
+%% $(S^0, S^1, \hdots, S^{k_0},s,\hdots)$, verifies the following properties:
+%% \begin{itemize}
+%% \item $X'$ is in $\mathcal{B}\left(X,\varepsilon\right)$,
+%% \item the state of $G_f^{k_0+1}(X')$ is the state of $Y$.
+%% \end{itemize}
+%% Finally the point $\left(\left(S^0, S^1, \hdots, S^{k_0},s,s^0, s^1, \hdots\right); E\right)$,
+%% where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties
+%% claimed in the lemma.
+%% \end{proof}
+
+%% We can now prove the Theorem~\ref{t:chaos des general}.
+
+%% \begin{proof}[Theorem~\ref{t:chaos des general}]
+%% Firstly, strong transitivity implies transitivity.
+
+%% Let $(S,E) \in\mathcal{X}$ and $\varepsilon >0$. To
+%% prove that $G_f$ is regular, it is sufficient to prove that
+%% there exists a strategy $\tilde S$ such that the distance between
+%% $(\tilde S,E)$ and $(S,E)$ is less than $\varepsilon$, and such that
+%% $(\tilde S,E)$ is a periodic point.
+
+%% Let $t_1=\lfloor-\log_{10}(\varepsilon)\rfloor$, and let $E'$ be the
+%% configuration that we obtain from $(S,E)$ after $t_1$ iterations of
+%% $G_f$. As $G_f$ is strongly transitive, there exists a strategy $S'$
+%% and $t_2\in\mathds{N}$ such
+%% that $E$ is reached from $(S',E')$ after $t_2$ iterations of $G_f$.
+
+%% Consider the strategy $\tilde S$ that alternates the first $t_1$ terms
+%% of $S$ and the first $t_2$ terms of $S'$:
+%% %%RAPH : j'ai coupé la ligne en 2
+%% $$\tilde
+%% S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,$$$$\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
+%% is clear that $(\tilde S,E)$ is obtained from $(\tilde S,E)$ after
+%% $t_1+t_2$ iterations of $G_f$. So $(\tilde S,E)$ is a periodic
+%% point. Since $\tilde S_t=S_t$ for $t<t_1$, by the choice of $t_1$, we
+%% have $d((S,E),(\tilde S,E))<\epsilon$.
+%% \end{proof}
+
+
+
+
+%%RAF : mis en supplementary
+
+
+%\section{Statistical Improvements Using Chaotic Iterations}
+%\label{The generation of pseudorandom sequence}
+%The content is this section is given in Section~\ref{A-The generation of pseudorandom sequence} of the annex document.
+The reasons to desire chaos to achieve randomness are given in Annex~\ref{A-The generation of pseudorandom sequence}.
+
+%% \label{The generation of pseudorandom sequence}
+
+
+%% Let us now explain why we have reasonable ground to believe that chaos
+%% can improve statistical properties.
+%% 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 close one to another) in the tested sequence that would indicate a deviation from the assumption of randomness.
+%% \end{itemize}
+
+%% \item \textbf{Transitivity}. This topological property previously introduced 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 the 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 the 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 oscillate 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. Once again, a similar objective has led to two different
+%% rewritting 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 the complexity of the topological dynamical system, whereas
+%% the Shannon approach comes to mind when defining the following test~\cite{Nist10}:
+%% \begin{itemize}
+%% \item \textbf{Approximate Entropy Test}. Compare the frequency of the 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{guyeux12:bc} 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 generators (LCGs) will be used.
+%% They are defined by the following recurrence:
+%% \begin{equation}
+%% 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 inferior to
+%% $m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer to two (resp. three)
+%% combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
-Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the results on the batteries recalled above, indicating that almost all the PRNGs cannot pass all their tests. In other words, the statistical quality of these PRNGs cannot fulfill the up-to-date standards presented previously. We will show that the CIPRNG can solve this issue.
+%% Secondly, the multiple recursive generators (MRGs) which will be used,
+%% 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 .
+%% \label{MRG}
+%% \end{equation}
+%% The combination of two MRGs (referred as 2MRGs) is also used in these experiments.
-To illustrate the effects of this CIPRNG in detail, experiments will be divided in three parts:
-\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 $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket,$
-\begin{equation}
-\begin{array}{l}
-x_i^n=\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}
+%% 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:
+%% \begin{equation}
+%% \label{AWC}
+%% \begin{array}{l}
+%% x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
+%% c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
+%% the SWB generator, having the recurrence:
+%% \begin{equation}
+%% \label{SWB}
+%% \begin{array}{l}
+%% x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
+%% c^n=\left\{
+%% \begin{array}{l}
+%% 1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
+%% 0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
+%% and the SWC generator, which is based on the following recurrence:
+%% \begin{equation}
+%% \label{SWC}
+%% \begin{array}{l}
+%% x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
+%% c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}
+%% Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
+%% \begin{equation}
+%% x^n = x^{n-r} \oplus x^{n-k} .
+%% \label{GFSR}
+%% \end{equation}
-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.
-
-\subsubsection{Tests based on the Single CIPRNG}
-
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
-\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
- \hline
-Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
-\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
-Old CIPRNG\\ \hline \hline
-NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
-DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
-New CIPRNG\\ \hline \hline
-NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
-DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
-Xor CIPRNG\\ \hline\hline
-NIST & 14/15*& \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} \\ \hline
-DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
-\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}
+%% Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:
-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}
+%% \begin{equation}
+%% \label{INV}
+%% \begin{array}{l}
+%% x^n=\left\{
+%% \begin{array}{ll}
+%% (a^1 + a^2 / z^{n-1})~mod~m & \text{if}~ z^{n-1} \neq 0 \\
+%% a^1 & \text{if}~ z^{n-1} = 0 .\end{array} \right. \end{array}\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}
+%% \begin{table}
+%% \renewcommand{\arraystretch}{1.3}
+%% \caption{TestU01 Statistical Test Failures}
+%% \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}
-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}.
+%% \begin{table}
+%% \renewcommand{\arraystretch}{1.3}
+%% \caption{TestU01 Statistical Test Failures 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}
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{Functional power $m$ making it possible to pass the whole NIST battery}
-\label{threshold}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
- \hline
-Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3 & MRG2 \\ \hline\hline
-Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline
-\end{tabular}
-\end{table*}
-\subsubsection{Results Summary}
-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}
-\end{color}
-\section{Efficient PRNG based on Chaotic Iterations}
+%% \subsection{Statistical tests}
+%% \label{Security analysis}
+
+%% Three batteries of tests are reputed and regularly used
+%% to evaluate the statistical properties of newly designed pseudorandom
+%% number generators. These batteries are named DieHard~\cite{Marsaglia1996},
+%% the NIST suite~\cite{ANDREW2008}, and the most stringent one called
+%% TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
+
+
+
+%% \label{Results and discussion}
+%% \begin{table*}
+%% \renewcommand{\arraystretch}{1.3}
+%% \caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
+%% \label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
+%% \centering
+%% \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
+%% \hline\hline
+%% Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%% \backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
+%% NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15} & 14/15 & 14/15 & 14/15 & 14/15& 14/15& 14/15 \\ \hline
+%% DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
+%% \end{tabular}
+%% \end{table*}
+
+%% Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the
+%% results on the two first batteries recalled above, indicating that all the PRNGs presented
+%% in the previous section
+%% 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}
+%% %
+%% 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 the ``New CI'' generators.
+%% Concerning the ``Xor CI PRNG'', the score is less spectacular. Because of a large speed improvement, the 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).
+
+
+%% \begin{table*}
+%% \renewcommand{\arraystretch}{1.3}
+%% \caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
+%% \label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
+%% \centering
+%% \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
+%% \hline
+%% Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%% \backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
+%% Old CIPRNG\\ \hline \hline
+%% NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%% DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
+%% New CIPRNG\\ \hline \hline
+%% NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%% DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
+%% Xor CIPRNG\\ \hline\hline
+%% NIST & 14/15*& \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} \\ \hline
+%% DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
+%% \end{tabular}
+%% \end{table*}
+
+
+%% We have then investigated in~\cite{bfg12a:ip} if it were 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{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|}
+%% \hline
+%% Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3 & MRG2 \\ \hline\hline
+%% Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline
+%% \end{tabular}
+%% \end{table*}
+
+%% 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.
+
+
+%% The 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 to as CIPRNG, or ``the proposed PRNG'', if this statement does not
+%% raise ambiguity.
+
+
+\section{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}.
+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.
+
+
\section{Efficient PRNGs based on Chaotic Iterations on GPU}
\label{sec:efficient PRNG gpu}
\label{algo:gpu_kernel2}
\end{algorithm}
-\subsection{Theoretical Evaluation of the Improved Version}
+\subsection{Chaos Evaluation of the Improved Version}
A run of Algorithm~\ref{algo:gpu_kernel2} consists in an operation ($x=x\oplus t$) having
the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
\section{Security Analysis}
-\label{sec:security analysis}
+This section is dedicated to the security analysis of the
+ proposed PRNGs.%, both from a theoretical and from a practical point of view.
+
+%\subsection{Theoretical Proof of Security}
+\label{sec:security analysis}
+
+The standard definition
+ of {\it indistinguishability} used here is the classical one as defined for
+ instance in~\cite[chapter~3]{Goldreich}.
+ This property shows that predicting the future results of the PRNG
+ cannot be done in a reasonable time compared to the generation time. It is important to emphasize that this
+ is a relative notion between breaking time and the sizes of the
+ keys/seeds. Of course, if small keys or seeds are chosen, the system can
+ be broken in practice. But it also means that if the keys/seeds are large
+ enough, the system is secured.
+As a complement, an example of a concrete practical evaluation of security
+is outlined in Annex~\ref{A-sec:Practicak evaluation}.
In this section the concatenation of two strings $u$ and $v$ is classically
denoted by $uv$.
internal coin tosses of $D$.
\end{definition}
-Intuitively, it means that there is no polynomial time algorithm that can
-distinguish a perfect uniform random generator from $G$ with a non
-negligible probability. The interested reader is referred
-to~\cite[chapter~3]{Goldreich} for more information. Note that it is
-quite easily possible to change the function $\ell$ into any polynomial
-function $\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}.
+Intuitively, it means that there is no polynomial time algorithm that can
+distinguish a perfect uniform random generator from $G$ with a non negligible
+probability. An equivalent formulation of this well-known security property
+means that it is possible \emph{in practice} to predict the next bit of the
+generator, knowing all the previously produced ones. The interested reader is
+referred to~\cite[chapter~3]{Goldreich} for more information. Note that it is
+quite easily possible to change the function $\ell$ into any polynomial function
+$\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}.
The generation schema developed in (\ref{equation Oplus}) is based on a
pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,
\end{proposition}
\begin{proof}
-The proposition is proved by contraposition. Assume that $X$ is not
+The proposition is proven by contraposition. Assume that $X$ is not
secure. By Definition, there exists a polynomial time probabilistic
algorithm $D$, a positive polynomial $p$, such that for all $k_0$ there exists
$N\geq \frac{k_0}{2}$ satisfying
\end{proof}
+
+%\subsection{Practical Security Evaluation}
+%\label{sec:Practicak evaluation}
+%This subsection is given in Section
+A example of a practical security evaluation is outlined in
+Annex~\ref{A-sec:Practicak evaluation}.
+%%RAF mis en annexe
+
+
+%% Pseudorandom generators based on Eq.~\eqref{equation Oplus} are thus cryptographically secure when
+%% they are XORed with an already cryptographically
+%% secure PRNG. But, as stated previously,
+%% such a property does not mean that, whatever the
+%% key size, no attacker can predict the next bit
+%% knowing all the previously released ones.
+%% However, given a key size, it is possible to
+%% measure in practice the minimum duration needed
+%% for an attacker to break a cryptographically
+%% secure PRNG, if we know the power of his/her
+%% machines. Such a concrete security evaluation
+%% is related to the $(T,\varepsilon)-$security
+%% notion, which is recalled and evaluated in what
+%% follows, for the sake of completeness.
+
+%% Let us firstly recall that,
+%% \begin{definition}
+%% Let $\mathcal{D} : \mathds{B}^M \longrightarrow \mathds{B}$ be a probabilistic algorithm that runs
+%% in time $T$.
+%% Let $\varepsilon > 0$.
+%% $\mathcal{D}$ is called a $(T,\varepsilon)-$distinguishing attack on pseudorandom
+%% generator $G$ if
+
+%% \begin{flushleft}
+%% $\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right.$
+%% \end{flushleft}
+
+%% \begin{flushright}
+%% $ - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$
+%% \end{flushright}
+
+%% \noindent where the probability is taken over the internal coin flips of $\mathcal{D}$, and the notation
+%% ``$\in_R$'' indicates the process of selecting an element at random and uniformly over the
+%% corresponding set.
+%% \end{definition}
+
+%% Let us recall that the running time of a probabilistic algorithm is defined to be the
+%% maximum of the expected number of steps needed to produce an output, maximized
+%% over all inputs; the expected number is averaged over all coin flips made by the algorithm~\cite{Knuth97}.
+%% We are now able to define the notion of cryptographically secure PRNGs:
+
+%% \begin{definition}
+%% A pseudorandom generator is $(T,\varepsilon)-$secure if there exists no $(T,\varepsilon)-$distinguishing attack on this pseudorandom generator.
+%% \end{definition}
+
+
+
+
+
+
+
+%% Suppose now that the PRNG of Eq.~\eqref{equation Oplus} will work during
+%% $M=100$ time units, and that during this period,
+%% an attacker can realize $10^{12}$ clock cycles.
+%% We thus wonder whether, during the PRNG's
+%% lifetime, the attacker can distinguish this
+%% sequence from a truly random one, with a probability
+%% greater than $\varepsilon = 0.2$.
+%% We consider that $N$ has 900 bits.
+
+%% Predicting the next generated bit knowing all the
+%% previously released ones by Eq.~\eqref{equation Oplus} is obviously equivalent to predicting the
+%% next bit in the BBS generator, which
+%% is cryptographically secure. More precisely, it
+%% is $(T,\varepsilon)-$secure: no
+%% $(T,\varepsilon)-$distinguishing attack can be
+%% successfully realized on this PRNG, if~\cite{Fischlin}
+%% \begin{equation}
+%% T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
+%% \label{mesureConcrete}
+%% \end{equation}
+%% where $M$ is the length of the output ($M=100$ in
+%% our example), and $L(N)$ is equal to
+%% $$
+%% 2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln~ 2)^\frac{1}{3} \times (ln(N~ln~ 2))^\frac{2}{3}\right)
+%% $$
+%% is the number of clock cycles to factor a $N-$bit
+%% integer.
+
+
+
+
+%% A direct numerical application shows that this attacker
+%% cannot achieve its $(10^{12},0.2)$ distinguishing
+%% attack in that context.
+
+
+
\section{Cryptographical Applications}
\subsection{A Cryptographically Secure PRNG for GPU}
where $S^n$ is referred in this algorithm as $t$: each iteration of this
PRNG ends with $x = x \wedge t$. This $S^n$ is only constituted
by secure bits produced by the BBS generator, and thus, due to
-Proposition~\ref{cryptopreuve}, the resulted PRNG is cryptographically
-secure.
-
-
+Proposition~\ref{cryptopreuve}, the resulted PRNG is
+cryptographically secure.
+
+As stated before, even if the proposed PRNG is cryptocaphically
+secure, it does not mean that such a generator
+can be used as described here when attacks are
+awaited. The problem is to determine the minimum
+time required for an attacker, with a given
+computational power, to predict under a probability
+lower than 0.5 the $n+1$th bit, knowing the $n$
+previous ones. The proposed GPU generator will be
+useful in a security context, at least in some
+situations where a secret protected by a pseudorandom
+keystream is rapidly obsolete, if this time to
+predict the next bit is large enough when compared
+to both the generation and transmission times.
+It is true that the prime numbers used in the last
+section are very small compared to up-to-date
+security recommendations. However the attacker has not
+access to each BBS, but to the output produced
+by Algorithm~\ref{algo:bbs_gpu}, which is far
+more complicated than a simple BBS. Indeed, to
+determine if this cryptographically secure PRNG
+on GPU can be useful in security context with the
+proposed parameters, or if it is only a very fast
+and statistically perfect generator on GPU, its
+$(T,\varepsilon)-$security must be determined, and
+a formulation similar to Annex~\ref{A-sec:Practicak evaluation} %.Eq.\eqref{mesureConcrete}
+must be established. Authors
+hope to achieve this difficult task in a future
+work.
-\begin{color}{red}
-\subsection{Practical Security Evaluation}
-
-Suppose now that the PRNG will work during
-$M=100$ time units, and that during this period,
-an attacker can realize $10^{12}$ clock cycles.
-We thus wonder whether, during the PRNG's
-lifetime, the attacker can distinguish this
-sequence from truly random one, with a probability
-greater than $\varepsilon = 0.2$.
-We consider that $N$ has 900 bits.
-
-The random process is the BBS generator, which
-is cryptographically secure. More precisely, it
-is $(T,\varepsilon)-$secure: no
-$(T,\varepsilon)-$distinguishing attack can be
-successfully realized on this PRNG, if~\cite{Fischlin}
-$$
-T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
-$$
-where $M$ is the length of the output ($M=100$ in
-our example), and $L(N)$ is equal to
-$$
-2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right)
-$$
-is the number of clock cycles to factor a $N-$bit
-integer.
-
-A direct numerical application shows that this attacker
-cannot achieve its $(10^{12},0.2)$ distinguishing
-attack in that context.
-
-\end{color}
\subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
\label{Blum-Goldwasser}
Furthermore, we have shown that when the inputted generator is cryptographically
secure, then it is the case too for the PRNG we propose, thus leading to
the possibility to develop fast and secure PRNGs using the GPU architecture.
-\begin{color}{red} An improvement of the Blum-Goldwasser cryptosystem, making it
-behaves chaotically, has finally been proposed. \end{color}
+An improvement of the Blum-Goldwasser cryptosystem, making it
+behave chaotically, has finally been proposed.
In future work we plan to extend this research, building a parallel PRNG for clusters or
grid computing. Topological properties of the various proposed generators will be investigated,
