-\documentclass{article}
+%\documentclass{article}
+\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{fullpage}
\usepackage{amscd}
\usepackage{moreverb}
\usepackage{commath}
-\usepackage{algorithm2e}
+\usepackage[ruled,vlined]{algorithm2e}
\usepackage{listings}
\usepackage[standard]{ntheorem}
+\usepackage{algorithmic}
+\usepackage{slashbox}
% Pour mathds : les ensembles IR, IN, etc.
\usepackage{dsfont}
\begin{document}
\author{Jacques M. Bahi, Rapha\"{e}l Couturier, Christophe
-Guyeux, and Pierre-Cyrille Heam\thanks{Authors in alphabetic order}}
+Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}
-\maketitle
+\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
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 billions of random numbers per second on Tesla C1060 and NVidia GTX280
+about 20 billion of random numbers per second on Tesla C1060 and NVidia GTX280
cards.
-It is finally established that, under reasonable assumptions, the proposed PRNG can be cryptographically
+It is then established that, under reasonable assumptions, the proposed PRNG can be cryptographically
secure.
+A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is finally proposed.
\end{abstract}
+}
+
+\maketitle
+
+\IEEEdisplaynotcompsoctitleabstractindextext
+\IEEEpeerreviewmaketitle
+
\section{Introduction}
-Randomness is of importance in many fields as scientific simulations or cryptography.
+Randomness is of importance in many fields such as scientific simulations or cryptography.
``Random numbers'' can mainly be generated either by a deterministic and reproducible algorithm
called a pseudorandom number generator (PRNG), or by a physical non-deterministic
process having all the characteristics of a random noise, called a truly random number
In this paper, we focus on reproducible generators, useful for instance in
Monte-Carlo based simulators or in several cryptographic schemes.
These domains need PRNGs that are statistically irreproachable.
-On some fields as in numerical simulations, speed is a strong requirement
+In some fields such as in numerical simulations, speed is a strong requirement
that is usually attained by using parallel architectures. In that case,
-a recurrent problem is that a deflate of the statistical qualities is often
+a recurrent problem is that a deflation of the statistical qualities is often
reported, when the parallelization of a good PRNG is realized.
This is why ad-hoc PRNGs for each possible architecture must be found to
achieve both speed and randomness.
On the other side, speed is not the main requirement in cryptography: the great
-need is to define \emph{secure} generators being able to withstand malicious
+need is to define \emph{secure} generators able to withstand malicious
attacks. Roughly speaking, an attacker should not be able in practice to make
the distinction between numbers obtained with the secure generator and a true random
sequence.
-Finally, a small part of the community working in this domain focus on a
+Finally, a small part of the community working in this domain focuses on a
third requirement, that is to define chaotic generators.
The main idea is to take benefits from a chaotic dynamical system to obtain a
-generator that is unpredictable, disordered, sensible to its seed, or in other words chaotic.
+generator that is unpredictable, disordered, sensible to its seed, or in other word chaotic.
Their desire is to map a given chaotic dynamics into a sequence that seems random
and unassailable due to chaos.
However, the chaotic maps used as a pattern are defined in the real line
The authors' opinion is that topological properties of disorder, as they are
properly defined in the mathematical theory of chaos, can reinforce the quality
of a PRNG. But they are not substitutable for security or statistical perfection.
-Indeed, to the authors' point of view, such properties can be useful in the two following situations. On the
+Indeed, to the authors' mind, such properties can be useful in the two following situations. On the
one hand, a post-treatment based on a chaotic dynamical system can be applied
to a PRNG statistically deflective, in order to improve its statistical
properties. Such an improvement can be found, for instance, in~\cite{bgw09:ip,bcgr11:ip}.
statistical perfection refers to the ability to pass the whole
{\it BigCrush} battery of tests, which is widely considered as the most
stringent statistical evaluation of a sequence claimed as random.
-This battery can be found into the well-known TestU01 package~\cite{LEcuyerS07}.
+This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}.
Chaos, for its part, refers to the well-established definition of a
chaotic dynamical system proposed by Devaney~\cite{Devaney}.
numbers inside a GPU when a scientific application runs in it. This remark
motivates our proposal of a chaotic and statistically perfect PRNG for GPU.
Such device
-allows us to generated almost 20 billions of pseudorandom numbers per second.
-Last, but not least, we show that the proposed post-treatment preserves the
+allows us to generate almost 20 billion of pseudorandom numbers per second.
+Furthermore, we show that the proposed post-treatment preserves the
cryptographical security of the inputted PRNG, when this last has such a
property.
+Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric
+key encryption protocol by using the proposed method.
The remainder of this paper is organized as follows. In Section~\ref{section:related
works} we review some GPU implementations of PRNGs. Section~\ref{section:BASIC
RECALLS} gives some basic recalls on the well-known Devaney's formulation of chaos,
and on an iteration process called ``chaotic
iterations'' on which the post-treatment is based.
-Proofs of chaos are given in Section~\ref{sec:pseudorandom}.
+The proposed PRNG and its proof of chaos are given in Section~\ref{sec:pseudorandom}.
Section~\ref{sec:efficient PRNG} presents an efficient
implementation of this chaotic PRNG on a CPU, whereas Section~\ref{sec:efficient PRNG
- gpu} describes the GPU implementation.
+ gpu} describes and evaluates theoretically the GPU implementation.
Such generators are experimented in
Section~\ref{sec:experiments}.
We show in Section~\ref{sec:security analysis} that, if the inputted
generator is cryptographically secure, then it is the case too for the
generator provided by the post-treatment.
Such a proof leads to the proposition of a cryptographically secure and
-chaotic generator on GPU based on the famous Blum Blum Shum
-in Section~\ref{sec:CSGPU}.
+chaotic generator on GPU based on the famous Blum Blum Shub
+in Section~\ref{sec:CSGPU}, and to an improvement of the
+Blum-Goldwasser protocol in Sect.~\ref{Blum-Goldwasser}.
This research work ends by a conclusion section, in which the contribution is
summarized and intended future work is presented.
\section{Related works on GPU based PRNGs}
\label{section:related works}
-Numerous research works on defining GPU based PRNGs have yet been proposed in the
-literature, so that completeness is impossible.
+Numerous research works on defining GPU based PRNGs have already been proposed in the
+literature, so that exhaustivity is impossible.
This is why authors of this document only give reference to the most significant attempts
in this domain, from their subjective point of view.
The quantity of pseudorandom numbers generated per second is mentioned here
In \cite{ZRKB10}, the authors propose different versions of efficient GPU PRNGs
based on Lagged Fibonacci or Hybrid Taus. They have used these
PRNGs for Langevin simulations of biomolecules fully implemented on
-GPU. Performance of the GPU versions are far better than those obtained with a
+GPU. Performances of the GPU versions are far better than those obtained with a
CPU, and these PRNGs succeed to pass the {\it BigCrush} battery of TestU01.
However the evaluations of the proposed PRNGs are only statistical ones.
FPGA appears as the fastest and the most
efficient architecture, providing the fastest number of generated pseudorandom numbers
per joule.
-However, we can notice that authors can ``only'' generate between 11 and 16GSamples/s
+However, we notice that authors can ``only'' generate between 11 and 16GSamples/s
with a GTX 280 GPU, which should be compared with
the results presented in this document.
We can remark too that the PRNGs proposed in~\cite{conf/fpga/ThomasHL09} are only
-able to pass the {\it Crush} battery, which is very easy compared to the {\it Big Crush} one.
+able to pass the {\it Crush} battery, which is far easier than the {\it Big Crush} one.
Lastly, Cuda has developed a library for the generation of pseudorandom numbers called
Curand~\cite{curand11}. Several PRNGs are implemented, among
But their PRNGs cannot pass the whole TestU01 battery (only one test is failed).
\newline
\newline
-We can finally remark that, to the best of our knowledge, no GPU implementation have been proven to be chaotic, and the cryptographically secure property is surprisingly never regarded.
+We can finally remark that, to the best of our knowledge, no GPU implementation has been proven to be chaotic, and the cryptographically secure property has surprisingly never been considered.
\section{Basic Recalls}
\label{section:BASIC RECALLS}
This section is devoted to basic definitions and terminologies in the fields of
-topological chaos and chaotic iterations.
+topological chaos and chaotic iterations. We assume the reader is familiar
+with basic notions on topology (see for instance~\cite{Devaney}).
+
+
\subsection{Devaney's Chaotic Dynamical Systems}
In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
\mathcal{X} \rightarrow \mathcal{X}$.
\begin{definition}
-$f$ is said to be \emph{topologically transitive} if, for any pair of open sets
+The function $f$ is said to be \emph{topologically transitive} if, for any pair of open sets
$U,V \subset \mathcal{X}$, there exists $k>0$ such that $f^k(U) \cap V \neq
\varnothing$.
\end{definition}
\begin{definition}[Devaney's formulation of chaos~\cite{Devaney}]
-$f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and
+The function $f$ is said to be \emph{chaotic} on $(\mathcal{X},\tau)$ if $f$ is regular and
topologically transitive.
\end{definition}
on a metric space $(\mathcal{X},d)$ by:
\begin{definition}
-\label{sensitivity} $f$ has \emph{sensitive dependence on initial conditions}
+\label{sensitivity} The function $f$ has \emph{sensitive dependence on initial conditions}
if there exists $\delta >0$ such that, for any $x\in \mathcal{X}$ and any
neighborhood $V$ of $x$, there exist $y\in V$ and $n > 0$ such that
$d\left(f^{n}(x), f^{n}(y)\right) >\delta $.
-$\delta$ is called the \emph{constant of sensitivity} of $f$.
+The constant $\delta$ is called the \emph{constant of sensitivity} of $f$.
\end{definition}
Indeed, Banks \emph{et al.} have proven in~\cite{Banks92} that when $f$ is
are continuous. For further explanations, see, e.g., \cite{guyeux10}.
Let $\delta $ be the \emph{discrete Boolean metric}, $\delta
-(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function:
-\begin{equation}
+(x,y)=0\Leftrightarrow x=y.$ Given a function $f$, define the function
+$F_{f}: \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}}
+\longrightarrow \mathds{B}^{\mathsf{N}}$
+\begin{equation*}
\begin{array}{lrll}
-F_{f}: & \llbracket1;\mathsf{N}\rrbracket\times \mathds{B}^{\mathsf{N}} &
-\longrightarrow & \mathds{B}^{\mathsf{N}} \\
-& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+f(E)_{k}.\overline{\delta
-(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+& (k,E) & \longmapsto & \left( E_{j}.\delta (k,j)+ f(E)_{k}.\overline{\delta
+(k,j)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket}%
\end{array}%
-\end{equation}%
+\end{equation*}%
\noindent where + and . are the Boolean addition and product operations.
Consider the phase space:
\begin{equation}
\item In addition, if two systems present the same cells and their respective
strategies start with the same terms, then the distance between these two points
must be small because the evolution of the two systems will be the same for a
-while. Indeed, the two dynamical systems start with the same initial condition,
-use the same update function, and as strategies are the same for a while, then
-components that are updated are the same too.
+while. Indeed, both dynamical systems start with the same initial condition,
+use the same update function, and as strategies are the same for a while, furthermore
+updated components are the same as well.
\end{itemize}
The distance presented above follows these recommendations. Indeed, if the floor
value $\lfloor d(X,Y)\rfloor $ is equal to $n$, then the systems $E, \check{E}$
precisely, this floating part is less than $10^{-k}$ if and only if the first
$k$ terms of the two strategies are equal. Moreover, if the $k^{th}$ digit is
nonzero, then the $k^{th}$ terms of the two strategies are different.
-The impact of this choice for a distance will be investigate at the end of the document.
+The impact of this choice for a distance will be investigated at the end of the document.
Finally, it has been established in \cite{guyeux10} that,
path from $x$ to $x'$ in $\Gamma(f)$ if and only if there exists a
strategy $s$ such that the parallel iteration of $G_f$ from the
initial point $(s,x)$ reaches the point $x'$.
-
-We have finally proven in \cite{bcgr11:ip} that,
+We have then proven in \cite{bcgr11:ip} that,
\begin{theorem}
if and only if $\Gamma(f)$ is strongly connected.
\end{theorem}
-This result of chaos has lead us to study the possibility to build a
+Finally, we have established in \cite{bcgr11:ip} that,
+\begin{theorem}
+ Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
+ iteration graph, $\check{M}$ its adjacency
+ matrix and $M$
+ a $n\times n$ matrix defined by
+ $
+ M_{ij} = \frac{1}{n}\check{M}_{ij}$ %\textrm{
+ if $i \neq j$ and
+ $M_{ii} = 1 - \frac{1}{n} \sum\limits_{j=1, j\neq i}^n \check{M}_{ij}$ otherwise.
+
+ If $\Gamma(f)$ is strongly connected, then
+ the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows
+ a law that tends to the uniform distribution
+ if and only if $M$ is a double stochastic matrix.
+\end{theorem}
+
+
+These results of chaos and uniform distribution have led us to study the possibility of building a
pseudorandom number generator (PRNG) based on the chaotic iterations.
As $G_f$, defined on the domain $\llbracket 1 ; \mathsf{N} \rrbracket^{\mathds{N}}
-\times \mathds{B}^\mathsf{N}$, is build from Boolean networks $f : \mathds{B}^\mathsf{N}
+\times \mathds{B}^\mathsf{N}$, is built from Boolean networks $f : \mathds{B}^\mathsf{N}
\rightarrow \mathds{B}^\mathsf{N}$, we can preserve the theoretical properties on $G_f$
-during implementations (due to the discrete nature of $f$). It is as if
+during implementations (due to the discrete nature of $f$). Indeed, it is as if
$\mathds{B}^\mathsf{N}$ represents the memory of the computer whereas $\llbracket 1 ; \mathsf{N}
\rrbracket^{\mathds{N}}$ is its input stream (the seeds, for instance, in PRNG, or a physical noise in TRNG).
+Let us finally remark that the vectorial negation satisfies the hypotheses of both theorems above.
\section{Application to Pseudorandomness}
\label{sec:pseudorandom}
possesses various chaos properties that none of the generators used as input
present.
+
\begin{algorithm}[h!]
-%\begin{scriptsize}
+\begin{small}
\KwIn{a function $f$, an iteration number $b$, an initial configuration $x^0$
($n$ bits)}
\KwOut{a configuration $x$ ($n$ bits)}
$x\leftarrow{F_f(s,x)}$\;
}
return $x$\;
-%\end{scriptsize}
+\end{small}
\caption{PRNG with chaotic functions}
\label{CI Algorithm}
\end{algorithm}
+
+
+
\begin{algorithm}[h!]
+\begin{small}
\KwIn{the internal configuration $z$ (a 32-bit word)}
\KwOut{$y$ (a 32-bit word)}
$z\leftarrow{z\oplus{(z\ll13)}}$\;
$z\leftarrow{z\oplus{(z\ll5)}}$\;
$y\leftarrow{z}$\;
return $y$\;
-\medskip
+\end{small}
\caption{An arbitrary round of \textit{XORshift} algorithm}
\label{XORshift}
\end{algorithm}
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 returns integers
+\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,
$2^{32}-1=4.29\times10^9$, is summed up in Algorithm~\ref{XORshift}. It is used
in our PRNG to compute the strategy length and the strategy elements.
-
-We have proven in \cite{bcgr11:ip} that,
-\begin{theorem}
- Let $f: \mathds{B}^{n} \rightarrow \mathds{B}^{n}$, $\Gamma(f)$ its
- iteration graph, $\check{M}$ its adjacency
- matrix and $M$ a $n\times n$ matrix defined as in the previous lemma.
- If $\Gamma(f)$ is strongly connected, then
- the output of the PRNG detailed in Algorithm~\ref{CI Algorithm} follows
- a law that tends to the uniform distribution
- if and only if $M$ is a double stochastic matrix.
-\end{theorem}
-
-This former generator as successively passed various batteries of statistical tests, as the NIST~\cite{bcgr11:ip}, DieHARD~\cite{Marsaglia1996}, and TestU01~\cite{LEcuyerS07}.
+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}
\label{equation Oplus}
\end{equation}
where $\oplus$ is for the bitwise exclusive or between two integers.
-This rewritten can be understood as follows. The $n-$th term $S^n$ of the
+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
$\mathcal{S}^n \subset \llbracket 1, \mathsf{N} \rrbracket$ 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} for
-the fact that, instead of updating only one term at each iteration,
+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}, possible when the iteration function is
+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
\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:
+the general form: $\forall n\in \mathds{N}^{\ast }$, $ \forall i\in
+\llbracket1;\mathsf{N}\rrbracket $,
\begin{equation}
-\forall n\in \mathds{N}^{\ast }, \forall i\in
-\llbracket1;\mathsf{N}\rrbracket ,x_i^n=\left\{
+ 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.
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} $, define the function:
-\begin{equation}
-\begin{array}{lrll}
-F_{f}: & \mathcal{P}\left(\llbracket1;\mathsf{N}\rrbracket \right) \times \mathds{B}^{\mathsf{N}} &
-\longrightarrow & \mathds{B}^{\mathsf{N}} \\
-& (P,E) & \longmapsto & \left( E_{j}.\chi (j,P)+f(E)_{j}.\overline{\chi
-(j,P)}\right) _{j\in \llbracket1;\mathsf{N}\rrbracket},%
+$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}%
+\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:
\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}
+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
\right.
\end{equation}%
-Another time, a shift function appears as a component of these general chaotic
+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
d(X,Y)=d_{e}(E,\check{E})+d_{s}(S,\check{S}),
\label{nouveau d}
\end{equation}
-\noindent where
-\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 another time 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}
+\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{proof}
$d_e$ is the Hamming distance. We will prove that $d_s$ is a distance
-too, thus $d$ will be a distance as sum of two distances.
+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
Before being able to study the topological behavior of the general
-chaotic iterations, we must firstly establish that:
+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
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 distance
+\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
\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 $.\bigskip \newline
+10^{-(k+1)}\leqslant \varepsilon $.
+
In conclusion,
-$$
-\forall \varepsilon >0,\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}%
-,\forall n\geqslant N_{0},
- d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
+%%RAPH : ici j'ai rajouté une ligne
+$
+\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
+,$ $\forall n\geqslant N_{0},$
+$ d\left( G_{f}(S^n,E^n);G_{f}(S,E)\right)
\leqslant \varepsilon .
-$$
+$
$G_{f}$ is consequently continuous.
\end{proof}
claimed in the lemma.
\end{proof}
-We can now prove the Theorem~\ref{t:chaos des general}...
+We can now prove the Theorem~\ref{t:chaos des general}.
\begin{proof}[Theorem~\ref{t:chaos des general}]
Firstly, strong transitivity implies transitivity.
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 $S'$: $$\tilde
-S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ It
+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
\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 $x_i^n = \overline{x_i^{n-1}}$.
+Finally, some $x^n$ are selected
+by a sequence $m^n$ as the pseudo-random 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}.
+
+\begin{equation}
+\label{Formula}
+m^n = g_1(y^n)=
+\left\{
+\begin{array}{l}
+0 \text{ if }0 \leqslant{y^n}<{C^0_{32}},\\
+1 \text{ if }{C^0_{32}} \leqslant{y^n}<\sum_{i=0}^1{C^i_{32}},\\
+2 \text{ if }\sum_{i=0}^1{C^i_{32}} \leqslant{y^n}<\sum_{i=0}^2{C^i_{32}},\\
+\vdots~~~~~ ~~\vdots~~~ ~~~~\\
+N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
+\end{array}
+\right.
+\end{equation}
+
+\begin{algorithm}
+\textbf{Input:} the internal state $x$ (32 bits)\\
+\textbf{Output:} a state $r$ of 32 bits
+\begin{algorithmic}[1]
+\FOR{$i=0,\dots,N$}
+{
+\STATE$d_i\leftarrow{0}$\;
+}
+\ENDFOR
+\STATE$a\leftarrow{PRNG_1()}$\;
+\STATE$m\leftarrow{f(a)}$\;
+\STATE$k\leftarrow{m}$\;
+\WHILE{$i=0,\dots,k$}
+
+\STATE$b\leftarrow{PRNG_2()~mod~\mathsf{N}}$\;
+\STATE$S\leftarrow{b}$\;
+ \IF{$d_S=0$}
+ {
+\STATE $x_S\leftarrow{ \overline{x_S}}$\;
+\STATE $d_S\leftarrow{1}$\;
+
+ }
+ \ELSIF{$d_S=1$}
+ {
+\STATE $k\leftarrow{ k+1}$\;
+ }\ENDIF
+\ENDWHILE\\
+\STATE $r\leftarrow{x}$\;
+\STATE return $r$\;
+\medskip
+\caption{An arbitrary round of the new CI generator}
+\label{Chaotic iteration1}
+\end{algorithmic}
+\end{algorithm}
+
+
+\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}:
+
+\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}
+%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}
+
+
+\subsubsection{INV}
+Finally, this module implements the nonlinear inversive generator, as defined in~\cite{testU01}, which is:
+
+\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}
+
+
+
+
+
+\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}
+
+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.
+
+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.
+
+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.
+
+
+
+
+
+\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.
+
+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 $P>0.9999$.
+
+
+\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*}
+
+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.
+
+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}
+
+
+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}
+
+Let us now denote by $n_{P1}$ and $n_{P2}$ the periods of respectively the $PRNG_1$ and $PRNG_2$ generators, then the period of the Mixed Xor CIPRNG will be:
+\begin{equation}
+n_{XXORCI}=
+\left\{
+\begin{array}{ll}
+LCM(n_{P1},n_{P2})&\text{if~}x^0=x^{LCM(n_{P1},n_{P2})}\\
+2LCM(n_{P1},n_{P2})&\text{if~}x^0\neq x^{LCM(n_{P1},n_{P2})}.\\
+\end{array}
+\right.
+\label{equation Oplus}
+\end{equation}
+
+In Table~\ref{DieHARD fail mixex CIPRNG}, we only show the results for the Mixed CIPRNGs that cannot pass all DieHARD suites (the NIST tests are all passed). It demonstrates that Mixed Xor CIPRNG involving LCG, MRG, LCG2, LCG3, MRG2, or INV cannot pass the two following tests, namely the ``Matrix Rank 32x32'' and the ``COUNT-THE-1's'' tests contained into the DieHARD battery. Let us recall their definitions:
+
+\begin{itemize}
+ \item \textbf{Matrix Rank 32x32.} A random 32x32 binary matrix is formed, each row having a 32-bit random vector. Its rank is an integer that ranges from 0 to 32. Ranks less than 29 must be rare, and their occurences must be pooled with those of rank 29. To achieve the test, ranks of 40,000 such random matrices are obtained, and a chisquare test is performed on counts for ranks 32,31,30 and for ranks $\leq29$.
+
+ \item \textbf{COUNT-THE-1's TEST} Consider the file under test as a stream of bytes (four per 2 bit integer). Each byte can contain from 0 to 8 1's, with probabilities 1,8,28,56,70,56,28,8,1 over 256. Now let the stream of bytes provide a string of overlapping 5-letter words, each ``letter'' taking values A,B,C,D,E. The letters are determined by the number of 1's in a byte: 0,1, or 2 yield A, 3 yields B, 4 yields C, 5 yields D and 6,7, or 8 yield E. Thus we have a monkey at a typewriter hitting five keys with various probabilities (37,56,70,56,37 over 256). There are $5^5$ possible 5-letter words, and from a string of 256,000 (over-lapping) 5-letter words, counts are made on the frequencies for each word. The quadratic form in the weak inverse of the covariance matrix of the cell counts provides a chisquare test: Q5-Q4, the difference of the naive Pearson sums of $(OBS-EXP)^2/EXP$ on counts for 5- and 4-letter cell counts.
+\end{itemize}
+
+The reason of these fails is that the output of LCG, LCG2, LCG3, MRG, and MRG2 under the experiments are in 31-bit. Compare with the Single CIPRNG, using different PRNGs to build CIPRNG seems more efficient in improving random number quality (mixed Xor CI can 100\% pass NIST, but single cannot).
+
+\begin{table*}
+\renewcommand{\arraystretch}{1.3}
+\caption{Scores of mixed Xor CIPRNGs when considering the DieHARD battery}
+\label{DieHARD fail mixex CIPRNG}
+\centering
+ \begin{tabular}{|l||c|c|c|c|c|c|}
+ \hline
+\backslashbox{\textbf{$PRNG_1$}} {\textbf{$PRNG_0$}} & LCG & MRG & INV & LCG2 & LCG3 & MRG2 \\ \hline\hline
+LCG &\backslashbox{} {} &16/18&16/18 &16/18 &16/18 &16/18\\ \hline
+MRG &16/18 &\backslashbox{} {} &16/18&16/18 &16/18 &16/18\\ \hline
+INV &16/18 &16/18&\backslashbox{} {} &16/18 &16/18&16/18 \\ \hline
+LCG2 &16/18 &16/18 &16/18 &\backslashbox{} {} &16/18&16/18\\ \hline
+LCG3 &16/18 &16/18 &16/18&16/18&\backslashbox{} {} &16/18\\ \hline
+MRG2 &16/18 &16/18 &16/18&16/18 &16/18 &\backslashbox{} {} \\ \hline
+\end{tabular}
+\end{table*}
+
+\subsubsection{Tests based on the Multiple CIPRNG}
+\label{Tests based on Multiple CIPRNG}
+
+Until now, the combination of at most two input PRNGs has been investigated.
+We now regard the possibility to use a larger number of generators to improve the statistics of the generated pseudorandom numbers, leading to the multiple functional power approach.
+For the CIPRNGs which have already pass both the NIST and DieHARD suites with 2 inputted PRNGs (all the Old and New CIPRNGs, and some of the Xor CIPRNGs), it is not meaningful to consider their adaption of this multiple CIPRNG method, hence only the Multiple Xor CIPRNGs, having the following form, will be investigated.
+\begin{equation}
+\left\{
+\begin{array}{l}
+x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket^\mathds{N} \\
+\forall n \in \mathds{N}^*, x^n = x^{n-1} \oplus S^{nm}\oplus S^{nm+1}\ldots \oplus S^{nm+m-1} ,
+\end{array}
+\right.
+\label{equation Oplus}
+\end{equation}
+
+The question is now to determine the value of the threshold $m$ (the functional power) making the multiple CIPRNG being able to pass the whole NIST battery.
+Such a question is answered in Table~\ref{threshold}.
+
+
+\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}
\label{sec:efficient 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, hoping by doing so to
+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
+%% done.
+%% Suppose that $x$ and the strategy $S^i$ are given as
+%% binary vectors.
+%% Table~\ref{TableExemple} shows the result of $x \oplus S^i$.
+
+%% \begin{table}
+%% \begin{scriptsize}
+%% $$
+%% \begin{array}{|cc|cccccccccccccccc|}
+%% \hline
+%% x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
+%% \hline
+%% S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
+%% \hline
+%% x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
+%% \hline
+
+%% \hline
+%% \end{array}
+%% $$
+%% \end{scriptsize}
+%% \caption{Example of an arbitrary round of the proposed generator}
+%% \label{TableExemple}
+%% \end{table}
-Let us give an example using 16-bits numbers, to clearly understand how the bitwise xor operations
-are
-done.
-Suppose that $x$ and the strategy $S^i$ are given as
-binary vectors.
-Table~\ref{TableExemple} shows the result of $x \oplus S^i$.
-
-\begin{table}
-$$
-\begin{array}{|cc|cccccccccccccccc|}
-\hline
-x &=&1&0&1&1&1&0&1&0&1&0&0&1&0&0&1&0\\
-\hline
-S^i &=&0&1&1&0&0&1&1&0&1&1&1&0&0&1&1&1\\
-\hline
-x \oplus S^i&=&1&1&0&1&1&1&0&0&0&1&1&1&0&1&0&1\\
-\hline
-
-\hline
- \end{array}
-$$
-\caption{Example of an arbitrary round of the proposed generator}
-\label{TableExemple}
-\end{table}
\lstset{language=C,caption={C code of the sequential PRNG based on chaotic iterations},label=algo:seqCIPRNG}
+\begin{small}
\begin{lstlisting}
+
unsigned int CIPRNG() {
static unsigned int x = 123123123;
unsigned long t1 = xorshift();
return x;
}
\end{lstlisting}
+\end{small}
+In Listing~\ref{algo:seqCIPRNG} a sequential version of the proposed PRNG based
+on chaotic iterations is presented. The xor operator is represented by
+\textasciicircum. This function uses three classical 64-bits PRNGs, namely the
+\texttt{xorshift}, the \texttt{xor128}, and the
+\texttt{xorwow}~\cite{Marsaglia2003}. In the following, we call them ``xor-like
+PRNGs''. As each xor-like PRNG uses 64-bits whereas our proposed generator
+works with 32-bits, we use the command \texttt{(unsigned int)}, that selects the
+32 least significant bits of a given integer, and the code \texttt{(unsigned
+ int)(t$>>$32)} in order to obtain the 32 most significant bits of \texttt{t}.
-
-In Listing~\ref{algo:seqCIPRNG} a sequential version of the proposed PRNG based on chaotic iterations
- is presented. The xor operator is represented by \textasciicircum.
-This function uses three classical 64-bits PRNGs, namely the \texttt{xorshift}, the
-\texttt{xor128}, and the \texttt{xorwow}~\cite{Marsaglia2003}. In the following, we call them
-``xor-like PRNGs''.
-As
-each xor-like PRNG uses 64-bits whereas our proposed generator works with 32-bits,
-we use the command \texttt{(unsigned int)}, that selects the 32 least significant bits of a given integer, and the code
-\texttt{(unsigned int)(t3$>>$32)} in order to obtain the 32 most significant bits of \texttt{t}.
-
-So 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
+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}.
\section{Efficient PRNGs based on Chaotic Iterations on GPU}
\label{sec:efficient PRNG gpu}
-In order to take benefits from the computing power of GPU, a program needs to have
-independent blocks of threads that can be computed simultaneously. In general,
-the larger the number of threads is, the more local memory is used, and the less
-branching instructions are used (if, while, ...), the better the performances on GPU is.
-Obviously, having these requirements in mind, it is possible to build a program similar to
-the one presented in Algorithm \ref{algo:seqCIPRNG}, which computes pseudorandom numbers
-on GPU.
-To do so, we must firstly recall that in
- the CUDA~\cite{Nvid10} environment, threads have a local
-identifier called \texttt{ThreadIdx}, which is relative to the block containing them.
+In order to take benefits from the computing power of GPU, a program
+needs to have independent blocks of threads that can be computed
+simultaneously. In general, the larger the number of threads is, the
+more local memory is used, and the less branching instructions are
+used (if, while, ...), the better the performances on GPU is.
+Obviously, having these requirements in mind, it is possible to build
+a program similar to the one presented in Listing
+\ref{algo:seqCIPRNG}, which computes pseudorandom numbers on GPU. To
+do so, we must firstly recall that in the CUDA~\cite{Nvid10}
+environment, threads have a local identifier called
+\texttt{ThreadIdx}, which is relative to the block containing
+them. Furthermore, in CUDA, parts of the code that are executed by the GPU, are
+called {\it kernels}.
\subsection{Naive Version for GPU}
It is possible to deduce from the CPU version a quite similar version adapted to GPU.
-The simple principle consists to make each thread of the GPU computing the CPU version of our PRNG.
+The simple principle consists in making each thread of the GPU computing the CPU version of our PRNG.
Of course, the three xor-like
PRNGs used in these computations must have different parameters.
-In a given thread, these lasts are
+In a given thread, these parameters are
randomly picked from another PRNGs.
The initialization stage is performed by the CPU.
To do it, the ISAAC PRNG~\cite{Jenkins96} is used to set all the
implementation of the xor128, the xorshift, and the xorwow respectively require
4, 5, and 6 unsigned long as internal variables.
-\begin{algorithm}
+\begin{algorithm}
+\begin{small}
\KwIn{InternalVarXorLikeArray: array with internal variables of the 3 xor-like
PRNGs in global memory\;
NumThreads: number of threads\;}
}
store internal variables in InternalVarXorLikeArray[threadIdx]\;
}
-
+\end{small}
\caption{Main kernel of the GPU ``naive'' version of the PRNG based on chaotic iterations}
\label{algo:gpu_kernel}
\end{algorithm}
+
+
Algorithm~\ref{algo:gpu_kernel} presents a naive implementation of the proposed PRNG on
GPU. Due to the available memory in the GPU and the number of threads
-used simultenaously, the number of random numbers that a thread can generate
+used simultaneously, the number of random numbers that a thread can generate
inside a kernel is limited (\emph{i.e.}, the variable \texttt{n} in
algorithm~\ref{algo:gpu_kernel}). For instance, if $100,000$ threads are used and
if $n=100$\footnote{in fact, we need to add the initial seed (a 32-bits number)},
This generator is able to pass the whole BigCrush battery of tests, for all
the versions that have been tested depending on their number of threads
-(called \texttt{NumThreads} in our algorithm, tested until $10$ millions).
+(called \texttt{NumThreads} in our algorithm, tested up to $5$ million).
\begin{remark}
-The proposed algorithm has the advantage to manipulate independent
+The proposed algorithm has the advantage of manipulating independent
PRNGs, so this version is easily adaptable on a cluster of computers too. The only thing
to ensure is to use a single ISAAC PRNG. To achieve this requirement, a simple solution consists in
using a master node for the initialization. This master node computes the initial parameters
-for all the differents nodes involves in the computation.
+for all the different nodes involved in the computation.
\end{remark}
\subsection{Improved Version for GPU}
i.e., to use less than 3 xor-like PRNGs. The solution consists in computing only
one xor-like PRNG by thread, saving it into the shared memory, and then to use the results
of some other threads in the same block of threads. In order to define which
-thread uses the result of which other one, we can use a permutation array that
-contains the indexes of all threads and for which a permutation has been
+thread uses the result of which other one, we can use a combination array that
+contains the indexes of all threads and for which a combination has been
performed.
-In Algorithm~\ref{algo:gpu_kernel2}, two permutations arrays are used.
-The variable \texttt{offset} is computed using the value of
-\texttt{permutation\_size}. Then we can compute \texttt{o1} and \texttt{o2}
-representing the indexes of the other threads whose results are used
-by the current one. In this algorithm, we consider that a 64-bits xor-like
-PRNG has been chosen, and so its two 32-bits parts are used.
+In Algorithm~\ref{algo:gpu_kernel2}, two combination arrays are used. The
+variable \texttt{offset} is computed using the value of
+\texttt{combination\_size}. Then we can compute \texttt{o1} and \texttt{o2}
+representing the indexes of the other threads whose results are used by the
+current one. In this algorithm, we consider that a 32-bits xor-like PRNG has
+been chosen. In practice, we use the xor128 proposed in~\cite{Marsaglia2003} in
+which unsigned longs (64 bits) have been replaced by unsigned integers (32
+bits).
-This version also can pass the whole {\it BigCrush} battery of tests.
+This version can also pass the whole {\it BigCrush} battery of tests.
\begin{algorithm}
-
+\begin{small}
\KwIn{InternalVarXorLikeArray: array with internal variables of 1 xor-like PRNGs
in global memory\;
NumThreads: Number of threads\;
-tab1, tab2: Arrays containing permutations of size permutation\_size\;}
+array\_comb1, array\_comb2: Arrays containing combinations of size combination\_size\;}
\KwOut{NewNb: array containing random numbers in global memory}
\If{threadId is concerned} {
retrieve data from InternalVarXorLikeArray[threadId] in local variables including shared memory and x\;
- offset = threadIdx\%permutation\_size\;
- o1 = threadIdx-offset+tab1[offset]\;
- o2 = threadIdx-offset+tab2[offset]\;
+ offset = threadIdx\%combination\_size\;
+ o1 = threadIdx-offset+array\_comb1[offset]\;
+ o2 = threadIdx-offset+array\_comb2[offset]\;
\For{i=1 to n} {
t=xor-like()\;
- t=t$\oplus$shmem[o1]$\oplus$shmem[o2]\;
+ t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\;
shared\_mem[threadId]=t\;
- x = x $\oplus$ t\;
+ x = x\textasciicircum t\;
store the new PRNG in NewNb[NumThreads*threadId+i]\;
}
store internal variables in InternalVarXorLikeArray[threadId]\;
}
-
-\caption{main kernel for the chaotic iterations based PRNG GPU efficient
-version}
-\label{algo:gpu_kernel2}
+\end{small}
+\caption{Main kernel for the chaotic iterations based PRNG GPU efficient
+version\label{IR}}
+\label{algo:gpu_kernel2}
\end{algorithm}
\subsection{Theoretical Evaluation of the Improved Version}
To be certain that we are in the framework of Theorem~\ref{t:chaos des general},
we must guarantee that this dynamical system iterates on the space
$\mathcal{X} = \mathcal{P}\left(\llbracket 1, \mathsf{N} \rrbracket\right)^\mathds{N}\times\mathds{B}^\mathsf{N}$.
-The left term $x$ obviously belongs into $\mathds{B}^ \mathsf{N}$.
+The left term $x$ obviously belongs to $\mathds{B}^ \mathsf{N}$.
To prevent from any flaws of chaotic properties, we must check that the right
term (the last $t$), corresponding to the strategies, can possibly be equal to any
integer of $\llbracket 1, \mathsf{N} \rrbracket$.
prove by an immediate mathematical induction that, as the initial $x$
is uniformly distributed (it is provided by a cryptographically secure PRNG),
the two other stored values shmem[o1] and shmem[o2] are uniformly distributed too,
-(this can be stated by an immediate mathematical
-induction), and thus the next $x$ is finally uniformly distributed.
+(this is the induction hypothesis), and thus the next $x$ is finally uniformly distributed.
Thus Algorithm~\ref{algo:gpu_kernel2} is a concrete realization of the general
chaotic iterations presented previously, and for this reason, it satisfies the
cards have 240 cores.
In Figure~\ref{fig:time_xorlike_gpu} we compare the quantity of pseudorandom numbers
-generated per second with various xor-like based PRNG. In this figure, the optimized
+generated per second with various xor-like based PRNGs. In this figure, the optimized
versions use the {\it xor64} described in~\cite{Marsaglia2003}, whereas the naive versions
embed the three xor-like PRNGs described in Listing~\ref{algo:seqCIPRNG}. In
order to obtain the optimal performances, the storage of pseudorandom numbers
generation. Moreover this storage is completely
useless, in case of applications that consume the pseudorandom
numbers directly after generation. We can see that when the number of threads is greater
-than approximately 30,000 and lower than 5 millions, the number of pseudorandom numbers generated
+than approximately 30,000 and lower than 5 million, the number of pseudorandom numbers generated
per second is almost constant. With the naive version, this value ranges from 2.5 to
3GSamples/s. With the optimized version, it is approximately equal to
20GSamples/s. Finally we can remark that both GPU cards are quite similar, but in
\begin{figure}[htbp]
\begin{center}
- \includegraphics[scale=.7]{curve_time_xorlike_gpu.pdf}
+ \includegraphics[width=\columnwidth]{curve_time_xorlike_gpu.pdf}
\end{center}
\caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG}
\label{fig:time_xorlike_gpu}
-In Figure~\ref{fig:time_bbs_gpu} we highlight the performances of the optimized
-BBS-based PRNG on GPU. On the Tesla C1060 we
-obtain approximately 1.8GSample/s and on the GTX 280 about 1.6GSample/s, which is
-obviously slower than the xorlike-based PRNG on GPU. However, we will show in the
-next sections that
-this new PRNG has a strong level of security, which is necessary paid by a speed
-reduction.
+In Figure~\ref{fig:time_bbs_gpu} we highlight the performances of the optimized
+BBS-based PRNG on GPU. On the Tesla C1060 we obtain approximately 700MSample/s
+and on the GTX 280 about 670MSample/s, which is obviously slower than the
+xorlike-based PRNG on GPU. However, we will show in the next sections that this
+new PRNG has a strong level of security, which is necessarily paid by a speed
+reduction.
\begin{figure}[htbp]
\begin{center}
- \includegraphics[scale=.7]{curve_time_bbs_gpu.pdf}
+ \includegraphics[width=\columnwidth]{curve_time_bbs_gpu.pdf}
\end{center}
\caption{Quantity of pseudorandom numbers generated per second using the BBS-based PRNG}
\label{fig:time_bbs_gpu}
All these experiments allow us to conclude that it is possible to
generate a very large quantity of pseudorandom numbers statistically perfect with the xor-like version.
-In a certain extend, it is the case too with the secure BBS-based version, the speed deflation being
+To a certain extend, it is also the case with the secure BBS-based version, the speed deflation being
explained by the fact that the former version has ``only''
chaotic properties and statistical perfection, whereas the latter is also cryptographically secure,
as it is shown in the next sections.
denoted by $uv$.
In a cryptographic context, a pseudorandom generator is a deterministic
algorithm $G$ transforming strings into strings and such that, for any
-seed $w$ of length $N$, $G(w)$ (the output of $G$ on the input $w$) has size
-$\ell_G(N)$ with $\ell_G(N)>N$.
+seed $s$ of length $m$, $G(s)$ (the output of $G$ on the input $s$) has size
+$\ell_G(m)$ with $\ell_G(m)>m$.
The notion of {\it secure} PRNGs can now be defined as follows.
\begin{definition}
A cryptographic PRNG $G$ is secure if for any probabilistic polynomial time
algorithm $D$, for any positive polynomial $p$, and for all sufficiently
-large $k$'s,
-$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(N)},$$
+large $m$'s,
+$$| \mathrm{Pr}[D(G(U_m))=1]-Pr[D(U_{\ell_G(m)})=1]|< \frac{1}{p(m)},$$
where $U_r$ is the uniform distribution over $\{0,1\}^r$ and the
-probabilities are taken over $U_N$, $U_{\ell_G(N)}$ as well as over the
+probabilities are taken over $U_m$, $U_{\ell_G(m)}$ as well as over the
internal coin tosses of $D$.
\end{definition}
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 $\ell^\prime(N)>N)$~\cite[Chapter 3.3]{Goldreich}.
+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,
the $S_i$'s). The cryptographic PRNG $X$ defined in (\ref{equation Oplus})
is the algorithm mapping any string of length $2N$ $x_0S_0$ into the string
$(x_0\oplus S_0 \oplus S_1)(x_0\oplus S_0 \oplus S_1\oplus S_2)\ldots
-(x_o\bigoplus_{i=0}^{i=k}S_i)$. Particularly one has $\ell_{X}(2N)=kN=\ell_H(N)$.
+(x_o\bigoplus_{i=0}^{i=k}S_i)$. One in particular has $\ell_{X}(2N)=kN=\ell_H(N)$.
We claim now that if this PRNG is secure,
then the new one is secure too.
by a direct induction, that $w_i=w_i^\prime$. Furthermore, since $\mathbb{B}^{kN}$
is finite, each $\varphi_y$ is bijective. Therefore, and using (\ref{PCH-1}),
one has
+$\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]$ and,
+therefore,
\begin{equation}\label{PCH-2}
-\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(\varphi_y(U_{kN}))=1]=\mathrm{Pr}[D(U_{kN})=1].
+\mathrm{Pr}[D^\prime(U_{kN})=1]=\mathrm{Pr}[D(U_{kN})=1].
\end{equation}
Now, using (\ref{PCH-1}) again, one has for every $x$,
\end{equation}
where $y$ is randomly generated. By construction, $\varphi_y(H(x))=X(yx)$,
thus
-\begin{equation}\label{PCH-3}
+\begin{equation}%\label{PCH-3} %%RAPH : j'ai viré ce label qui existe déjà, il est 3 ligne avant
D^\prime(H(x))=D(yx),
\end{equation}
where $y$ is randomly generated.
\mathrm{Pr}[D^\prime(H(U_{N}))=1]=\mathrm{Pr}[D(U_{2N})=1].
\end{equation}
From (\ref{PCH-2}) and (\ref{PCH-4}), one can deduce that
-there exist a polynomial time probabilistic
+there exists a polynomial time probabilistic
algorithm $D^\prime$, a positive polynomial $p$, such that for all $k_0$ there exists
$N\geq \frac{k_0}{2}$ satisfying
$$| \mathrm{Pr}[D(H(U_{N}))=1]-\mathrm{Pr}[D(U_{kN}=1]|\geq \frac{1}{p(2N)},$$
-proving that $H$ is not secure, a contradiction.
+proving that $H$ is not secure, which is a contradiction.
\end{proof}
algorithm (Algorithm~\ref{algo:gpu_kernel2}). Due to Proposition~\ref{cryptopreuve},
it simply consists in replacing
the {\it xor-like} PRNG by a cryptographically secure one.
-We have chosen the Blum Blum Shum generator~\cite{BBS} (usually denoted by BBS) having the form:
-$$x_{n+1}=x_n^2~ mod~ M$$ where $M$ is the product of two prime numbers. These
-prime numbers need to be congruent to 3 modulus 4. BBS is
+We have chosen the Blum Blum Shub generator~\cite{BBS} (usually denoted by BBS) having the form:
+$$x_{n+1}=x_n^2~ mod~ M$$ where $M$ is the product of two prime numbers (these
+prime numbers need to be congruent to 3 modulus 4). BBS is known to be
very slow and only usable for cryptographic applications.
-The modulus operation is the most time consuming operation for
-current GPU cards.
-So in order to obtain quite reasonable performances, it is required
-to use only modulus on 32 bits integer numbers. Consequently $x_n^2$ need to be
-less than $2^{32}$ and the number $M$ need to be less than $2^{16}$. So in
-practice we can choose prime numbers around 256 that are congruent to 3 modulus
-4. With 32 bits numbers, only the 4 least significant bits of $x_n$ can be
-chosen (the maximum number of indistinguishable bits is lesser than or equals to
-$log_2(log_2(x_n))$). So to generate a 32 bits number, we need to use 8 times
-the BBS algorithm with different combinations of $M$.
+The modulus operation is the most time consuming operation for current
+GPU cards. So in order to obtain quite reasonable performances, it is
+required to use only modulus on 32-bits integer numbers. Consequently
+$x_n^2$ need to be lesser than $2^{32}$, and thus the number $M$ must be
+lesser than $2^{16}$. So in practice we can choose prime numbers around
+256 that are congruent to 3 modulus 4. With 32-bits numbers, only the
+4 least significant bits of $x_n$ can be chosen (the maximum number of
+indistinguishable bits is lesser than or equals to
+$log_2(log_2(M))$). In other words, to generate a 32-bits number, we need to use
+8 times the BBS algorithm with possibly different combinations of $M$. This
+approach is not sufficient to be able to pass all the tests of TestU01,
+as small values of $M$ for the BBS lead to
+ small periods. So, in order to add randomness we have proceeded with
+the followings modifications.
+\begin{itemize}
+\item
+Firstly, we define 16 arrangement arrays instead of 2 (as described in
+Algorithm \ref{algo:gpu_kernel2}), but only 2 of them are used at each call of
+the PRNG kernels. In practice, the selection of combination
+arrays to be used is different for all the threads. It is determined
+by using the three last bits of two internal variables used by BBS.
+%This approach adds more randomness.
+In Algorithm~\ref{algo:bbs_gpu},
+character \& is for the bitwise AND. Thus using \&7 with a number
+gives the last 3 bits, thus providing a number between 0 and 7.
+\item
+Secondly, after the generation of the 8 BBS numbers for each thread, we
+have a 32-bits number whose period is possibly quite small. So
+to add randomness, we generate 4 more BBS numbers to
+shift the 32-bits numbers, and add up to 6 new bits. This improvement is
+described in Algorithm~\ref{algo:bbs_gpu}. In practice, the last 2 bits
+of the first new BBS number are used to make a left shift of at most
+3 bits. The last 3 bits of the second new BBS number are added to the
+strategy whatever the value of the first left shift. The third and the
+fourth new BBS numbers are used similarly to apply a new left shift
+and add 3 new bits.
+\item
+Finally, as we use 8 BBS numbers for each thread, the storage of these
+numbers at the end of the kernel is performed using a rotation. So,
+internal variable for BBS number 1 is stored in place 2, internal
+variable for BBS number 2 is stored in place 3, ..., and finally, internal
+variable for BBS number 8 is stored in place 1.
+\end{itemize}
-Currently this PRNG does not succeed to pass all the tests of TestU01.
+\begin{algorithm}
+\begin{small}
+\KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS
+in global memory\;
+NumThreads: Number of threads\;
+array\_comb: 2D Arrays containing 16 combinations (in first dimension) of size combination\_size (in second dimension)\;
+array\_shift[4]=\{0,1,3,7\}\;
+}
+\KwOut{NewNb: array containing random numbers in global memory}
+\If{threadId is concerned} {
+ retrieve data from InternalVarBBSArray[threadId] in local variables including shared memory and x\;
+ we consider that bbs1 ... bbs8 represent the internal states of the 8 BBS numbers\;
+ offset = threadIdx\%combination\_size\;
+ o1 = threadIdx-offset+array\_comb[bbs1\&7][offset]\;
+ o2 = threadIdx-offset+array\_comb[8+bbs2\&7][offset]\;
+ \For{i=1 to n} {
+ t$<<$=4\;
+ t|=BBS1(bbs1)\&15\;
+ ...\;
+ t$<<$=4\;
+ t|=BBS8(bbs8)\&15\;
+ \tcp{two new shifts}
+ shift=BBS3(bbs3)\&3\;
+ t$<<$=shift\;
+ t|=BBS1(bbs1)\&array\_shift[shift]\;
+ shift=BBS7(bbs7)\&3\;
+ t$<<$=shift\;
+ t|=BBS2(bbs2)\&array\_shift[shift]\;
+ t=t\textasciicircum shmem[o1]\textasciicircum shmem[o2]\;
+ shared\_mem[threadId]=t\;
+ x = x\textasciicircum t\;
-\subsection{A Secure Asymetric Cryptosystem}
+ store the new PRNG in NewNb[NumThreads*threadId+i]\;
+ }
+ store internal variables in InternalVarXorLikeArray[threadId] using a rotation\;
+}
+\end{small}
+\caption{main kernel for the BBS based PRNG GPU}
+\label{algo:bbs_gpu}
+\end{algorithm}
+In Algorithm~\ref{algo:bbs_gpu}, $n$ is for the quantity of random numbers that
+a thread has to generate. The operation t<<=4 performs a left shift of 4 bits
+on the variable $t$ and stores the result in $t$, and $BBS1(bbs1)\&15$ selects
+the last four bits of the result of $BBS1$. Thus an operation of the form
+$t<<=4; t|=BBS1(bbs1)\&15\;$ realizes in $t$ a left shift of 4 bits, and then
+puts the 4 last bits of $BBS1(bbs1)$ in the four last positions of $t$. Let us
+remark that the initialization $t$ is not a necessity as we fill it 4 bits by 4
+bits, until having obtained 32-bits. The two last new shifts are realized in
+order to enlarge the small periods of the BBS used here, to introduce a kind of
+variability. In these operations, we make twice a left shift of $t$ of \emph{at
+ most} 3 bits, represented by \texttt{shift} in the algorithm, and we put
+\emph{exactly} the \texttt{shift} last bits from a BBS into the \texttt{shift}
+last bits of $t$. For this, an array named \texttt{array\_shift}, containing the
+correspondence between the shift and the number obtained with \texttt{shift} 1
+to make the \texttt{and} operation is used. For example, with a left shift of 0,
+we make an and operation with 0, with a left shift of 3, we make an and
+operation with 7 (represented by 111 in binary mode).
+
+It should be noticed that this generator has once more the form $x^{n+1} = x^n \oplus S^n$,
+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.
-\section{Conclusion}
+\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.
-In this paper we have presented a new class of PRNGs based on chaotic
-iterations. We have proven that these PRNGs are chaotic in the sense of Devaney.
-We also propose a PRNG cryptographically secure and its implementation on GPU.
+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}
+We finish this research work by giving some thoughts about the use of
+the proposed PRNG in an asymmetric cryptosystem.
+This first approach will be further investigated in a future work.
+
+\subsubsection{Recalls of the Blum-Goldwasser Probabilistic Cryptosystem}
+
+The Blum-Goldwasser cryptosystem is a cryptographically secure asymmetric key encryption algorithm
+proposed in 1984~\cite{Blum:1985:EPP:19478.19501}. The encryption algorithm
+implements a XOR-based stream cipher using the BBS PRNG, in order to generate
+the keystream. Decryption is done by obtaining the initial seed thanks to
+the final state of the BBS generator and the secret key, thus leading to the
+ reconstruction of the keystream.
+
+The key generation consists in generating two prime numbers $(p,q)$,
+randomly and independently of each other, that are
+ congruent to 3 mod 4, and to compute the modulus $N=pq$.
+The public key is $N$, whereas the secret key is the factorization $(p,q)$.
+
+
+Suppose Bob wishes to send a string $m=(m_0, \dots, m_{L-1})$ of $L$ bits to Alice:
+\begin{enumerate}
+\item Bob picks an integer $r$ randomly in the interval $\llbracket 1,N\rrbracket$ and computes $x_0 = r^2~mod~N$.
+\item He uses the BBS to generate the keystream of $L$ pseudorandom bits $(b_0, \dots, b_{L-1})$, as follows. For $i=0$ to $L-1$,
+\begin{itemize}
+\item $i=0$.
+\item While $i \leqslant L-1$:
+\begin{itemize}
+\item Set $b_i$ equal to the least-significant\footnote{As signaled previously, BBS can securely output up to $\mathsf{N} = \lfloor log(log(N)) \rfloor$ of the least-significant bits of $x_i$ during each round.} bit of $x_i$,
+\item $i=i+1$,
+\item $x_i = (x_{i-1})^2~mod~N.$
+\end{itemize}
+\end{itemize}
+\item The ciphertext is computed by XORing the plaintext bits $m$ with the keystream: $ c = (c_0, \dots, c_{L-1}) = m \oplus b$. This ciphertext is $[c, y]$, where $y=x_{0}^{2^{L}}~mod~N.$
+\end{enumerate}
+
+
+When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$ as follows:
+\begin{enumerate}
+\item Using the secret key $(p,q)$, she computes $r_p = y^{((p+1)/4)^{L}}~mod~p$ and $r_q = y^{((q+1)/4)^{L}}~mod~q$.
+\item The initial seed can be obtained using the following procedure: $x_0=q(q^{-1}~{mod}~p)r_p + p(p^{-1}~{mod}~q)r_q~{mod}~N$.
+\item She recomputes the bit-vector $b$ by using BBS and $x_0$.
+\item Alice finally computes the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$.
+\end{enumerate}
+
+
+\subsubsection{Proposal of a new Asymmetric Cryptosystem Adapted from Blum-Goldwasser}
+
+We propose to adapt the Blum-Goldwasser protocol as follows.
+Let $\mathsf{N} = \lfloor log(log(N)) \rfloor$ be the number of bits that can
+be obtained securely with the BBS generator using the public key $N$ of Alice.
+Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too, and
+her new public key will be $(S^0, N)$.
+
+To encrypt his message, Bob will compute
+%%RAPH : ici, j'ai mis un simple $
+%\begin{equation}
+$c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, \right.$
+$ \left. m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)$
+%%\end{equation}
+instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$.
+
+The same decryption stage as in Blum-Goldwasser leads to the sequence
+$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$.
+Thus, with a simple use of $S^0$, Alice can obtain the plaintext.
+By doing so, the proposed generator is used in place of BBS, leading to
+the inheritance of all the properties presented in this paper.
+
+\section{Conclusion}
-An efficient implementation on GPU based on a xor-like PRNG allows us to
-generate a huge number of pseudorandom numbers per second (about
-20Gsamples/s). This PRNG succeeds to pass the hardest batteries of TestU01.
-In future work we plan to extend this work for parallel PRNG for clusters or
-grid computing. We also plan to improve the BBS version in order to succeed all
-the tests of TestU01.
+In this paper, a formerly proposed PRNG based on chaotic iterations
+has been generalized to improve its speed. It has been proven to be
+chaotic according to Devaney.
+Efficient implementations on GPU using xor-like PRNGs as input generators
+have shown that a very large quantity of pseudorandom numbers can be generated per second (about
+20Gsamples/s), and that these proposed PRNGs succeed to pass the hardest battery in TestU01,
+namely the BigCrush.
+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}
+
+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,
+and the use of other categories of PRNGs as input will be studied too. The improvement
+of Blum-Goldwasser will be deepened. Finally, we
+will try to enlarge the quantity of pseudorandom numbers generated per second either
+in a simulation context or in a cryptographic one.
