X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/4ad2ccae91afa1f83fad9be3c87213a9b8d81734..3abda4f59d76446238e6b891bc14e1ac7b44c34a:/prng_gpu.tex diff --git a/prng_gpu.tex b/prng_gpu.tex index a57e2a0..f357476 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -11,7 +11,12 @@ \usepackage[ruled,vlined]{algorithm2e} \usepackage{listings} \usepackage[standard]{ntheorem} - +\usepackage{algorithmic} +\usepackage{slashbox} +\usepackage{ctable} +\usepackage{cite} +\usepackage{tabularx} +\usepackage{multirow} % Pour mathds : les ensembles IR, IN, etc. \usepackage{dsfont} @@ -21,8 +26,11 @@ \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} @@ -35,6 +43,8 @@ \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}} + + \title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU} \begin{document} @@ -45,8 +55,8 @@ Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}} \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 @@ -85,7 +95,13 @@ On the other side, speed is not the main requirement in cryptography: the great need is to define \emph{secure} generators able to withstand malicious attacks. Roughly speaking, an attacker should not be able in practice to make the distinction between numbers obtained with the secure generator and a true random -sequence. +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 @@ -119,10 +135,17 @@ statistical perfection refers to the ability to pass the whole {\it BigCrush} battery of tests, which is widely considered as the most stringent statistical evaluation of a sequence claimed as random. This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}. +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 @@ -146,23 +169,46 @@ property. 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. @@ -170,7 +216,7 @@ 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 @@ -229,7 +275,7 @@ with basic notions on topology (see for instance~\cite{Devaney}). \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 @@ -416,7 +462,7 @@ the metric space $(\mathcal{X},d)$. \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 @@ -473,10 +519,11 @@ Let us finally remark that the vectorial negation satisfies the hypotheses of bo 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!] @@ -485,21 +532,43 @@ present. ($n$ 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)} @@ -515,31 +584,92 @@ return $y$\; \end{algorithm} +\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 pseudorandom bit sequence of our generator. +$(m^n)_{n \in \mathds{N}} \in \mathcal{M}^\mathds{N}$ is computed from $PRNG_1$, where $\mathcal{M}\subset \mathds{N}^*$ is a finite nonempty set of integers. + +The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}. +The internal state is $x$, the output state is $r$. $a$ and $b$ are those computed by the two input +PRNGs. Lastly, the value $g(a)$ is an integer defined as in Eq.~\ref{Formula}. +This function 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(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} - -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. +\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$k\leftarrow{g(a)}$\; +\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} \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, +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} @@ -580,276 +710,626 @@ than the ones presented in Definition \ref{Def:chaotic iterations} because, inst we select a subset of components to change. -Obviously, replacing Algorithm~\ref{CI Algorithm} by +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. However, proofs +the vectorial negation, leads to a speed improvement +(the resulting generator will be referred as ``Xor CI PRNG'' +in what follows). +However, proofs of chaos obtained in~\cite{bg10:ij} have been established only for chaotic iterations of the form presented in Definition -\ref{Def:chaotic iterations}. The question is now to determine whether the +\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. +faster, does not deflate their topological chaos properties, has been +investigated in Annex~\ref{A-deuxième def}, leading to the following result. -\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{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} -\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. +%%RAF proof en supplementary, j'ai mis le theorem. +% A vérifier -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. +% \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 $, -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, $Y \in \mathcal{P}\left(X\right) \Longleftrightarrow Y \subset X$. +%% \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} -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}% +%% In other words, at the $n^{th}$ iteration, only the cells whose id is +%% contained into the set $S^{n}$ are iterated. -Once more, a shift function appears as a component of these general chaotic -iterations. +%% 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. -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 +%% 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}{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}}}.% +%% \begin{array}{l} +%% X^0 \in \mathcal{X} \\ +%% X^{k+1}=G_{f}(X^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 -$A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$. +%% \end{equation}% +%% Once more, a shift function appears as a component of these general chaotic +%% iterations. -\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} +%% 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 $t0$ 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}. +%% 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. -Before being able to study the topological behavior of the general -chaotic iterations, we must first establish that: +%% 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} -\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} +%% 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} -\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} +%% Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, 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} -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{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} -\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 $tm)$~\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, @@ -1198,7 +1702,7 @@ PRNG too. \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 @@ -1261,6 +1765,103 @@ proving that $H$ is not secure, which is a contradiction. \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} @@ -1384,9 +1985,37 @@ It should be noticed that this generator has once more the form $x^{n+1} = x^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. - +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. \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem} @@ -1472,7 +2101,7 @@ 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. An improvement of the Blum-Goldwasser cryptosystem, making it -behaves chaotically, has finally been proposed. +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,