X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/00f33bfae729c2256b647502d7cbd542ba0892e9..9e057cd5768916849c2767ef4bd0f54dd9adc3b4:/prng_gpu.tex diff --git a/prng_gpu.tex b/prng_gpu.tex index ed7e927..90f00f8 100644 --- a/prng_gpu.tex +++ b/prng_gpu.tex @@ -1,4 +1,5 @@ -\documentclass{article} +%\documentclass{article} +\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage{fullpage} @@ -10,6 +11,8 @@ \usepackage[ruled,vlined]{algorithm2e} \usepackage{listings} \usepackage[standard]{ntheorem} +\usepackage{algorithmic} +\usepackage{slashbox} % Pour mathds : les ensembles IR, IN, etc. \usepackage{dsfont} @@ -38,10 +41,10 @@ \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 @@ -56,6 +59,13 @@ A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is fin \end{abstract} +} + +\maketitle + +\IEEEdisplaynotcompsoctitleabstractindextext +\IEEEpeerreviewmaketitle + \section{Introduction} @@ -135,7 +145,7 @@ 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 rewritting of the Blum-Goldwasser asymmetric +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 @@ -153,7 +163,7 @@ 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 +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 @@ -216,7 +226,10 @@ We can finally remark that, to the best of our knowledge, no GPU implementation \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 $V_{i}$ @@ -229,7 +242,7 @@ Consider a topological space $(\mathcal{X},\tau)$ and a continuous function $f : \mathcal{X} \rightarrow \mathcal{X}$. \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} @@ -248,7 +261,7 @@ necessarily the same period). \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} @@ -256,12 +269,12 @@ The chaos property is strongly linked to the notion of ``sensitivity'', defined 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 @@ -320,15 +333,15 @@ Let us now recall how to define a suitable metric space where chaotic iterations 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} @@ -467,8 +480,9 @@ generator taken alone. Furthermore, our generator 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)} @@ -480,12 +494,16 @@ $s\leftarrow{\textit{XORshift}(n)}$\; $x\leftarrow{F_f(s,x)}$\; } 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)} $z\leftarrow{z\oplus{(z\ll13)}}$\; @@ -493,7 +511,7 @@ $z\leftarrow{z\oplus{(z\gg17)}}$\; $z\leftarrow{z\oplus{(z\ll5)}}$\; $y\leftarrow{z}$\; return $y$\; -\medskip +\end{small} \caption{An arbitrary round of \textit{XORshift} algorithm} \label{XORshift} \end{algorithm} @@ -536,7 +554,7 @@ x^0 \in \llbracket 0, 2^\mathsf{N}-1 \rrbracket, S \in \llbracket 0, 2^\mathsf{N \label{equation Oplus} \end{equation} where $\oplus$ is for the bitwise exclusive or between two integers. -This rewritting 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 @@ -576,11 +594,11 @@ faster, does not deflate their topological chaos properties. \subsection{Proofs of Chaos of the General Formulation of the Chaotic Iterations} \label{deuxième def} Let us consider the discrete dynamical systems in chaotic iterations having -the general form: +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. @@ -605,14 +623,13 @@ Let us introduce the following function: 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: @@ -622,7 +639,7 @@ 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 @@ -649,17 +666,21 @@ Let us introduce: 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 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} +\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 $A\,\Delta\,B = (A \setminus B) \cup (B \setminus A)$. @@ -738,14 +759,16 @@ thus after $n_{2}$, the $k+2$ first terms of $S^n$ and $S$ are equal. \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} @@ -785,7 +808,7 @@ where $(s^0,s^1, \hdots)$ is the strategy of $Y$, satisfies the properties claimed in the lemma. \end{proof} -We can now prove 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. @@ -803,8 +826,10 @@ 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 $S'$: $$\tilde -S=(S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots,S_{t_1-1},S'_0,\dots,S'_{t_2-1},S_0,\dots).$$ 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 $t0.9999$. + + +\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} @@ -824,41 +1277,44 @@ given PRNG. An iteration of the system is simply the bitwise exclusive or between the last computed state and the current strategy. Topological properties of disorder exhibited by chaotic -iterations can be inherited by the inputted generator, 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. - -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 iteration\ -s},label=algo:seqCIPRNG} +%%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} + + + + +\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(); @@ -873,7 +1329,7 @@ unsigned int CIPRNG() { return x; } \end{lstlisting} - +\end{small} @@ -902,7 +1358,7 @@ 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 remind that in the CUDA~\cite{Nvid10} +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 @@ -916,7 +1372,7 @@ It is possible to deduce from the CPU version a quite similar version adapted to 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 @@ -929,8 +1385,9 @@ number $x$ that saves the last generated pseudorandom number. Additionally 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\;} @@ -943,14 +1400,16 @@ 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)}, @@ -994,7 +1453,7 @@ bits). 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\; @@ -1016,7 +1475,7 @@ array\_comb1, array\_comb2: Arrays containing combinations of size combination\_ } store internal variables in InternalVarXorLikeArray[threadId]\; } - +\end{small} \caption{Main kernel for the chaotic iterations based PRNG GPU efficient version\label{IR}} \label{algo:gpu_kernel2} @@ -1081,7 +1540,7 @@ As a comparison, Listing~\ref{algo:seqCIPRNG} leads to the generation of \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} @@ -1095,12 +1554,12 @@ 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 necessary paid by a speed +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} @@ -1108,7 +1567,7 @@ reduction. 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. @@ -1128,17 +1587,17 @@ In this section the concatenation of two strings $u$ and $v$ is classically 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 $k$ of length $k$, $G(k)$ (the output of $G$ on the input $k$) has size -$\ell_G(k)$ with $\ell_G(k)>k$. +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, -$$| \mathrm{Pr}[D(G(U_k))=1]-Pr[D(U_{\ell_G(k)})=1]|< \frac{1}{p(k)},$$ +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} @@ -1147,7 +1606,7 @@ distinguish a perfect uniform random generator from $G$ with a non negligible probability. The interested reader is referred to~\cite[chapter~3]{Goldreich} for more information. Note that it is quite easily possible to change the function $\ell$ into any polynomial -function $\ell^\prime$ satisfying $\ell^\prime(N)>N)$~\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, @@ -1158,7 +1617,7 @@ strings of length $N$ such that $H(S_0)=S_1 \ldots S_k$ ($H(S_0)$ is the concate 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. @@ -1202,8 +1661,10 @@ $y\bigoplus_{i=1}^{i=j} w_i^\prime=y\bigoplus_{i=1}^{i=j} w_i$. It follows, 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 $x$, @@ -1212,7 +1673,7 @@ D^\prime(H(x))=D(\varphi_y(H(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. @@ -1222,11 +1683,11 @@ It follows that \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} @@ -1239,7 +1700,7 @@ It is possible to build a cryptographically secure PRNG based on the previous 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: +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. @@ -1255,21 +1716,21 @@ lesser than $2^{16}$. So in practice we can choose prime numbers around 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 TestU01, +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 proceed with + 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 combinations +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, providing so a number between 0 and 7. +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 @@ -1277,7 +1738,7 @@ 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 add to the +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. @@ -1290,7 +1751,7 @@ variable for BBS number 8 is stored in place 1. \end{itemize} \begin{algorithm} - +\begin{small} \KwIn{InternalVarBBSArray: array with internal variables of the 8 BBS in global memory\; NumThreads: Number of threads\; @@ -1326,7 +1787,7 @@ array\_shift[4]=\{0,1,3,7\}\; } 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} @@ -1344,7 +1805,7 @@ 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 -correspondance between the shift and the number obtained with \texttt{shift} 1 +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). @@ -1358,6 +1819,40 @@ secure. +\begin{color}{red} +\subsection{Practical Security Evaluation} + +Suppose now that the PRNG will work during +$M=100$ time units, and that during this period, +an attacker can realize $10^{12}$ clock cycles. +We thus wonder whether, during the PRNG's +lifetime, the attacker can distinguish this +sequence from truly random one, with a probability +greater than $\varepsilon = 0.2$. +We consider that $N$ has 900 bits. + +The random process is the BBS generator, which +is cryptographically secure. More precisely, it +is $(T,\varepsilon)-$secure: no +$(T,\varepsilon)-$distinguishing attack can be +successfully realized on this PRNG, if~\cite{Fischlin} +$$ +T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M) +$$ +where $M$ is the length of the output ($M=100$ in +our example), and $L(N)$ is equal to +$$ +2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right) +$$ +is the number of clock cycles to factor a $N-$bit +integer. + +A direct numerical application shows that this attacker +cannot achieve its $(10^{12},0.2)$ distinguishing +attack in that context. + +\end{color} + \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem} \label{Blum-Goldwasser} We finish this research work by giving some thoughts about the use of @@ -1401,7 +1896,7 @@ When Alice receives $\left[(c_0, \dots, c_{L-1}), y\right]$, she can recover $m$ \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 computes finally the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$. +\item Alice finally computes the plaintext by XORing the keystream with the ciphertext: $ m = c \oplus b$. \end{enumerate} @@ -1414,14 +1909,16 @@ Alice will pick randomly $S^0$ in $\llbracket 0, 2^{\mathsf{N}-1}\rrbracket$ too her new public key will be $(S^0, N)$. To encrypt his message, Bob will compute -\begin{equation} -c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right) -\end{equation} +%%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 obtained the plaintext. +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. @@ -1432,16 +1929,16 @@ 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 -shown that a very large quantity of pseudorandom numbers can be generated per second (about +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. -Thoughts about an improvement of the Blum-Goldwasser cryptosystem, using the -proposed method, has been finally proposed. +\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 these researches, building a parallel PRNG for clusters or +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