-%\documentclass{article}
-\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
+\documentclass{article}
+%\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
+%\documentclass[preprint,12pt]{elsarticle}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{fullpage}
\newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
-
+\begin{document}
\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
-\begin{document}
-\author{Jacques M. Bahi, Rapha\"{e}l Couturier, Christophe
-Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}
-
-\IEEEcompsoctitleabstractindextext{
+%% \author{Jacques M. Bahi}
+%% \ead{jacques.bahi@univ-fcomte.fr}
+%% \author{ Rapha\"{e}l Couturier \corref{cor1}}
+%% \ead{raphael.couturier@univ-fcomte.fr}
+%% \cortext[cor1]{Corresponding author}
+%% \author{ Christophe Guyeux}
+%% \ead{christophe.guyeux@univ-fcomte.fr}
+%% \author{ Pierre-Cyrille Héam }
+%% \ead{pierre-cyrille.heam@univ-fcomte.fr}
+
+\author{Christophe Guyeux \and Rapha\"{e}l Couturier \and Pierre-Cyrille Héam \and Jacques M. Bahi\\
+FEMTO-ST Institute, UMR 6174 CNRS,\\ University of Franche Comte, Belfort, France}
+
+\maketitle
+
+
+%\begin{frontmatter}
+%\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
\end{abstract}
-}
+%}
+%\begin{keyword}
+% pseudo random number\sep parallelization\sep GPU\sep cryptography\sep chaos
+%\end{keyword}
+%\end{frontmatter}
-\maketitle
-\IEEEdisplaynotcompsoctitleabstractindextext
-\IEEEpeerreviewmaketitle
+%\IEEEdisplaynotcompsoctitleabstractindextext
+%\IEEEpeerreviewmaketitle
\section{Introduction}
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. However, in an equivalent formulation, he or she should not be
+sequence. Or, in an equivalent formulation, he or she should not be
able (in practice) to predict the next bit of the generator, having the knowledge of all the
binary digits that have been already released. ``Being able in practice'' refers here
to the possibility to achieve this attack in polynomial time, and to the exponential growth
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}.
+chaotic dynamical system defined 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
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.
+%Note also that an original qualitative comparison between topological chaotic
+%properties and statistical tests is also proposed.
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{The generation of pseudorandom sequence} illustrates the statistical
-improvement related to the chaotic iteration based post-treatment, for
-our previously released PRNGs and a new efficient
-implementation on CPU.
-
- Section~\ref{sec:efficient PRNG
- gpu} describes and evaluates theoretically the GPU implementation.
+%Section~\ref{The generation of pseudorandom sequence} illustrates the statistical
+%improvement related to the chaotic iteration based post-treatment, for
+%our previously released PRNGs and a new efficient
+%implementation on CPU.
+ Section~\ref{sec:efficient PRNG} %{sec:efficient PRNG
+% gpu}
+ describes and evaluates theoretically new effective versions of
+our pseudorandom generators, in particular with a GPU implementation.
Such generators are experimented in
Section~\ref{sec:experiments}.
We show in Section~\ref{sec:security analysis} that, if the inputted
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.
+Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as input present.
\end{algorithmic}
\end{algorithm}
+
+We have shown in~\cite{bfg12a:ip} that the use of chaotic iterations
+implies an improvement of the statistical properties for all the
+inputted defective generators we have investigated.
+For instance, when considering the TestU01 battery with its 588 tests, we obtained 261
+failures for a PRNG based on the logistic map alone, and
+this number of failures falls below 138 in the Old CI(Logistic,Logistic) generator.
+In the XORshift case (146 failures when considering it alone), the results are more amazing,
+as the chaotic iterations post-treatment makes it fails only 8 tests.
+Further investigations have been systematically realized in \cite{bfg12a:ip}
+using a large set of inputted defective PRNGs, the three most used batteries of
+tests (DieHARD, NIST, and TestU01), and for all the versions of generators we have proposed.
+In all situations, an obvious improvement of the statistical behavior has
+been obtained, reinforcing the impression that chaos leads to statistical
+enhancement~\cite{bfg12a:ip}.
+
\subsection{Improving the Speed of the Former Generator}
Instead of updating only one cell at each iteration, we now propose to choose a
\end{proof}
-\section{Statistical Improvements Using Chaotic Iterations}
-
-\label{The generation of pseudorandom sequence}
-
-
-Let us now explain why we have reasonable ground to believe that chaos
-can improve statistical properties.
-We will show in this section that chaotic properties as defined in the
-mathematical theory of chaos are related to some statistical tests that can be found
-in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with
-chaotic iterations, the new generator presents better statistical properties
-(this section summarizes and extends the work of~\cite{bfg12a:ip}).
-
-
-
-\subsection{Qualitative relations between topological properties and statistical tests}
-
-
-There are various relations between topological properties that describe an unpredictable behavior for a discrete
-dynamical system on the one
-hand, and statistical tests to check the randomness of a numerical sequence
-on the other hand. These two mathematical disciplines follow a similar
-objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a
-recurrent sequence), with two different but complementary approaches.
-It is true that the following illustrative links give only qualitative arguments,
-and proofs should be provided later to make such arguments irrefutable. However
-they give a first understanding of the reason why we think that chaotic properties should tend
-to improve the statistical quality of PRNGs.
-%
-Let us now list some of these relations between topological properties defined in the mathematical
-theory of chaos and tests embedded into the NIST battery. %Such relations need to be further
-%investigated, but they presently give a first illustration of a trend to search similar properties in the
-%two following fields: mathematical chaos and statistics.
-
-
-\begin{itemize}
- \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney}, a chaotic dynamical system must
-have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of
-a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity
-is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a
-knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in
-the two following NIST tests~\cite{Nist10}:
- \begin{itemize}
- \item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern.
- \item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are close one to another) in the tested sequence that would indicate a deviation from the assumption of randomness.
- \end{itemize}
-
-\item \textbf{Transitivity}. This topological property previously introduced states that the dynamical system is intrinsically complicated: it cannot be simplified into
-two subsystems that do not interact, as we can find in any neighborhood of any point another point whose orbit visits the whole phase space.
-This focus on the places visited by the orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory
-of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention
-is brought on the states visited during a random walk in the two tests below~\cite{Nist10}:
- \begin{itemize}
- \item \textbf{Random Excursions Variant Test}. Detect deviations from the expected number of visits to various states in the random walk.
- \item \textbf{Random Excursions Test}. Determine if the number of visits to a particular state within a cycle deviates from what one would expect for a random sequence.
- \end{itemize}
-
-\item \textbf{Chaos according to Li and Yorke}. Two points of the phase space $(x,y)$ define a couple of Li-Yorke when $\limsup_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))>0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillate as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according
-to Li-Yorke~\cite{Li75,Ruette2001}. A similar property is regarded in the following NIST test~\cite{Nist10}.
- \begin{itemize}
- \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow.
- \end{itemize}
- \item \textbf{Topological entropy}. The desire to formulate an equivalency of the thermodynamics entropy
-has emerged both in the topological and statistical fields. Once again, a similar objective has led to two different
-rewritting of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach,
-whereas topological entropy is defined as follows:
-$x,y \in \mathcal{X}$ are $\varepsilon-$\emph{separated in time $n$} if there exists $k \leqslant n$ such that $d\left(f^{(k)}(x),f^{(k)}(y)\right)>\varepsilon$. Then $(n,\varepsilon)-$separated sets are sets of points that are all $\varepsilon-$separated in time $n$, which
-leads to the definition of $s_n(\varepsilon,Y)$, being the maximal cardinality of all $(n,\varepsilon)-$separated sets. Using these notations,
-the topological entropy is defined as follows: $$h_{top}(\mathcal{X},f) = \displaystyle{\lim_{\varepsilon \rightarrow 0} \Big[ \limsup_{n \rightarrow +\infty} \dfrac{1}{n} \log s_n(\varepsilon,\mathcal{X})\Big]}.$$
-This value measures the average exponential growth of the number of distinguishable orbit segments.
-In this sense, it measures the complexity of the topological dynamical system, whereas
-the Shannon approach comes to mind when defining the following test~\cite{Nist10}:
- \begin{itemize}
-\item \textbf{Approximate Entropy Test}. Compare the frequency of the overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence.
- \end{itemize}
-
- \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are
-not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}.
- \begin{itemize}
-\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence.
-\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random.
- \end{itemize}
-\end{itemize}
-
-
-We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques} are, among other
-things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke,
-and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$,
-where $\mathsf{N}$ is the size of the iterated vector.
-These topological properties make that we are ground to believe that a generator based on chaotic
-iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like
-the NIST one. The following subsections, in which we prove that defective generators have their
-statistical properties improved by chaotic iterations, show that such an assumption is true.
-
-\subsection{Details of some Existing Generators}
-
-The list of defective PRNGs we will use
-as inputs for the statistical tests to come is introduced here.
-
-Firstly, the simple linear congruency generators (LCGs) will be used.
-They are defined by the following recurrence:
-\begin{equation}
-x^n = (ax^{n-1} + c)~mod~m,
-\label{LCG}
-\end{equation}
-where $a$, $c$, and $x^0$ must be, among other things, non-negative and inferior to
-$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer to two (resp. three)
-combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
+%\section{Statistical Improvements Using Chaotic Iterations}
+
+%\label{The generation of pseudorandom sequence}
+
+
+%Let us now explain why we have reasonable ground to believe that chaos
+%can improve statistical properties.
+%We will show in this section that chaotic properties as defined in the
+%mathematical theory of chaos are related to some statistical tests that can be found
+%in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with
+%chaotic iterations, the new generator presents better statistical properties
+%(this section summarizes and extends the work of~\cite{bfg12a:ip}).
+
+
+
+%\subsection{Qualitative relations between topological properties and statistical tests}
+
+
+%There are various relations between topological properties that describe an unpredictable behavior for a discrete
+%dynamical system on the one
+%hand, and statistical tests to check the randomness of a numerical sequence
+%on the other hand. These two mathematical disciplines follow a similar
+%objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a
+%recurrent sequence), with two different but complementary approaches.
+%It is true that the following illustrative links give only qualitative arguments,
+%and proofs should be provided later to make such arguments irrefutable. However
+%they give a first understanding of the reason why we think that chaotic properties should tend
+%to improve the statistical quality of PRNGs.
+%%
+%Let us now list some of these relations between topological properties defined in the mathematical
+%theory of chaos and tests embedded into the NIST battery. %Such relations need to be further
+%%investigated, but they presently give a first illustration of a trend to search similar properties in the
+%%two following fields: mathematical chaos and statistics.
+
+
+%\begin{itemize}
+% \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney}, a chaotic dynamical system must
+%have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of
+%a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity
+%is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a
+%knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in
+%the two following NIST tests~\cite{Nist10}:
+% \begin{itemize}
+% \item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern.
+% \item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are close one to another) in the tested sequence that would indicate a deviation from the assumption of randomness.
+% \end{itemize}
+
+%\item \textbf{Transitivity}. This topological property previously introduced states that the dynamical system is intrinsically complicated: it cannot be simplified into
+%two subsystems that do not interact, as we can find in any neighborhood of any point another point whose orbit visits the whole phase space.
+%This focus on the places visited by the orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory
+%of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention
+%is brought on the states visited during a random walk in the two tests below~\cite{Nist10}:
+% \begin{itemize}
+% \item \textbf{Random Excursions Variant Test}. Detect deviations from the expected number of visits to various states in the random walk.
+% \item \textbf{Random Excursions Test}. Determine if the number of visits to a particular state within a cycle deviates from what one would expect for a random sequence.
+% \end{itemize}
+
+%\item \textbf{Chaos according to Li and Yorke}. Two points of the phase space $(x,y)$ define a couple of Li-Yorke when $\limsup_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))>0$ et $\liminf_{n \rightarrow +\infty} d(f^{(n)}(x), f^{(n)}(y))=0$, meaning that their orbits always oscillate as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according
+%to Li-Yorke~\cite{Li75,Ruette2001}. A similar property is regarded in the following NIST test~\cite{Nist10}.
+% \begin{itemize}
+% \item \textbf{Runs Test}. To determine whether the number of runs of ones and zeros of various lengths is as expected for a random sequence. In particular, this test determines whether the oscillation between such zeros and ones is too fast or too slow.
+% \end{itemize}
+% \item \textbf{Topological entropy}. The desire to formulate an equivalency of the thermodynamics entropy
+%has emerged both in the topological and statistical fields. Once again, a similar objective has led to two different
+%rewritting of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach,
+%whereas topological entropy is defined as follows:
+%$x,y \in \mathcal{X}$ are $\varepsilon-$\emph{separated in time $n$} if there exists $k \leqslant n$ such that $d\left(f^{(k)}(x),f^{(k)}(y)\right)>\varepsilon$. Then $(n,\varepsilon)-$separated sets are sets of points that are all $\varepsilon-$separated in time $n$, which
+%leads to the definition of $s_n(\varepsilon,Y)$, being the maximal cardinality of all $(n,\varepsilon)-$separated sets. Using these notations,
+%the topological entropy is defined as follows: $$h_{top}(\mathcal{X},f) = \displaystyle{\lim_{\varepsilon \rightarrow 0} \Big[ \limsup_{n \rightarrow +\infty} \dfrac{1}{n} \log s_n(\varepsilon,\mathcal{X})\Big]}.$$
+%This value measures the average exponential growth of the number of distinguishable orbit segments.
+%In this sense, it measures the complexity of the topological dynamical system, whereas
+%the Shannon approach comes to mind when defining the following test~\cite{Nist10}:
+% \begin{itemize}
+%\item \textbf{Approximate Entropy Test}. Compare the frequency of the overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence.
+% \end{itemize}
+
+% \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are
+%not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}.
+% \begin{itemize}
+%\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence.
+%\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random.
+% \end{itemize}
+%\end{itemize}
+
+
+%We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques} are, among other
+%things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke,
+%and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$,
+%where $\mathsf{N}$ is the size of the iterated vector.
+%These topological properties make that we are ground to believe that a generator based on chaotic
+%iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like
+%the NIST one. The following subsections, in which we prove that defective generators have their
+%statistical properties improved by chaotic iterations, show that such an assumption is true.
+
+%\subsection{Details of some Existing Generators}
+
+%The list of defective PRNGs we will use
+%as inputs for the statistical tests to come is introduced here.
+
+%Firstly, the simple linear congruency generators (LCGs) will be used.
+%They are defined by the following recurrence:
+%\begin{equation}
+%x^n = (ax^{n-1} + c)~mod~m,
+%\label{LCG}
+%\end{equation}
+%where $a$, $c$, and $x^0$ must be, among other things, non-negative and inferior to
+%$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer to two (resp. three)
+%combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
-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.
+%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.
-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}
+%Generators based on linear recurrences with carry will be regarded too.
+%This family of generators includes the add-with-carry (AWC) generator, based on the recurrence:
+%\begin{equation}
+%\label{AWC}
+%\begin{array}{l}
+%x^n = (x^{n-r} + x^{n-s} + c^{n-1})~mod~m, \\
+%c^n= (x^{n-r} + x^{n-s} + c^{n-1}) / m, \end{array}\end{equation}
+%the SWB generator, having the recurrence:
+%\begin{equation}
+%\label{SWB}
+%\begin{array}{l}
+%x^n = (x^{n-r} - x^{n-s} - c^{n-1})~mod~m, \\
+%c^n=\left\{
+%\begin{array}{l}
+%1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
+%0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
+%and the SWC generator, which is based on the following recurrence:
+%\begin{equation}
+%\label{SWC}
+%\begin{array}{l}
+%x^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ mod ~ 2^w, \\
+%c^n = (a^1x^{n-1} \oplus ~...~ \oplus a^rx^{n-r} \oplus c^{n-1}) ~ / ~ 2^w. \end{array}\end{equation}
-Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
-\begin{equation}
-x^n = x^{n-r} \oplus x^{n-k} .
-\label{GFSR}
-\end{equation}
+%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}
-Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:
+%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}
-
-
-
-\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{table}
-\renewcommand{\arraystretch}{1.3}
-\caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
-\label{TestU01 for Old CI}
-\centering
- \begin{tabular}{lcccc}
- \toprule
-\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
-&Logistic& XORshift& ISAAC&ISAAC \\
-&+& +& + & + \\
-&Logistic& XORshift& XORshift&ISAAC \\ \cmidrule(r){2-5}
-Rabbit &7 &2 &0 &0 \\
-Alphabit & 3 &0 &0 &0 \\
-DieHARD &0 &0 &0 &0 \\
-FIPS\_140\_2 &0 &0 &0 &0 \\
-SmallCrush &2 &0 &0 &0 \\
-Crush &47 &4 &0 &0 \\
-Big Crush &79 &3 &0 &0 \\ \hline
-Failures &138 &9 &0 &0 \\
-\bottomrule
- \end{tabular}
-\end{table}
-
-
-
-
-
-\subsection{Statistical tests}
-\label{Security analysis}
-
-Three batteries of tests are reputed and regularly used
-to evaluate the statistical properties of newly designed pseudorandom
-number generators. These batteries are named DieHard~\cite{Marsaglia1996},
-the NIST suite~\cite{ANDREW2008}, and the most stringent one called
-TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
-
-
-
-\label{Results and discussion}
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
-\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
- \hline\hline
-Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
-\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
-NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15} & 14/15 & 14/15 & 14/15 & 14/15& 14/15& 14/15 \\ \hline
-DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
-\end{tabular}
-\end{table*}
-
-Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the
-results on the two first batteries recalled above, indicating that all the PRNGs presented
-in the previous section
-cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot
-fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
-iterations can solve this issue.
-%More precisely, to
-%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
-%\begin{enumerate}
-% \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
-% \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
-% \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
%\begin{equation}
+%\label{INV}
%\begin{array}{l}
-%\left\{
-%\begin{array}{l}
-%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
-%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
-%\end{equation}
-%$m$ is called the \emph{functional power}.
-%\end{enumerate}
-%
-The obtained results are reproduced in Table
-\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
-The scores written in boldface indicate that all the tests have been passed successfully, whereas an
-asterisk ``*'' means that the considered passing rate has been improved.
-The improvements are obvious for both the ``Old CI'' and the ``New CI'' generators.
-Concerning the ``Xor CI PRNG'', the score is less spectacular. Because of a large speed improvement, the statistics
- are not as good as for the two other versions of these CIPRNGs.
-However 8 tests have been improved (with no deflation for the other results).
-
-
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
-\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
- \hline
-Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
-\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
-Old CIPRNG\\ \hline \hline
-NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
-DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
-New CIPRNG\\ \hline \hline
-NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
-DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
-Xor CIPRNG\\ \hline\hline
-NIST & 14/15*& \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} \\ \hline
-DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
-\end{tabular}
-\end{table*}
-
-
-We have then investigated in~\cite{bfg12a:ip} if it were possible to improve
-the statistical behavior of the Xor CI version by combining more than one
-$\oplus$ operation. Results are summarized in Table~\ref{threshold}, illustrating
-the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in.
-Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained
-using chaotic iterations on defective generators.
-
-\begin{table*}
-\renewcommand{\arraystretch}{1.3}
-\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
-\label{threshold}
-\centering
- \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
- \hline
-Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3 & MRG2 \\ \hline\hline
-Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline
-\end{tabular}
-\end{table*}
-
-Finally, the TestU01 battery has been launched on three well-known generators
-(a logistic map, a simple XORshift, and the cryptographically secure ISAAC,
-see Table~\ref{TestU011}). These results can be compared with
-Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
-Old CI PRNG that has received these generators.
-The obvious improvement speaks for itself, and together with the other
-results recalled in this section, it reinforces the opinion that a strong
-correlation between topological properties and statistical behavior exists.
-
-
-The next subsection will now give a concrete original implementation of the Xor CI PRNG, the
-fastest generator in the chaotic iteration based family. In the remainder,
-this generator will be simply referred to as CIPRNG, or ``the proposed PRNG'', if this statement does not
-raise ambiguity.
-
+%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}
+
+
+
+%\begin{table}
+%%\renewcommand{\arraystretch}{1}
+%\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{table}
+%%\renewcommand{\arraystretch}{1}
+%\caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
+%\label{TestU01 for Old CI}
+%\centering
+% \begin{tabular}{lcccc}
+% \toprule
+%\multirow{3}*{Test name} & \multicolumn{4}{c}{Old CI}\\
+%&Logistic& XORshift& ISAAC&ISAAC \\
+%&+& +& + & + \\
+%&Logistic& XORshift& XORshift&ISAAC \\ \cmidrule(r){2-5}
+%Rabbit &7 &2 &0 &0 \\
+%Alphabit & 3 &0 &0 &0 \\
+%DieHARD &0 &0 &0 &0 \\
+%FIPS\_140\_2 &0 &0 &0 &0 \\
+%SmallCrush &2 &0 &0 &0 \\
+%Crush &47 &4 &0 &0 \\
+%Big Crush &79 &3 &0 &0 \\ \hline
+%Failures &138 &9 &0 &0 \\
+%\bottomrule
+% \end{tabular}
+%\end{table}
+
+
+
+
+
+%\subsection{Statistical tests}
+%\label{Security analysis}
+
+%Three batteries of tests are reputed and regularly used
+%to evaluate the statistical properties of newly designed pseudorandom
+%number generators. These batteries are named DieHard~\cite{Marsaglia1996},
+%the NIST suite~\cite{ANDREW2008}, and the most stringent one called
+%TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
+
+
+
+%\label{Results and discussion}
+%\begin{table*}
+%%\renewcommand{\arraystretch}{1}
+%\caption{NIST and DieHARD tests suite passing rates for PRNGs without CI}
+%\label{NIST and DieHARD tests suite passing rate the for PRNGs without CI}
+%\centering
+% \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|}
+% \hline\hline
+%Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%\backslashbox{\textbf{$Tests$}} {\textbf{$PRNG$}} & LCG& MRG& AWC & SWB & SWC & GFSR & INV & LCG2& LCG3& MRG2 \\ \hline
+%NIST & 11/15 & 14/15 &\textbf{15/15} & \textbf{15/15} & 14/15 & 14/15 & 14/15 & 14/15& 14/15& 14/15 \\ \hline
+%DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18& 16/18& 16/18\\ \hline
+%\end{tabular}
+%\end{table*}
+
+%Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the
+%results on the two first batteries recalled above, indicating that all the PRNGs presented
+%in the previous section
+%cannot pass all these tests. In other words, the statistical quality of these PRNGs cannot
+%fulfill the up-to-date standards presented previously. We have shown in~\cite{bfg12a:ip} that the use of chaotic
+%iterations can solve this issue.
+%%More precisely, to
+%%illustrate the effects of chaotic iterations on these defective PRNGs, experiments have been divided in three parts~\cite{bfg12a:ip}:
+%%\begin{enumerate}
+%% \item \textbf{Single CIPRNG}: The PRNGs involved in CI computing are of the same category.
+%% \item \textbf{Mixed CIPRNG}: Two different types of PRNGs are mixed during the chaotic iterations process.
+%% \item \textbf{Multiple CIPRNG}: The generator is obtained by repeating the composition of the iteration function as follows: $x^0\in \mathds{B}^{\mathsf{N}}$, and $\forall n\in \mathds{N}^{\ast },\forall i\in \llbracket1;\mathsf{N}\rrbracket, x_i^n=$
+%%\begin{equation}
+%%\begin{array}{l}
+%%\left\{
+%%\begin{array}{l}
+%%x_i^{n-1}~~~~~\text{if}~S^n\neq i \\
+%%\forall j\in \llbracket1;\mathsf{m}\rrbracket,f^m(x^{n-1})_{S^{nm+j}}~\text{if}~S^{nm+j}=i.\end{array} \right. \end{array}
+%%\end{equation}
+%%$m$ is called the \emph{functional power}.
+%%\end{enumerate}
+%%
+%The obtained results are reproduced in Table
+%\ref{NIST and DieHARD tests suite passing rate the for single CIPRNGs}.
+%The scores written in boldface indicate that all the tests have been passed successfully, whereas an
+%asterisk ``*'' means that the considered passing rate has been improved.
+%The improvements are obvious for both the ``Old CI'' and the ``New CI'' generators.
+%Concerning the ``Xor CI PRNG'', the score is less spectacular. Because of a large speed improvement, the statistics
+% are not as good as for the two other versions of these CIPRNGs.
+%However 8 tests have been improved (with no deflation for the other results).
+
+
+%\begin{table*}
+%%\renewcommand{\arraystretch}{1.3}
+%\caption{NIST and DieHARD tests suite passing rates for PRNGs with CI}
+%\label{NIST and DieHARD tests suite passing rate the for single CIPRNGs}
+%\centering
+% \begin{tabular}{|l||c|c|c|c|c|c|c|c|c|c|c|c|}
+% \hline
+%Types of PRNGs & \multicolumn{2}{c|}{Linear PRNGs} & \multicolumn{4}{c|}{Lagged PRNGs} & \multicolumn{1}{c|}{ICG PRNGs} & \multicolumn{3}{c|}{Mixed PRNGs}\\ \hline
+%\backslashbox{\textbf{$Tests$}} {\textbf{$Single~CIPRNG$}} & LCG & MRG & AWC & SWB & SWC & GFSR & INV& LCG2 & LCG3& MRG2 \\ \hline\hline
+%Old CIPRNG\\ \hline \hline
+%NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} * \\ \hline
+%New CIPRNG\\ \hline \hline
+%NIST & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} * & \textbf{15/15} * & \textbf{15/15} \\ \hline
+%DieHARD & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} * & \textbf{18/18} *& \textbf{18/18} *\\ \hline
+%Xor CIPRNG\\ \hline\hline
+%NIST & 14/15*& \textbf{15/15} * & \textbf{15/15} & \textbf{15/15} & 14/15 & \textbf{15/15} * & 14/15& \textbf{15/15} * & \textbf{15/15} *& \textbf{15/15} \\ \hline
+%DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18} & \textbf{18/18} * & 16/18 & 16/18 & 16/18& 16/18\\ \hline
+%\end{tabular}
+%\end{table*}
+
+
+%We have then investigated in~\cite{bfg12a:ip} if it were possible to improve
+%the statistical behavior of the Xor CI version by combining more than one
+%$\oplus$ operation. Results are summarized in Table~\ref{threshold}, illustrating
+%the progressive increasing effects of chaotic iterations, when giving time to chaos to get settled in.
+%Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained
+%using chaotic iterations on defective generators.
+
+%\begin{table*}
+%%\renewcommand{\arraystretch}{1.3}
+%\caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
+%\label{threshold}
+%\centering
+% \begin{tabular}{|l||c|c|c|c|c|c|c|c|}
+% \hline
+%Inputted $PRNG$ & LCG & MRG & SWC & GFSR & INV& LCG2 & LCG3 & MRG2 \\ \hline\hline
+%Threshold value $m$& 19 & 7 & 2& 1 & 11& 9& 3& 4\\ \hline\hline
+%\end{tabular}
+%\end{table*}
+
+%Finally, the TestU01 battery has been launched on three well-known generators
+%(a logistic map, a simple XORshift, and the cryptographically secure ISAAC,
+%see Table~\ref{TestU011}). These results can be compared with
+%Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
+%Old CI PRNG that has received these generators.
+%The obvious improvement speaks for itself, and together with the other
+%results recalled in this section, it reinforces the opinion that a strong
+%correlation between topological properties and statistical behavior exists.
+
+
+%The next subsection will now give a concrete original implementation of the Xor CI PRNG, the
+%fastest generator in the chaotic iteration based family. In the remainder,
+%this generator will be simply referred to as CIPRNG, or ``the proposed PRNG'', if this statement does not
+%raise ambiguity.
+
+
+\section{Toward Efficiency and Improvement for CI PRNG}
+\label{sec:efficient PRNG}
\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
-\label{sec:efficient PRNG}
%
%Based on the proof presented in the previous section, it is now possible to
%improve the speed of the generator formerly presented in~\cite{bgw09:ip,guyeux10}.
-\section{Efficient PRNGs based on Chaotic Iterations on GPU}
+\subsection{Efficient PRNGs based on Chaotic Iterations on GPU}
\label{sec:efficient PRNG gpu}
In order to take benefits from the computing power of GPU, a program
\begin{figure}[htbp]
\begin{center}
- \includegraphics[width=\columnwidth]{curve_time_xorlike_gpu.pdf}
+ \includegraphics[scale=0.7]{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}
\begin{figure}[htbp]
\begin{center}
- \includegraphics[width=\columnwidth]{curve_time_bbs_gpu.pdf}
+ \includegraphics[scale=0.7]{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}
$\mathcal{D}$ 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}
-
+$$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right. - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$$
\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.
To encrypt his message, Bob will compute
%%RAPH : ici, j'ai mis un simple $
-%\begin{equation}
-$c = \left(m_0 \oplus (b_0 \oplus S^0), m_1 \oplus (b_0 \oplus b_1 \oplus S^0), \hdots, \right.$
-$ \left. m_{L-1} \oplus (b_0 \oplus b_1 \hdots \oplus b_{L-1} \oplus S^0) \right)$
-%%\end{equation}
-instead of $\left(m_0 \oplus b_0, m_1 \oplus b_1, \hdots, m_{L-1} \oplus b_{L-1} \right)$.
+\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
-$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right)$.
+$$\left(m_0 \oplus S^0, m_1 \oplus S^0, \hdots, m_{L-1} \oplus S^0 \right).$$
Thus, with a simple use of $S^0$, Alice can obtain the plaintext.
By doing so, the proposed generator is used in place of BBS, leading to
the inheritance of all the properties presented in this paper.
