add format parallel comp

[prng_gpu.git] / prng_gpu.tex

diff --git a/prng_gpu.tex b/prng_gpu.tex
index f769cfb0e2b1511e9d0a6acdf7e984f27fe5ad61..b3a844a113d6826ba9d14acc7219f966e32520c1 100644 (file)
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -1,5 +1,6 @@
 %\documentclass{article}
 %\documentclass{article}

-\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
+%\documentclass[10pt,journal,letterpaper,compsoc]{IEEEtran}
+\documentclass[preprint,12pt]{elsarticle}

 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{fullpage}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
 \usepackage{fullpage}


@@ -27,7 +28,6 @@
 % Pour faire des sous-figures dans les figures
 \usepackage{subfigure}
 
 % Pour faire des sous-figures dans les figures
 \usepackage{subfigure}
 

-\usepackage{color}

 
 \newtheorem{notation}{Notation}
 
 
 \newtheorem{notation}{Notation}
 


@@ -41,7 +41,6 @@
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
 
 \newcommand{\alert}[1]{\begin{color}{blue}\textit{#1}\end{color}}
 
 

-\newcommand{\PCH}[1]{\begin{color}{blue}#1\end{color}}

 
 \title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
 \begin{document}
 
 \title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
 \begin{document}


@@ -50,7 +49,7 @@
 Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}
    
 
 Guyeux, and Pierre-Cyrille Héam\thanks{Authors in alphabetic order}}
    
 

-\IEEEcompsoctitleabstractindextext{
+%\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
 \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


@@ -65,12 +64,12 @@ A chaotic version of the Blum-Goldwasser asymmetric key encryption scheme is fin
 
 
 \end{abstract}
 
 
 \end{abstract}

-}
+%}

 
 \maketitle
 
 
 \maketitle
 

-\IEEEdisplaynotcompsoctitleabstractindextext
-\IEEEpeerreviewmaketitle
+%\IEEEdisplaynotcompsoctitleabstractindextext
+%\IEEEpeerreviewmaketitle

 
 
 \section{Introduction}
 
 
 \section{Introduction}


@@ -93,12 +92,12 @@ 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
 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. \begin{color}{red} However, in an equivalent formulation, he or she should not be
+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.
 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.

-\end{color}
+

 
 Finally, a small part of the community working in this domain focuses on a
 third requirement, that is to define chaotic generators.
 
 Finally, a small part of the community working in this domain focuses on a
 third requirement, that is to define chaotic generators.


@@ -133,7 +132,6 @@ 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}.
 {\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}.

-\begin{color}{red}

 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
 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


@@ -142,7 +140,6 @@ 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].
 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].

-\end{color}

 Chaos, for its part, refers to the well-established definition of a
 chaotic dynamical system proposed by Devaney~\cite{Devaney}.
 
 Chaos, for its part, refers to the well-established definition of a
 chaotic dynamical system proposed by Devaney~\cite{Devaney}.
 


@@ -170,7 +167,6 @@ Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric
 key encryption protocol by using the proposed method.
 
 
 key encryption protocol by using the proposed method.
 
 

-\PCH{

 {\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
 {\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


@@ -183,7 +179,7 @@ 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.
 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.

-}
+

 
 
 
 
 
 


@@ -193,12 +189,12 @@ The remainder of this paper  is organized as follows. In Section~\ref{section:re
   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}.
   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}.

-\begin{color}{red}
+

 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{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.

-\end{color}
+

  Section~\ref{sec:efficient PRNG
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 
  Section~\ref{sec:efficient PRNG
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 


@@ -206,8 +202,8 @@ 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.
 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.

-\begin{color}{red} A practical
-security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.\end{color}
+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
 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


@@ -522,7 +518,6 @@ 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 
 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 

-\begin{color}{red}

 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.
 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.


@@ -666,12 +661,11 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
 \label{Chaotic iteration1}
 \end{algorithmic}
 \end{algorithm}
 \label{Chaotic iteration1}
 \end{algorithmic}
 \end{algorithm}

-\end{color}

 
 \subsection{Improving the Speed of the Former Generator}
 
 
 \subsection{Improving the Speed of the Former Generator}
 

-Instead of updating only one cell at each iteration, \begin{color}{red} we now propose to choose a
-subset of components and to update them together, for speed improvement. Such a proposition leads \end{color}
+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:
 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:


@@ -974,7 +968,6 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 \end{proof}
 
 
 \end{proof}
 
 

-\begin{color}{red}

 \section{Statistical Improvements Using Chaotic Iterations}
 
 \label{The generation of pseudorandom sequence}
 \section{Statistical Improvements Using Chaotic Iterations}
 
 \label{The generation of pseudorandom sequence}


@@ -1136,7 +1129,7 @@ a^1 & \text{if}~  z^{n-1} = 0 .\end{array} \right. \end{array}\end{equation}
 
 
 \begin{table}
 
 
 \begin{table}

-\renewcommand{\arraystretch}{1.3}
+%\renewcommand{\arraystretch}{1}

 \caption{TestU01 Statistical Test Failures}
 \label{TestU011}
 \centering
 \caption{TestU01 Statistical Test Failures}
 \label{TestU011}
 \centering


@@ -1158,7 +1151,7 @@ Failures          &       &261            &146    &0       \\
 
 
 \begin{table}
 
 
 \begin{table}

-\renewcommand{\arraystretch}{1.3}
+%\renewcommand{\arraystretch}{1}

 \caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
 \label{TestU01 for Old CI}
 \centering
 \caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
 \label{TestU01 for Old CI}
 \centering


@@ -1197,7 +1190,7 @@ TestU01~\cite{LEcuyerS07}, which encompasses the two other batteries.
 
 \label{Results and discussion}
 \begin{table*}
 
 \label{Results and discussion}
 \begin{table*}

-\renewcommand{\arraystretch}{1.3}
+%\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
 \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


@@ -1243,7 +1236,7 @@ However 8 tests have been improved (with no deflation for the other results).
 
 
 \begin{table*}
 
 
 \begin{table*}

-\renewcommand{\arraystretch}{1.3}
+%\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
 \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


@@ -1272,7 +1265,7 @@ Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be o
 using chaotic iterations on defective generators.
 
 \begin{table*}
 using chaotic iterations on defective generators.
 
 \begin{table*}

-\renewcommand{\arraystretch}{1.3}
+%\renewcommand{\arraystretch}{1.3}

 \caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
 \label{threshold}
 \centering
 \caption{Number of $\oplus$ operations to pass the whole NIST and DieHARD batteries}
 \label{threshold}
 \centering


@@ -1297,7 +1290,7 @@ The next subsection will now give a concrete original implementation of the Xor
 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.
 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.

-\end{color}
+

 
 \subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
 \label{sec:efficient PRNG}
 
 \subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
 \label{sec:efficient PRNG}


@@ -1379,11 +1372,11 @@ works with 32-bits, we use the command \texttt{(unsigned int)}, that selects the
 Thus producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers
 that  are provided by  3 64-bits  PRNGs.  This  version successfully  passes the
 stringent BigCrush battery of tests~\cite{LEcuyerS07}. 
 Thus producing a pseudorandom number needs 6 xor operations with 6 32-bits numbers
 that  are provided by  3 64-bits  PRNGs.  This  version successfully  passes the
 stringent BigCrush battery of tests~\cite{LEcuyerS07}. 

-\begin{color}{red}At this point, we thus
+At this point, we thus

 have defined an efficient and statistically unbiased generator. Its speed is
 directly related to the use of linear operations, but for the same reason,
 this fast generator cannot be proven as secure.
 have defined an efficient and statistically unbiased generator. Its speed is
 directly related to the use of linear operations, but for the same reason,
 this fast generator cannot be proven as secure.

-\end{color}
+

 
 
 \section{Efficient PRNGs based on Chaotic Iterations on GPU}
 
 
 \section{Efficient PRNGs based on Chaotic Iterations on GPU}


@@ -1520,9 +1513,7 @@ version\label{IR}}
 \label{algo:gpu_kernel2} 
 \end{algorithm}
 
 \label{algo:gpu_kernel2} 
 \end{algorithm}
 

-\begin{color}{red}

 \subsection{Chaos Evaluation of the Improved Version}
 \subsection{Chaos Evaluation of the Improved Version}

-\end{color}

 
 A run of Algorithm~\ref{algo:gpu_kernel2} consists in an operation ($x=x\oplus t$) having 
 the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative
 
 A run of Algorithm~\ref{algo:gpu_kernel2} consists in an operation ($x=x\oplus t$) having 
 the form of Equation~\ref{equation Oplus}, which is equivalent to the iterative


@@ -1581,7 +1572,7 @@ As a  comparison,   Listing~\ref{algo:seqCIPRNG}  leads   to the  generation of
 
 \begin{figure}[htbp]
 \begin{center}
 
 \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}
 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second with the xorlike-based PRNG}
 \label{fig:time_xorlike_gpu}


@@ -1600,7 +1591,7 @@ reduction.
 
 \begin{figure}[htbp]
 \begin{center}
 
 \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}
 \end{center}
 \caption{Quantity of pseudorandom numbers generated per second using the BBS-based PRNG}
 \label{fig:time_bbs_gpu}


@@ -1622,7 +1613,6 @@ as it is shown in the next sections.
 \section{Security Analysis}
 
 
 \section{Security Analysis}
 
 

-\begin{color}{red}

 This section is dedicated to the security analysis of the
   proposed PRNGs, both from a theoretical and from a practical point of view.
 
 This section is dedicated to the security analysis of the
   proposed PRNGs, both from a theoretical and from a practical point of view.
 


@@ -1640,7 +1630,6 @@ The standard definition
   enough, the system is secured.
 As a complement, an example of a concrete practical evaluation of security
 is outlined in the next subsection.
   enough, the system is secured.
 As a complement, an example of a concrete practical evaluation of security
 is outlined in the next subsection.

-\end{color}

 
 In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.
 
 In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.


@@ -1660,20 +1649,14 @@ probabilities are taken over $U_m$, $U_{\ell_G(m)}$ as well as over the
 internal coin tosses of $D$. 
 \end{definition}
 
 internal coin tosses of $D$. 
 \end{definition}
 

-Intuitively, it means that there is no polynomial time algorithm that can
-distinguish a perfect uniform random generator from $G$ with a non
-negligible probability.
-\begin{color}{red}
- 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.
-\end{color}
-The interested reader is referred
-to~\cite[chapter~3]{Goldreich} for more information. Note that it is
-quite easily possible to change the function $\ell$ into any polynomial
-function $\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}.
+Intuitively,  it means  that  there is  no  polynomial time  algorithm that  can
+distinguish a  perfect uniform random generator  from $G$ with  a non negligible
+probability.   An equivalent  formulation of  this well-known  security property
+means that  it is  possible \emph{in practice}  to predict  the next bit  of the
+generator, knowing all  the previously produced ones.  The  interested reader is
+referred to~\cite[chapter~3]{Goldreich}  for more  information. Note that  it is
+quite easily possible to change the function $\ell$ into any polynomial function
+$\ell^\prime$ satisfying $\ell^\prime(m)>m)$~\cite[Chapter 3.3]{Goldreich}.

 
 The generation schema developed in (\ref{equation Oplus}) is based on a
 pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,
 
 The generation schema developed in (\ref{equation Oplus}) is based on a
 pseudorandom generator. Let $H$ be a cryptographic PRNG. We may assume,


@@ -1759,7 +1742,6 @@ proving that $H$ is not secure, which is a contradiction.
 
 
 
 
 
 

-\begin{color}{red}

 \subsection{Practical Security Evaluation}
 \label{sec:Practicak evaluation}
 
 \subsection{Practical Security Evaluation}
 \label{sec:Practicak evaluation}
 


@@ -1849,7 +1831,6 @@ A direct numerical application shows that this attacker
 cannot achieve its $(10^{12},0.2)$ distinguishing
 attack in that context.
 
 cannot achieve its $(10^{12},0.2)$ distinguishing
 attack in that context.
 

-\end{color}

 
 
 \section{Cryptographical Applications}
 
 
 \section{Cryptographical Applications}


@@ -1978,7 +1959,6 @@ 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.
 

-\begin{color}{red}

 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
 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


@@ -2007,7 +1987,6 @@ a formulation similar to Eq.\eqref{mesureConcrete}
 must be established. Authors
 hope to achieve this difficult task in a future
 work.
 must be established. Authors
 hope to achieve this difficult task in a future
 work.

-\end{color}

 
 
 \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
 
 
 \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}


@@ -2067,14 +2046,14 @@ her new public key will be $(S^0, N)$.
 
 To encrypt his message, Bob will compute
 %%RAPH : ici, j'ai mis un simple $
 
 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 
 
 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.
 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.


@@ -2092,8 +2071,8 @@ 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.
 Furthermore, we have shown that when the inputted generator is cryptographically
 secure, then it is the case too for the PRNG we propose, thus leading to
 the possibility to develop fast and secure PRNGs using the GPU architecture.

-\begin{color}{red} An improvement of the Blum-Goldwasser cryptosystem, making it 
-behave chaotically, has finally been proposed. \end{color}
+An improvement of the Blum-Goldwasser cryptosystem, making it 
+behave chaotically, has finally been proposed. 

 
 In future  work we plan to extend this research, building a parallel PRNG for  clusters or
 grid computing. Topological properties of the various proposed generators will be investigated,
 
 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,