X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/9bc351a3f4add087b29ed019fe4d1a0db25b0aa6..e4f33be63a2bad38b19204da8d8be5c7e2113e13:/prng_gpu.tex?ds=sidebyside

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 38431e5..0ab28a1 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -27,7 +27,6 @@
 % Pour faire des sous-figures dans les figures
 \usepackage{subfigure}
 
-\usepackage{color}
 
 \newtheorem{notation}{Notation}
 
@@ -41,7 +40,6 @@
 \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}
@@ -93,12 +91,12 @@ 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
-sequence. \begin{color}{red} Or, 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.
-\end{color}
+
 
 Finally, a small part of the community working in this domain focuses on a
 third requirement, that is to define chaotic generators.
@@ -133,7 +131,6 @@ 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}.
-\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
@@ -142,7 +139,6 @@ 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].
-\end{color}
 Chaos, for its part, refers to the well-established definition of a
 chaotic dynamical system proposed by Devaney~\cite{Devaney}.
 
@@ -170,21 +166,19 @@ Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric
 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
-the based PRNG is also cryptographically secured). From a practical point of
-view, experiments point out a very good statistical behavior. Optimized
-original implementation of this PRNG are also proposed and experimented.
+the initial PRNG is also cryptographically secured). From a practical point of
+view, experiments point out a very good statistical behavior. An optimized
+original implementation of this PRNG is also proposed and experimented.
 Pseudorandom numbers are generated at a rate of 20GSamples/s, which is faster
 than in~\cite{conf/fpga/ThomasHL09,Marsaglia2003} (and with a better
-statistical behavior). Experiments are also provided using BBS as the based
-random generator. The generation speed is significantly weaker but, as far
-as we know, it is the first cryptographically secured PRNG proposed on GPU.
-Note too that an original qualitative comparison between topological chaotic
+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.
-}
+
 
 
 
@@ -194,12 +188,12 @@ 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}.
-\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.
-\end{color}
+
  Section~\ref{sec:efficient PRNG
   gpu}   describes and evaluates theoretically  the  GPU   implementation. 
 Such generators are experimented in 
@@ -207,8 +201,8 @@ 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.
-\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
@@ -523,11 +517,9 @@ 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 
-\begin{color}{red}
 should improve the statistical properties of each
 generator taken alone. 
-Furthermore, the generator obtained by this way possesses various chaos properties that none of the generators used as input
-present.
+Furthermore, the generator obtained in this way possesses various chaos properties that none of the generators used as present input.
 
 
 
@@ -593,8 +585,7 @@ return $y$\;
 
 In order to make the Old CI PRNG usable in practice, we have proposed 
 an adapted version of the chaotic iteration based generator in~\cite{bg10:ip}.
-In this ``New CI PRNG'', we prevent from changing twice a given
-bit between two outputs.
+In this ``New CI PRNG'', we prevent a given bit from changing twice between two outputs.
 This new generator is designed by the following process. 
 
 First of all, some chaotic iterations have to be done to generate a sequence 
@@ -610,7 +601,7 @@ Algorithm~\ref{Chaotic iteration1}.
 
 Then, at each iteration, only the $S^n$-th component of state $x^n$ is 
 updated, as follows: $x_i^n = x_i^{n-1}$ if $i \neq S^n$, else $x_i^n = \overline{x_i^{n-1}}$.
-Such a procedure is equivalent to achieve chaotic iterations with
+Such a procedure is equivalent to achieving chaotic iterations with
 the Boolean vectorial negation $f_0$ and some well-chosen strategies.
 Finally, some $x^n$ are selected
 by a sequence $m^n$ as the pseudorandom bit sequence of our generator.
@@ -669,12 +660,11 @@ N \text{ if }\sum_{i=0}^{N-1}{C^i_{32}}\leqslant{y^n}<1.\\
 \label{Chaotic iteration1}
 \end{algorithmic}
 \end{algorithm}
-\end{color}
 
 \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 improvements. 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:
@@ -977,13 +967,12 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 \end{proof}
 
 
-\begin{color}{red}
 \section{Statistical Improvements Using Chaotic Iterations}
 
 \label{The generation of pseudorandom sequence}
 
 
-Let us now explain why we are reasonable grounds to believe that chaos 
+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
@@ -1022,36 +1011,36 @@ knowledge about the behavior of the system, that is, it never enters into a loop
 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 near each other) in the tested sequence that would indicate a deviation from the assumption of randomness.
+        \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 introduced previously states that the dynamical system is intrinsically complicated: it cannot be simplified into 
+\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 orbits of the dynamical system takes various nonequivalent formulations in the mathematical theory
+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 states visited during a random walk in the two tests below~\cite{Nist10}:
+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 oscillates as the iterations pass. When a system is compact and contains an uncountable set of such points, it is claimed as chaotic according
+\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. Another time, a similar objective has led to two different
-rewritten of an entropy based disorder: the famous Shannon definition of entropy is approximated in the statistical approach, 
-whereas topological entropy is defined as follows.
+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 complexity of the topological dynamical system, whereas 
-the Shannon approach is in mind when defining the following test~\cite{Nist10}:
+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 overlapping blocks of two consecutive/adjacent lengths ($m$ and $m+1$) against the expected result for a random sequence.
+\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 
@@ -1083,18 +1072,18 @@ They are defined by the following recurrence:
 x^n = (ax^{n-1} + c)~mod~m,
 \label{LCG}
 \end{equation}
-where $a$, $c$, and $x^0$ must be, among other things, non-negative and less than 
-$m$~\cite{LEcuyerS07}. In what follows, 2LCGs and 3LCGs refer as two (resp. three) 
+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) will be used, which
+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}
-Combination of two MRGs (referred as 2MRGs) is also used in these experiments.
+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:
@@ -1112,7 +1101,7 @@ c^n=\left\{
 \begin{array}{l}
 1 ~~~~~\text{if}~ (x^{i-r} - x^{i-s} - c^{i-1})<0\\
 0 ~~~~~\text{else},\end{array} \right. \end{array}\end{equation}
-and the SWC generator designed by R. Couture, which is based on the following recurrence:
+and the SWC generator, which is based on the following recurrence:
 \begin{equation}
 \label{SWC}
 \begin{array}{l}
@@ -1140,7 +1129,7 @@ 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}
+\caption{TestU01 Statistical Test Failures}
 \label{TestU011}
 \centering
   \begin{tabular}{lccccc}
@@ -1162,7 +1151,7 @@ Failures 		&	&261 	 	&146	&0	 \\
 
 \begin{table}
 \renewcommand{\arraystretch}{1.3}
-\caption{TestU01 Statistical Test for Old CI algorithms ($\mathsf{N}=4$)}
+\caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
 \label{TestU01 for Old CI}
 \centering
   \begin{tabular}{lcccc}
@@ -1190,7 +1179,7 @@ Failures 	 		 	&138 	&9 	&0 	&0	 \\
 \subsection{Statistical tests}
 \label{Security analysis}
 
-Three batteries of tests are reputed and usually used
+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
@@ -1214,7 +1203,7 @@ DieHARD & 16/18 & 16/18 & 15/18 & 16/18 & \textbf{18/18} & 16/18 & 16/18 & 16/18
 \end{table*}
 
 Table~\ref{NIST and DieHARD tests suite passing rate the for PRNGs without CI} shows the 
-results on the two firsts batteries recalled above, indicating that all the PRNGs presented
+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
@@ -1239,8 +1228,8 @@ 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 ``New CI'' generators.
-Concerning the ``Xor CI PRNG'', the score is less spectacular: a large speed improvement makes that statistics
+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).
 
@@ -1267,7 +1256,7 @@ DieHARD & 16/18 & 16/18 & 17/18* & \textbf{18/18} * & \textbf{18/18}  & \textbf{
 \end{table*}
 
 
-We have then investigate in~\cite{bfg12a:ip} if it is possible to improve
+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.
@@ -1296,11 +1285,11 @@ results recalled in this section, it reinforces the opinion that a strong
 correlation between topological properties and statistical behavior exists.
 
 
-Next subsection will now give a concrete original implementation of the Xor CI PRNG, the
+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 as CIPRNG, or ``the proposed PRNG'', if this statement does not
+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}
@@ -1382,11 +1371,11 @@ 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}. 
-\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.
-\end{color}
+
 
 
 \section{Efficient PRNGs based on Chaotic Iterations on GPU}
@@ -1523,9 +1512,7 @@ version\label{IR}}
 \label{algo:gpu_kernel2} 
 \end{algorithm}
 
-\begin{color}{red}
 \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
@@ -1625,9 +1612,8 @@ as it is shown in the next sections.
 \section{Security Analysis}
 
 
-\begin{color}{red}
 This section is dedicated to the security analysis of the
-  proposed PRNGs, both from a theoretical and a practical points of view.
+  proposed PRNGs, both from a theoretical and from a practical point of view.
 
 \subsection{Theoretical Proof of Security}
 \label{sec:security analysis}
@@ -1643,7 +1629,6 @@ 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.
-\end{color}
 
 In this section the concatenation of two strings $u$ and $v$ is classically
 denoted by $uv$.
@@ -1663,20 +1648,14 @@ probabilities are taken over $U_m$, $U_{\ell_G(m)}$ as well as over the
 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,
@@ -1762,7 +1741,6 @@ proving that $H$ is not secure, which is a contradiction.
 
 
 
-\begin{color}{red}
 \subsection{Practical Security Evaluation}
 \label{sec:Practicak evaluation}
 
@@ -1822,12 +1800,12 @@ $M=100$ 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
+sequence from a truly random one, with a probability
 greater than $\varepsilon = 0.2$.
 We consider that $N$ has 900 bits.
 
 Predicting the next generated bit knowing all the
-previously released ones by Eq.~\eqref{equation Oplus} is obviously equivalent to predict the
+previously released ones by Eq.~\eqref{equation Oplus} is obviously equivalent to predicting the
 next bit in the BBS generator, which
 is cryptographically secure. More precisely, it
 is $(T,\varepsilon)-$secure: no 
@@ -1840,7 +1818,7 @@ T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-
 where $M$ is the length of the output ($M=100$ in
 our example), and $L(N)$ is equal to
 $$
-2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right)
+2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln~ 2)^\frac{1}{3} \times (ln(N~ln~  2))^\frac{2}{3}\right)
 $$
 is the number of clock cycles to factor a $N-$bit
 integer.
@@ -1852,7 +1830,6 @@ A direct numerical application shows that this attacker
 cannot achieve its $(10^{12},0.2)$ distinguishing
 attack in that context.
 
-\end{color}
 
 
 \section{Cryptographical Applications}
@@ -1981,7 +1958,6 @@ by secure bits produced by the BBS generator, and thus, due to
 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
@@ -1997,9 +1973,9 @@ predict the next bit is large enough when compared
 to both the generation and transmission times.
 It is true that the prime numbers used in the last
 section are very small compared to up-to-date 
-security recommends. However the attacker has not
+security recommendations. However the attacker has not
 access to each BBS, but to the output produced 
-by Algorithm~\ref{algo:bbs_gpu}, which is quite
+by Algorithm~\ref{algo:bbs_gpu}, which is far
 more complicated than a simple BBS. Indeed, to
 determine if this cryptographically secure PRNG
 on GPU can be useful in security context with the 
@@ -2008,9 +1984,8 @@ and statistically perfect generator on GPU, its
 $(T,\varepsilon)-$security must be determined, and
 a formulation similar to Eq.\eqref{mesureConcrete}
 must be established. Authors
-hope to achieve to realize this difficult task in a future
+hope to achieve this difficult task in a future
 work.
-\end{color}
 
 
 \subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
@@ -2095,8 +2070,8 @@ 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.
-\begin{color}{red} An improvement of the Blum-Goldwasser cryptosystem, making it 
-behaves 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,
