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\title{Efficient and Cryptographically Secure Generation of Chaotic Pseudorandom Numbers on GPU}
\begin{document}
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
\end{abstract}
-}
+%}
\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. \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.
{\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
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}.
Last, but not least, we propose a rewriting of the Blum-Goldwasser asymmetric
key encryption protocol by using the proposed method.
+
+{\bf Main contributions.} In this paper a new PRNG using chaotic iteration
+is defined. From a theoretical point of view, it is proven that it has fine
+topological chaotic properties and that it is cryptographically secured (when
+the initial PRNG is also cryptographically secured). From a practical point of
+view, experiments point out a very good statistical behavior. An optimized
+original implementation of this PRNG is also proposed and experimented.
+Pseudorandom numbers are generated at a rate of 20GSamples/s, which is faster
+than in~\cite{conf/fpga/ThomasHL09,Marsaglia2003} (and with a better
+statistical behavior). Experiments are also provided using BBS as the initial
+random generator. The generation speed is significantly weaker.
+Note also that an original qualitative comparison between topological chaotic
+properties and statistical test is also proposed.
+
+
+
+
The remainder of this paper is organized as follows. In Section~\ref{section:related
works} we review some GPU implementations of PRNGs. Section~\ref{section:BASIC
RECALLS} gives some basic recalls on the well-known Devaney's formulation of chaos,
and on an iteration process called ``chaotic
iterations'' on which the post-treatment is based.
The proposed PRNG and its proof of chaos are given in Section~\ref{sec:pseudorandom}.
-\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
We show in Section~\ref{sec:security analysis} that, if the inputted
generator is cryptographically secure, then it is the case too for the
generator provided by the post-treatment.
+A practical
+security evaluation is also outlined in Section~\ref{sec:Practicak evaluation}.
Such a proof leads to the proposition of a cryptographically secure and
chaotic generator on GPU based on the famous Blum Blum Shub
-in Section~\ref{sec:CSGPU}, \begin{color}{red} to a practical
-security evaluation in Section~\ref{sec:Practicak evaluation}, \end{color} and to an improvement of the
+in Section~\ref{sec:CSGPU} and to an improvement of the
Blum-Goldwasser protocol in Sect.~\ref{Blum-Goldwasser}.
This research work ends by a conclusion section, in which the contribution is
summarized and intended future work is presented.
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.
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
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.
\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:
\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
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
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:
\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}
\begin{table}
-\renewcommand{\arraystretch}{1.3}
-\caption{TestU01 Statistical Test}
+%\renewcommand{\arraystretch}{1}
+\caption{TestU01 Statistical Test Failures}
\label{TestU011}
\centering
\begin{tabular}{lccccc}
\begin{table}
-\renewcommand{\arraystretch}{1.3}
-\caption{TestU01 Statistical Test for Old CI algorithms ($\mathsf{N}=4$)}
+%\renewcommand{\arraystretch}{1}
+\caption{TestU01 Statistical Test Failures for Old CI algorithms ($\mathsf{N}=4$)}
\label{TestU01 for Old CI}
\centering
\begin{tabular}{lcccc}
\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
\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
\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
\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).
\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
\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.
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
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}
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}
\label{algo:gpu_kernel2}
\end{algorithm}
-\subsection{Theoretical Evaluation of the Improved Version}
+\subsection{Chaos Evaluation of the Improved Version}
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
\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}
\section{Security Analysis}
+
+
+This section is dedicated to the security analysis of the
+ proposed PRNGs, both from a theoretical and from a practical point of view.
+
+\subsection{Theoretical Proof of Security}
\label{sec:security analysis}
-\PCH{This section is dedicated to the analysis of the security of the
- proposed PRNGs from a theoretical point of view. The standard definition
+The standard definition
of {\it indistinguishability} used is the classical one as defined for
- instance in~\cite[chapter~3]{Goldreich}. It is important to emphasize that
- this property shows that predicting the future results of the PRNG's
- cannot be done in a reasonable time compared to the generation time. This
+ instance in~\cite[chapter~3]{Goldreich}.
+ This property shows that predicting the future results of the PRNG
+ cannot be done in a reasonable time compared to the generation time. It is important to emphasize that this
is a relative notion between breaking time and the sizes of the
keys/seeds. Of course, if small keys or seeds are chosen, the system can
be broken in practice. But it also means that if the keys/seeds are large
- enough, the system is secured.}
+ enough, the system is secured.
+As a complement, an example of a concrete practical evaluation of security
+is outlined in the next subsection.
In this section the concatenation of two strings $u$ and $v$ is classically
denoted by $uv$.
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. 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,
\end{proposition}
\begin{proof}
-The proposition is proved by contraposition. Assume that $X$ is not
+The proposition is proven by contraposition. Assume that $X$ is not
secure. By Definition, there exists a polynomial time probabilistic
algorithm $D$, a positive polynomial $p$, such that for all $k_0$ there exists
$N\geq \frac{k_0}{2}$ satisfying
\end{proof}
+
+\subsection{Practical Security Evaluation}
+\label{sec:Practicak evaluation}
+
+Pseudorandom generators based on Eq.~\eqref{equation Oplus} are thus cryptographically secure when
+they are XORed with an already cryptographically
+secure PRNG. But, as stated previously,
+such a property does not mean that, whatever the
+key size, no attacker can predict the next bit
+knowing all the previously released ones.
+However, given a key size, it is possible to
+measure in practice the minimum duration needed
+for an attacker to break a cryptographically
+secure PRNG, if we know the power of his/her
+machines. Such a concrete security evaluation
+is related to the $(T,\varepsilon)-$security
+notion, which is recalled and evaluated in what
+follows, for the sake of completeness.
+
+Let us firstly recall that,
+\begin{definition}
+Let $\mathcal{D} : \mathds{B}^M \longrightarrow \mathds{B}$ be a probabilistic algorithm that runs
+in time $T$.
+Let $\varepsilon > 0$.
+$\mathcal{D}$ is called a $(T,\varepsilon)-$distinguishing attack on pseudorandom
+generator $G$ if
+
+\begin{flushleft}
+$\left| Pr[\mathcal{D}(G(k)) = 1 \mid k \in_R \{0,1\}^\ell ]\right.$
+\end{flushleft}
+
+\begin{flushright}
+$ - \left. Pr[\mathcal{D}(s) = 1 \mid s \in_R \mathds{B}^M ]\right| \geqslant \varepsilon,$
+\end{flushright}
+
+\noindent where the probability is taken over the internal coin flips of $\mathcal{D}$, and the notation
+``$\in_R$'' indicates the process of selecting an element at random and uniformly over the
+corresponding set.
+\end{definition}
+
+Let us recall that the running time of a probabilistic algorithm is defined to be the
+maximum of the expected number of steps needed to produce an output, maximized
+over all inputs; the expected number is averaged over all coin flips made by the algorithm~\cite{Knuth97}.
+We are now able to define the notion of cryptographically secure PRNGs:
+
+\begin{definition}
+A pseudorandom generator is $(T,\varepsilon)-$secure if there exists no $(T,\varepsilon)-$distinguishing attack on this pseudorandom generator.
+\end{definition}
+
+
+
+
+
+
+
+Suppose now that the PRNG of Eq.~\eqref{equation Oplus} will work during
+$M=100$ time units, and that during this period,
+an attacker can realize $10^{12}$ clock cycles.
+We thus wonder whether, during the PRNG's
+lifetime, the attacker can distinguish this
+sequence from a truly random one, with a probability
+greater than $\varepsilon = 0.2$.
+We consider that $N$ has 900 bits.
+
+Predicting the next generated bit knowing all the
+previously released ones by Eq.~\eqref{equation Oplus} is obviously equivalent to predicting the
+next bit in the BBS generator, which
+is cryptographically secure. More precisely, it
+is $(T,\varepsilon)-$secure: no
+$(T,\varepsilon)-$distinguishing attack can be
+successfully realized on this PRNG, if~\cite{Fischlin}
+\begin{equation}
+T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
+\label{mesureConcrete}
+\end{equation}
+where $M$ is the length of the output ($M=100$ in
+our example), and $L(N)$ is equal to
+$$
+2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln~ 2)^\frac{1}{3} \times (ln(N~ln~ 2))^\frac{2}{3}\right)
+$$
+is the number of clock cycles to factor a $N-$bit
+integer.
+
+
+
+
+A direct numerical application shows that this attacker
+cannot achieve its $(10^{12},0.2)$ distinguishing
+attack in that context.
+
+
+
\section{Cryptographical Applications}
\subsection{A Cryptographically Secure PRNG for GPU}
where $S^n$ is referred in this algorithm as $t$: each iteration of this
PRNG ends with $x = x \wedge t$. This $S^n$ is only constituted
by secure bits produced by the BBS generator, and thus, due to
-Proposition~\ref{cryptopreuve}, the resulted PRNG is cryptographically
-secure.
+Proposition~\ref{cryptopreuve}, the resulted PRNG is
+cryptographically secure.
+
+As stated before, even if the proposed PRNG is cryptocaphically
+secure, it does not mean that such a generator
+can be used as described here when attacks are
+awaited. The problem is to determine the minimum
+time required for an attacker, with a given
+computational power, to predict under a probability
+lower than 0.5 the $n+1$th bit, knowing the $n$
+previous ones. The proposed GPU generator will be
+useful in a security context, at least in some
+situations where a secret protected by a pseudorandom
+keystream is rapidly obsolete, if this time to
+predict the next bit is large enough when compared
+to both the generation and transmission times.
+It is true that the prime numbers used in the last
+section are very small compared to up-to-date
+security recommendations. However the attacker has not
+access to each BBS, but to the output produced
+by Algorithm~\ref{algo:bbs_gpu}, which is far
+more complicated than a simple BBS. Indeed, to
+determine if this cryptographically secure PRNG
+on GPU can be useful in security context with the
+proposed parameters, or if it is only a very fast
+and statistically perfect generator on GPU, its
+$(T,\varepsilon)-$security must be determined, and
+a formulation similar to Eq.\eqref{mesureConcrete}
+must be established. Authors
+hope to achieve this difficult task in a future
+work.
-
-\begin{color}{red}
-\subsection{Practical Security Evaluation}
-\label{sec:Practicak evaluation}
-
-Suppose now that the PRNG will work during
-$M=100$ time units, and that during this period,
-an attacker can realize $10^{12}$ clock cycles.
-We thus wonder whether, during the PRNG's
-lifetime, the attacker can distinguish this
-sequence from truly random one, with a probability
-greater than $\varepsilon = 0.2$.
-We consider that $N$ has 900 bits.
-
-The random process is the BBS generator, which
-is cryptographically secure. More precisely, it
-is $(T,\varepsilon)-$secure: no
-$(T,\varepsilon)-$distinguishing attack can be
-successfully realized on this PRNG, if~\cite{Fischlin}
-$$
-T \leqslant \dfrac{L(N)}{6 N (log_2(N))\varepsilon^{-2}M^2}-2^7 N \varepsilon^{-2} M^2 log_2 (8 N \varepsilon^{-1}M)
-$$
-where $M$ is the length of the output ($M=100$ in
-our example), and $L(N)$ is equal to
-$$
-2.8\times 10^{-3} exp \left(1.9229 \times (N ~ln(2)^\frac{1}{3}) \times ln(N~ln 2)^\frac{2}{3}\right)
-$$
-is the number of clock cycles to factor a $N-$bit
-integer.
-
-A direct numerical application shows that this attacker
-cannot achieve its $(10^{12},0.2)$ distinguishing
-attack in that context.
-
-\end{color}
-
\subsection{Toward a Cryptographically Secure and Chaotic Asymmetric Cryptosystem}
\label{Blum-Goldwasser}
We finish this research work by giving some thoughts about the use of
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.
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,
