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
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. Optimized
-original implementation of this PRNG are also proposed and experimented.
+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
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 but, as far
-as we know, it is the first cryptographically secured PRNG proposed on GPU.
+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 test is also proposed.
}
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 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}.
This new generator is designed by the following process.
First of all, some chaotic iterations have to be done to generate a sequence
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 $x_i^n = \overline{x_i^{n-1}}$.
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 $x_i^n = \overline{x_i^{n-1}}$.
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.
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.
\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
\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
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:
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
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.
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.
of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention
of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention
\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}
\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
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.
$x,y \in \mathcal{X}$ 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}:
\end{itemize}
\item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are
\end{itemize}
\item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are
-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)
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}
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}
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:
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}
\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}
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
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
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
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.
\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).
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).
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.
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.
-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
an attacker can realize $10^{12}$ clock cycles.
We thus wonder whether, during the PRNG's
lifetime, the attacker can distinguish this
an attacker can realize $10^{12}$ clock cycles.
We thus wonder whether, during the PRNG's
lifetime, the attacker can distinguish this
greater than $\varepsilon = 0.2$.
We consider that $N$ has 900 bits.
Predicting the next generated bit knowing all the
greater than $\varepsilon = 0.2$.
We consider that $N$ has 900 bits.
Predicting the next generated bit knowing all the
next bit in the BBS generator, which
is cryptographically secure. More precisely, it
is $(T,\varepsilon)-$secure: no
next bit in the BBS generator, which
is cryptographically secure. More precisely, it
is $(T,\varepsilon)-$secure: no
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
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
more complicated than a simple BBS. Indeed, to
determine if this cryptographically secure PRNG
on GPU can be useful in security context with the
more complicated than a simple BBS. Indeed, to
determine if this cryptographically secure PRNG
on GPU can be useful in security context with the
$(T,\varepsilon)-$security must be determined, and
a formulation similar to Eq.\eqref{mesureConcrete}
must be established. Authors
$(T,\varepsilon)-$security must be determined, and
a formulation similar to Eq.\eqref{mesureConcrete}
must be established. Authors
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
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
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,
