\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}
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 proved 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.
+Pseudo-random 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
+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,
In conclusion,
%%RAPH : ici j'ai rajouté une ligne
+%%TOF : ici j'ai rajouté un commentaire
+%%TOF : ici aussi
$
\forall \varepsilon >0,$ $\exists N_{0}=max(n_{0},n_{1},n_{2})\in \mathds{N}
,$ $\forall n\geqslant N_{0},$
on the other hand. These two mathematical disciplines follow a similar
objective in case of a recurrent sequence (to characterize an intrinsically complicated behavior for a
recurrent sequence), with two different but complementary approaches.
-It is true that these illustrative links give only qualitative arguments,
+It is true that the following illustrative links give only qualitative arguments,
and proofs should be provided later to make such arguments irrefutable. However
they give a first understanding of the reason why we think that chaotic properties should tend
to improve the statistical quality of PRNGs.
-
+%
Let us now list some of these relations between topological properties defined in the mathematical
-theory of chaos and tests embedded into the NIST battery. Such relations need to be further
-investigated, but they presently give a first illustration of a trend to search similar properties in the
-two following fields: mathematical chaos and statistics.
+theory of chaos and tests embedded into the NIST battery. %Such relations need to be further
+%investigated, but they presently give a first illustration of a trend to search similar properties in the
+%two following fields: mathematical chaos and statistics.
\begin{itemize}
have an element of regularity. Depending on the chosen definition of chaos, this element can be the existence of
a dense orbit, the density of periodic points, etc. The key idea is that a dynamical system with no periodicity
is not as chaotic as a system having periodic orbits: in the first situation, we can predict something and gain a
-knowledge about the behavior of the system, that is, it never enter into a loop. A similar importance for regularity is emphasized in
-the two following tests~\cite{Nist10}:
+knowledge about the behavior of the system, that is, it never enters into a loop. A similar importance for periodicity is emphasized in
+the two following NIST tests~\cite{Nist10}:
\begin{itemize}
\item \textbf{Non-overlapping Template Matching Test}. Detect generators that produce too many occurrences of a given non-periodic (aperiodic) pattern.
\item \textbf{Discrete Fourier Transform (Spectral) Test}. Detect periodic features (i.e., repetitive patterns that are near each other) in the tested sequence that would indicate a deviation from the assumption of randomness.
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
of chaos, namely: transitivity, strong transitivity, total transitivity, topological mixing, and so on~\cite{bg10:ij}. A similar attention
-is brought on stated visited during a random walk in the two tests below~\cite{Nist10}:
+is brought on 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
-to Li-Yorke~\cite{Li75,Ruette2001}. This property is related to the following test~\cite{Nist10}.
+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}. Both in topological and statistical fields.
+ \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.
+$x,y \in \mathcal{X}$ 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}:
\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 (m is the length of each block).
+\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.
\end{itemize}
- \item \textbf{Non-linearity, complexity}.
+ \item \textbf{Non-linearity, complexity}. Finally, let us remark that non-linearity and complexity are
+not only sought in general to obtain chaos, but they are also required for randomness, as illustrated by the two tests below~\cite{Nist10}.
\begin{itemize}
\item \textbf{Binary Matrix Rank Test}. Check for linear dependence among fixed length substrings of the original sequence.
-\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random (M is the length in bits of a block).
+\item \textbf{Linear Complexity Test}. Determine whether or not the sequence is complex enough to be considered random.
\end{itemize}
\end{itemize}
-
-
+We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques} are, among other
+things, strongly transitive, topologically mixing, chaotic as defined by Li and Yorke,
+and that they have a topological entropy and an exponent of Lyapunov both equal to $ln(\mathsf{N})$,
+where $\mathsf{N}$ is the size of the iterated vector.
+These topological properties make that we are ground to believe that a generator based on chaotic
+iterations will probably be able to pass all the existing statistical batteries for pseudorandomness like
+the NIST one. The following subsections, in which we prove that defective generators have their
+statistical properties improved by chaotic iterations, show that such an assumption is true.
\subsection{Details of some Existing Generators}
raise ambiguity.
\end{color}
-\subsection{Efficient Implementation of a PRNG based on Chaotic Iterations}
+\subsection{First Efficient Implementation of a PRNG based on Chaotic Iterations}
\label{sec:efficient PRNG}
%
%Based on the proof presented in the previous section, it is now possible to
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}.
+stringent BigCrush battery of tests~\cite{LEcuyerS07}.
+\begin{color}{red}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{sec:efficient PRNG gpu}
\section{Security Analysis}
\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
+ 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
+ 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.}
In this section the concatenation of two strings $u$ and $v$ is classically
denoted by $uv$.
