\subsection{Devaney's Chaotic Dynamical Systems}
-
+\label{subsec:Devaney}
In the sequel $S^{n}$ denotes the $n^{th}$ term of a sequence $S$ and $V_{i}$
denotes the $i^{th}$ component of a vector $V$. $f^{k}=f\circ ...\circ f$
is for the $k^{th}$ composition of a function $f$. Finally, the following
The basic design procedure of the New CI generator is summarized in Algorithm~\ref{Chaotic iteration1}.
The internal state is $x$, the output state is $r$. $a$ and $b$ are those computed by the two input
PRNGs. Lastly, the value $g(a)$ is an integer defined as in Eq.~\ref{Formula}.
-This function must be chosen such that the outputs of the resulted PRNG is uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$. Function of \eqref{Formula} achieves this
+This function must be chosen such that the outputs of the resulted PRNG are uniform in $\llbracket 0, 2^\mathsf{N}-1 \rrbracket$. Function of \eqref{Formula} achieves this
goal (other candidates and more information can be found in ~\cite{bg10:ip}).
\begin{equation}
Let us now explain why we are reasonable grounds to believe that chaos
can improve statistical properties.
-We will show in this section that, when mixing defective PRNGs with
-chaotic iterations, the result presents better statistical properties
-(this section summarizes the work of~\cite{bfg12a:ip}).
+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
+in the NIST battery. Furthermore, we will check that, when mixing defective PRNGs with
+chaotic iterations, the new generator presents better statistical properties
+(this section summarizes and extends the work of~\cite{bfg12a:ip}).
+
+
+
+\subsection{Qualitative relations between topological properties and statistical tests}
+
+
+There are various relations between topological properties that describe an unpredictable behavior for a discrete
+dynamical system on the one
+hand, and statistical tests to check the randomness of a numerical sequence
+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,
+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.
+
+
+\begin{itemize}
+ \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney}, a chaotic dynamical system must
+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}:
+ \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.
+ \end{itemize}
+
+\item \textbf{Transitivity}. This topological property introduced previously 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
+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}:
+ \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}.
+ \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.
+ \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).
+ \end{itemize}
+
+ \item \textbf{Non-linearity, complexity}.
+ \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).
+ \end{itemize}
+\end{itemize}
+
+
+
+
\subsection{Details of some Existing Generators}
The list of defective PRNGs we will use
as inputs for the statistical tests to come is introduced here.
-Firstly, the simple linear congruency generator (LCGs) will be used.
-It is defined by the following recurrence:
+Firstly, the simple linear congruency generators (LCGs) will be used.
+They are defined by the following recurrence:
\begin{equation}
-x^n = (ax^{n-1} + c)~mod~m
+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)
combinations of such LCGs. For further details, see~\cite{bfg12a:ip,combined_lcg}.
-Secondly, the multiple recursive generators (MRGs) will be used too, which
+Secondly, the multiple recursive generators (MRGs) will be used, which
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
+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 experimentations.
+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:
Then the generalized feedback shift register (GFSR) generator has been implemented, that is:
\begin{equation}
-x^n = x^{n-r} \oplus x^{n-k}
+x^n = x^{n-r} \oplus x^{n-k} .
\label{GFSR}
\end{equation}
-Finally, the nonlinear inversive generator~\cite{LEcuyerS07} has been regarded too, which is:
+Finally, the nonlinear inversive (INV) generator~\cite{LEcuyerS07} has been studied, which is:
\begin{equation}
\label{INV}
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 speed improvement makes that statistical
-results are not as good as for the two other versions of these CIPRNGs.
+Concerning the ``Xor CI PRNG'', the score is less spectacular: a large speed improvement makes that 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*}
We have then investigate in~\cite{bfg12a:ip} if it is possible to improve
the statistical behavior of the Xor CI version by combining more than one
-$\oplus$ operation. Results are summarized in~\ref{threshold}, showing
-that rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained
+$\oplus$ 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.
+Thus rapid and perfect PRNGs, regarding the NIST and DieHARD batteries, can be obtained
using chaotic iterations on defective generators.
\begin{table*}
\end{tabular}
\end{table*}
-Finally, the TestU01 battery as been launched on three well-known generators
+Finally, the TestU01 battery has been launched on three well-known generators
(a logistic map, a simple XORshift, and the cryptographically secure ISAAC,
see Table~\ref{TestU011}). These results can be compared with
Table~\ref{TestU01 for Old CI}, which gives the scores obtained by the
Old CI PRNG that has received these generators.
+The obvious improvement speaks for itself, and together with the other
+results recalled in this section, it reinforces the opinion that a strong
+correlation between topological properties and statistical behavior exists.
-Next subsection gives a concrete implementation of this Xor CI PRNG, which will
-new be simply called CIPRNG, or ``the proposed PRNG'', if this statement does not
+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
raise ambiguity.
\end{color}
