Randomness is of importance in many fields such as scientific
simulations or cryptography. ``Random numbers'' can mainly be
generated either by a deterministic and reproducible algorithm called
-a pseudorandom number generator (PRNG), or by a physical non-deterministic
+a pseudorandom number generator (PRNG)\index{PRNG}, or by a physical non-deterministic
process having all the characteristics of a random noise, called a truly random number
generator (TRNG). In this chapter, we focus on
reproducible generators, useful for instance in Monte-Carlo based
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}.
+This battery can be found in the well-known TestU01 package~\cite{LEcuyerS07}\index{TestU01}.
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
Chaos theory studies the behavior of dynamical systems that are perfectly predictable, yet appear to be wildly amorphous and without meaningful.
-Chaotic systems are highly sensitive to initial conditions,
+Chaotic systems\index{chaotic systems} are highly sensitive to initial conditions,
which is popularly referred to as the butterfly effect.
In other words, small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes,
rendering long-term prediction impossible in general \cite{kellert1994wake}. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved \cite{kellert1994wake}. That is, the deterministic nature of these systems does not make them predictable \cite{kellert1994wake,Werndl01032009}. This behavior is known as deterministic chaos, or simply chaos. It has been well-studied in mathematics and
-\subsection{On Devaney's Definition of Chaos}
+\subsection{On Devaney's Definition of Chaos}\index{chaos}
\label{sec:dev}
Consider a metric space $(\mathcal{X},d)$ and a continuous function $f:\mathcal{X}\longrightarrow \mathcal{X}$, for one-dimensional dynamical systems of the form:
\begin{equation}
\begin{definition}
A dynamical system of Form~(\ref{Devaney}) is said to be chaotic if the following conditions hold.
\begin{itemize}
-\item Topological transitivity:
+\item Topological transitivity\index{topological transitivity}:
\begin{equation}
\forall U,V \textrm{ open sets of } \mathcal{X},~\exists k>0, f^k(U) \cap V \neq \varnothing .
\end{equation}
Intuitively, a topologically transitive map has points that eventually move under iteration from one arbitrarily small neighborhood to any other. Consequently, the dynamical system cannot be decomposed into two disjoint open sets that are invariant under the map. Note that if a map possesses a dense orbit, then it is clearly topologically transitive.
-\item Density of periodic points in $\mathcal{X}$.
+\item Density of periodic points in $\mathcal{X}$\index{density of periodic points}.
Let $P=\{p\in \mathcal{X}|\exists n \in \mathds{N}^{\ast}:f^n(p)=p\}$ the set of periodic points of $f$. Then $P$ is dense in $\mathcal{X}$:
\end{equation}
Density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. Topologically mixing systems failing this condition may not display sensitivity to initial conditions presented below, and hence may not be chaotic.
-\item Sensitive dependence on initial conditions:
+\item Sensitive dependence on initial conditions\index{sensitive dependence on initial conditions}:
$\exists \varepsilon>0,$ $\forall x \in \mathcal{X},$ $\forall \delta >0,$ $\exists y \in \mathcal{X},$ $\exists n \in \mathbb{N},$ $d(x,y)<\delta$ and $d\left(f^n(x),f^n(y)\right) \geqslant \varepsilon.$
-\subsection{Chaotic iterations}
+\subsection{Chaotic iterations}\index{chaotic iterations}
\label{subsection:Chaotic iterations}
Let us now introduce an example of a dynamical systems family that has
-In Figure~\ref{fig:time_bbs_gpu} we highlight the performances of the optimized
-BBS-based PRNG on GPU. On the Tesla C1060 we obtain approximately 700MSample/s
-and on the GTX 280 about 670MSample/s, which is obviously slower than the
-xorlike-based PRNG on GPU. However, we will show in the next sections that this
-new PRNG has a strong level of security, which is necessarily paid by a speed
-reduction.
-\begin{figure}[htbp]
-\begin{center}
- \includegraphics[scale=0.65]{Chapters/chapter18/figures/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}
-\end{figure}
-
-All these experiments allow us to conclude that it is possible to
+These experiments allow us to conclude that it is possible to
generate a very large quantity of pseudorandom numbers statistically perfect with the xor-like version.
-To a certain extend, it is also the case with the secure BBS-based version, the speed deflation being
-explained by the fact that the former version has ``only''
-chaotic properties and statistical perfection, whereas the latter is also cryptographically secure,
-as it is shown in the next sections.
+
