\usepackage{subfigure}
\usepackage{xr-hyper}
\usepackage{hyperref}
-\externaldocument{prng_gpu}
+\externaldocument[M-]{prng_gpu}
%\usepackage{hyperref}
\begin{itemize}
- \item \textbf{Regularity}. As stated in Section~\ref{subsec:Devaney} of the main document, a chaotic dynamical system must
+ \item \textbf{Regularity}. As stated in Section~\ref{M-subsec:Devaney} of the main document, 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
\end{itemize}
-We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation des IC chaotiques} of the main document are, among other
+We have proven in our previous works~\cite{guyeux12:bc} that chaotic iterations satisfying Theorem~\ref{M-Th:Caractérisation des IC chaotiques} of the main document 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.
\section{Practical Security Evaluation}
\label{sec:Practicak evaluation}
-Pseudorandom generators based on Eq.~\eqref{equation Oplus} of the main document are thus cryptographically secure when
+Pseudorandom generators based on Eq.~\eqref{M-equation Oplus} of the main document 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
-Suppose now that the PRNG of Eq.~\eqref{equation Oplus} of the main document will work during
+Suppose now that the PRNG of Eq.~\eqref{M-equation Oplus} of the main document 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
We consider that $N$ has 900 bits.
Predicting the next generated bit knowing all the
-previously released ones by Eq.~\eqref{equation Oplus} of the main document is obviously equivalent to predicting the
+previously released ones by Eq.~\eqref{M-equation Oplus} of the main document is obviously equivalent to predicting the
next bit in the BBS generator, which
is cryptographically secure. More precisely, it
is $(T,\varepsilon)-$secure: no
