X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/a17d0a6c41e0afab24e5a0c8c806533db3ad753a..3abda4f59d76446238e6b891bc14e1ac7b44c34a:/supplementary.tex

diff --git a/supplementary.tex b/supplementary.tex
index 012cfca..1ad62e6 100644
--- a/supplementary.tex
+++ b/supplementary.tex
@@ -27,7 +27,7 @@
 \usepackage{subfigure}
 \usepackage{xr-hyper}
 \usepackage{hyperref}
-\externaldocument{prng_gpu}
+\externaldocument[M-]{prng_gpu}
 %\usepackage{hyperref}
 
 
@@ -311,7 +311,7 @@ have $d((S,E),(\tilde S,E))<\epsilon$.
 \label{The generation of pseudorandom sequence}
 
 
-Let us now explain why we have reasonable ground to believe that chaos 
+Let us explain in this annex why we have reasonable ground to believe that chaos 
 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
@@ -342,7 +342,7 @@ theory of chaos and tests embedded into the NIST battery. %Such relations need t
 
 
 \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
@@ -391,7 +391,7 @@ not only sought in general to obtain chaos, but they are also required for rando
 \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.
@@ -634,7 +634,7 @@ raise ambiguity.
 \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
@@ -685,7 +685,7 @@ A pseudorandom generator is $(T,\varepsilon)-$secure if there exists no $(T,\var
 
 
 
-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 
@@ -695,7 +695,7 @@ greater than $\varepsilon = 0.2$.
 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 
@@ -717,7 +717,7 @@ integer.
 
 
 A direct numerical application shows that this attacker 
-cannot achieve its $(10^{12},0.2)$ distinguishing
+cannot achieve his $(10^{12},0.2)$ distinguishing
 attack in that context.