X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/prng_gpu.git/blobdiff_plain/a0cdbfd7933130eaba676883427b46743a6cf7ea..e413aa9f2f3893a394428e26368d44eaa851a986:/prng_gpu.tex?ds=inline

diff --git a/prng_gpu.tex b/prng_gpu.tex
index 39cee40..f7499d9 100644
--- a/prng_gpu.tex
+++ b/prng_gpu.tex
@@ -1040,7 +1040,7 @@ 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{} that chaotic iterations satisfying Theorem~\ref{Th:Caractérisation   des   IC   chaotiques} are, among other
+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.
@@ -1279,7 +1279,7 @@ this generator will be simply referred as CIPRNG, or ``the proposed PRNG'', if t
 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 
@@ -1358,7 +1358,13 @@ works with 32-bits, we use the command \texttt{(unsigned int)}, that selects the
 
 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}