X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/book_gpu.git/blobdiff_plain/013c063fb5cb99cc33b413d937f330acc42ca73e..59263d81c5f07e8eff22c9091a0847e79b4fbf2c:/BookGPU/Chapters/chapter18/ch18.tex

diff --git a/BookGPU/Chapters/chapter18/ch18.tex b/BookGPU/Chapters/chapter18/ch18.tex
index 3808995..b4401c9 100755
--- a/BookGPU/Chapters/chapter18/ch18.tex
+++ b/BookGPU/Chapters/chapter18/ch18.tex
@@ -50,7 +50,21 @@ Furthermore, authors of such chaotic generators often claim their PRNG
 as secure due to their chaos properties, but there is no obvious relation
 between chaos and security as it is understood in cryptography.
 This is why the use of chaos for PRNG still remains marginal and disputable.
-
+Let us finish this paragraph by noticing that, in this paper, 
+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}.
+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
+outside this interval but inside [0.001, 0.999], so that is why a
+second run allows us to confirm that the values outside are not for
+the same test. With this approach all our PRNGs pass the {\it
+  BigCrush} successfully and all $p-$values are at least once inside
+[0.01, 0.99].
+Chaos, for its part, refers to the well-established definition of a
+chaotic dynamical system defined by Devaney~\cite{Devaney}.
 
 The remainder of this paper is organized as follows. 
 A COMPLETER