X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/blobdiff_plain/e1fe6e435ee452003a7135763d26e2320756398c..79f46c777f6ef3202c2fb25194822518cad3e32c:/intro.tex

diff --git a/intro.tex b/intro.tex
index df2a0ee..83b0a5a 100644
--- a/intro.tex
+++ b/intro.tex
@@ -1,5 +1,5 @@
 The exploitation of chaotic systems to generate pseudorandom sequences
-is  an   hot  topic~\cite{915396,915385,5376454}.  Such   systems  are
+is  a    hot  topic~\cite{915396,915385,5376454}.  Such   systems  are
 fundamentally chosen  due to  their unpredictable character  and their
 sensitiveness to initial conditions.   In most cases, these generators
 simply  consist in  iterating  a chaotic  function  like the  logistic
@@ -35,7 +35,7 @@ function  (\textit{i.e.},  $f  :  \{0,1\}^{\mathsf{N}}
 \rightarrow  \{0,1\}^{\mathsf{N}}$).
 These         generators          are         $\textit{CIPRNG}_f^1(u)$
 \cite{guyeuxTaiwan10,bcgr11:ip},            $\textit{CIPRNG}_f^2(u,v)$
-\cite{wbg10ip}             and             $\chi_{\textit{14Secrypt}}$
+\cite{wbg10ip},             and             $\chi_{\textit{14Secrypt}}$
 \cite{DBLP:conf/secrypt/CouchotHGWB14}    where   \textit{CI}    means
 \emph{Chaotic Iterations}.  We have firstly proven in~\cite{bcgr11:ip}
 that,    to    establish    the   chaotic    nature    of    algorithm
@@ -71,12 +71,12 @@ a doubly stochastic Markov matrix.
 
 However, the removed Hamiltonian cycle  
 has a great influence in the quality of the output.
-For instance, if this one one is not balanced (\textit{i.e.},
+For instance, if this one is not balanced (\textit{i.e.},
 the number of changes in different bits are completely different),
 some bits would be hard to switch.
 This article shows an effective algorithm that efficiently 
 implements the previous scheme and provides thus functions issued
-from removing in the $\mathsf{N}$-cube a \emph{balanced} Hamiltonian cycle. 
+from removing, in the $\mathsf{N}$-cube, a \emph{balanced} Hamiltonian cycle. 
 
 The length $b$ of the walk to reach a
 distribution close to the uniform one would be dramatically long.
@@ -88,8 +88,9 @@ suite is evaluated.
 
 
 
-
-The  remainder of  this  article  is organized  as  follows. The  next
+This article, which 
+is an extension of~\cite{DBLP:conf/secrypt/CouchotHGWB14}, 
+is organized  as  follows. The  next
 section   is   devoted   to  preliminaries,   basic   notations,   and
 terminologies   regarding   Boolean    map   iterations.    Then,   in
 Section~\ref{sec:proofOfChaos},  Devaney's  definition   of  chaos  is
@@ -99,15 +100,21 @@ Section~\ref{sec:SCCfunc}  recalls a general scheme
 to obtain functions with awaited behavior. Main theorems are recalled
 to make the document self-content. 
 The  next  section (Sect.~\ref{sec:hamilton}) presents an algorithm that
-implements this scheme and proves it always produces a solution.  
-This  is   the   second   major  contribution
-The  later section
-(Sect.~\ref{sec:hypercube}) defines the theoretical framework to study
-the  mixing-time,  \textit{i.e.},  time  until  reaching  a  uniform
+implements this scheme and proves that it always produces a solution.  
+This  is   the   second   major  contribution.
+Then, Section~\ref{sec:hypercube} defines the theoretical framework to study
+the  mixing-time,  \textit{i.e.},  time  until  reaching  an uniform
 distribution. It proves that this one is at worth quadratic in the number
-of elements. Experiments show that the bound is practically largely much
-lower. This  is   the   third   major  contribution.    The
+of elements. Experiments show that the bound is in practice largely much
+lower. This  is   the   third   major  contribution.  
 Section~\ref{sec:prng} gives practical results  on evaluating the PRNG
 against  the NIST  suite.  This  research  work ends  by a  conclusion
 section, where the contribution is summarized and intended future work
 is outlined.
+
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