From 236f25b2f3a081b11c71bedad6d044d695ce2cca Mon Sep 17 00:00:00 2001 From: Sylvain C-V Date: Thu, 23 Jun 2016 17:24:35 +0200 Subject: [PATCH 1/1] Ajouts variables LaTeX aux fins de fichiers + corrections quelques typos --- chaos.tex | 13 +++++++++---- conclusion.tex | 7 +++++++ generating.tex | 7 +++++++ hamilton.tex | 8 +++++++- intro.tex | 9 ++++++++- main.pdf | Bin 495003 -> 627209 bytes main.tex | 6 ++++++ preliminaries.tex | 6 ++++++ prng.tex | 9 ++++++++- stopping.tex | 13 ++++++++++--- 10 files changed, 68 insertions(+), 10 deletions(-) diff --git a/chaos.tex b/chaos.tex index 122c0ad..511a11a 100644 --- a/chaos.tex +++ b/chaos.tex @@ -247,7 +247,7 @@ $\check{u}^{v^0}$ (on $n$ digits), ..., $\check{u}^{\check{v}^0-1}$ (on $n$ digi \begin{xpl} -Consider for instance that $\mathsf{N}=13$, $\mathcal{P}=\{1,2,11\}$ (so $\mathsf{p}=3$), and that +Consider for instance that $\mathsf{N}=13$, $\mathcal{P}=\{1,2,11\}$ (so $\mathsf{p}=2$), and that $s=\left\{ \begin{array}{l} u=\underline{6,} ~ \underline{11,5}, ...\\ @@ -425,7 +425,7 @@ $\mathcal{P}=\{2,3\}$. The graphs of iterations are given in The \textsc{Figure~\ref{graphe1}} shows what happens when displaying each iteration result. On the contrary, the \textsc{Figure~\ref{graphe2}} explicits the behaviors -when always applying 2 or 3 modification and next outputing results. +when always applying either 2 or 3 modifications before generating results. Notice that here, orientations of arcs are not necessary since the function $f_0$ is equal to its inverse $f_0^{-1}$. \end{xpl} @@ -521,7 +521,7 @@ and only if its iteration graph $\Gamma_{\mathcal{P}}(f)$ is strongly connected. In this context, $\mathcal{P}$ is the singleton $\{b\}$. If $b$ is even, any vertex $e$ of $\Gamma_{\{b\}}(f_0)$ cannot reach its neighborhood and thus $\Gamma_{\{b\}}(f_0)$ is not strongly connected. - If $b$ is even, any vertex $e$ of $\Gamma_{\{b\}}(f_0)$ cannot reach itself + If $b$ is odd, any vertex $e$ of $\Gamma_{\{b\}}(f_0)$ cannot reach itself and thus $\Gamma_{\{b\}}(f_0)$ is not strongly connected. \end{proof} @@ -530,4 +530,9 @@ functions and a iteration number $b$ such that $\Gamma_{\{b\}}$ is strongly connected. - \ No newline at end of file +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "main" +%%% ispell-dictionary: "american" +%%% mode: flyspell +%%% End: diff --git a/conclusion.tex b/conclusion.tex index fcf2bad..a9bf636 100644 --- a/conclusion.tex +++ b/conclusion.tex @@ -33,3 +33,10 @@ Conditions allowing the reduction of the stopping-time will be investigated too, while other modifications of the hypercube will be regarded in order to enlarge the set of known chaotic and random iterations. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "main" +%%% ispell-dictionary: "american" +%%% mode: flyspell +%%% End: diff --git a/generating.tex b/generating.tex index ace2f7a..3d22441 100644 --- a/generating.tex +++ b/generating.tex @@ -76,3 +76,10 @@ from the The next section presents how to build balanced Hamiltonian cycles in the $\mathsf{N}$-cube with the objective to embed them into the pseudorandom number generator. + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "main" +%%% ispell-dictionary: "american" +%%% mode: flyspell +%%% End: diff --git a/hamilton.tex b/hamilton.tex index 17d9380..dc19f08 100644 --- a/hamilton.tex +++ b/hamilton.tex @@ -23,7 +23,7 @@ $\mathsf{N}$ cube. Obviously, the number of iterations $b$ has to be sufficiently large to provide a uniform output distribution. To reduce the number of iterations, the provided Gray code -should ideally possess the both balanced and locally balanced properties. +should ideally possess both balanced and locally balanced properties. However, none of the two algorithms is compatible with the second one: balanced Gray codes that are generated by state of the art works~\cite{ZanSup04,DBLP:journals/combinatorics/BhatS96} are not locally balanced. Conversely, locally balanced Gray codes yielded by Igor Bykov approach~\cite{Bykov2016} @@ -328,3 +328,9 @@ Notice that all such choices lead to a hamiltonian path. +%%% Local Variables: +%%% mode: latex +%%% TeX-master: "main" +%%% ispell-dictionary: "american" +%%% mode: flyspell +%%% End: diff --git a/intro.tex b/intro.tex index df2a0ee..10463ea 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 @@ -111,3 +111,10 @@ Section~\ref{sec:prng} gives practical results on evaluating the PRNG against the NIST suite. 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