From: Jean-François Couchot Date: Thu, 7 Apr 2016 18:23:02 +0000 (+0200) Subject: khkhj X-Git-Url: https://bilbo.iut-bm.univ-fcomte.fr/and/gitweb/16dcc.git/commitdiff_plain/1b5eed47c5d9fada5fd208ab9a007f7d4355102e khkhj --- diff --git a/hamilton.tex b/hamilton.tex index 694bbcf..f4dc64a 100644 --- a/hamilton.tex +++ b/hamilton.tex @@ -133,24 +133,62 @@ the code is neither balanced nor totally balanced. \begin{thrm}\label{prop:balanced} Let $\mathsf{N}$ in $\Nats$, and $a_{\mathsf{N}}$ be defined by $a_{\mathsf{N}}= 2 \lfloor \dfrac{2^{\mathsf{N}}}{2\mathsf{N}} \rfloor$. -Then, there exists a sequence $l$ in +There exists then a sequence $l$ in step~(\ref{item:nondet}) of the \emph{Robinson-Cohn extension} algorithm such that all the transition counts $\textit{TC}_{\mathsf{N}}(i)$ are $a_{\mathsf{N}}$ or $a_{\mathsf{N}}+2$ for any $i$, $1 \le i \le \mathsf{N}$. +\end{thrm} -\end{thrm} - - +The proof is done by induction on $\mathsf{N}$. Let us imadialty verify +that it is established for both odd and even smallest values, \textit{i.e.} +$3$ and $4$. +For the initial case where $\mathsf{N}=3$, \textit{i.e.} $\mathsf{N-2}=1$ we successively have: $S_1=1,1$, $l=2$, $u_0 = \emptyset$, and $v=\emptyset$. +Thus again the algorithm successively produces +$U= 1,2,1$, $V = 3$, $W= 2, 1, 1,3$, and $W' = 1,2,1,3$. +Finally, $S_3$ is $1,2,1,3,1,2,1,3$ which obviously verifies the theorem. + For the initial case where $\mathsf{N}=4$, \textit{i.e.} $\mathsf{N-2}=2$ +we successively have: $S_1=1,2,1,2$, $l=4$, +$u_0,u_1,u_2 = \emptyset,\emptyset,\emptyset$, and $v=\emptyset$. +Thus again the algorithm successively produces +$U= 1,3,2,3,4,1,4,3,2$, $V = 4$, $W= 3, 1, 2, 1,2, 4$, and $W' = 1, 3, 2, 1,2, 4 $. +Finally, $S_4$ is +$ +2,3,4,1,4,3,2,3,1,4,1,3,2,1,2,4 +$ +such that $\textit{TC}_4(i) = 4$ and the theorem is established for +odd and even initial values. -\begin{proof} +For the inductive case, let us first define some variables. +Let $c_{\mathsf{N}}$ (resp. $d_{\mathsf{N}}$) be the number of elements +whose transistion count is exactly $a_{\mathsf{N}}$ (resp $a_{\mathsf{N}} +2$). +These two variables are defined by the system + +$$ +\left\{ +\begin{array}{lcl} +c_{\mathsf{N}} + d_{\mathsf{N}} & = & \mathsf{N} \\ +c_{\mathsf{N}}a_{\mathsf{N}} + d_{\mathsf{N}}(a_{\mathsf{N}}+2) & = & 2^{\mathsf{N}} +\end{array} +\right. +\qquad +\Leftrightarrow +\qquad +\left\{ +\begin{array}{lcl} +d_{\mathsf{N}} & = & \dfrac{2^{\mathsf{N} -n.a_{\mathsf{N}}}{2} \\ +c_{\mathsf{N}} = \mathsf{N} - d_{\mathsf{N}} +\end{array} +\right. +$$ +Since $a_{\mathsf{N}$ is even, $d_{\mathsf{N}}$ is defined. +Both $c_{\mathsf{N}}$ and $d_{\mathsf{N}}$ are obviously positves. -\end{proof} \subsection{Toward a local uniform distribution of switches} diff --git a/main.aux b/main.aux index 6f13901..af7621c 100644 --- a/main.aux +++ b/main.aux @@ -38,13 +38,13 @@ \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{A Metric Space for PRNG Iterations}}{5}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{A metric on $\mathcal {X}_{\mathsf {N},\mathcal {P}}$}}{6}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.4}{$\Gamma _{\mathcal {P}}(f)$ as an extension of $\Gamma (f)$}}{8}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.5}{Proofs of chaos}}{8}} \newlabel{graphe1}{{2a}{9}} \newlabel{sub@graphe1}{{a}{9}} \newlabel{graphe2}{{2b}{9}} \newlabel{sub@graphe2}{{b}{9}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Iterating $f_0:(x_1,x_2) \DOTSB \mapstochar \rightarrow (\overline {x_1}, \overline {x_2})$\relax }}{9}} \newlabel{fig:itg}{{2}{9}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.5}{Proofs of chaos}}{9}} \newlabel{prop:trans}{{3.8}{9}} \citation{bcgr11:ip} \citation{DBLP:conf/secrypt/CouchotHGWB14} @@ -69,51 +69,4 @@ \citation{ZanSup04} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Analysis of the Robinson-Cohn extension algorithm}}{12}} \newlabel{item:nondet}{{1}{12}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Balanced Codes}}{13}} -\newlabel{prop:balanced}{{5.1}{13}} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.3}{Toward a local uniform distribution of switches}}{13}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Stopping Time}}{13}} -\newlabel{sec:hypercube}{{6}{13}} -\citation{LevinPeresWilmer2006} -\newlabel{eq:Markov:rairo}{{3}{15}} -\newlabel{lm:h}{{6.2}{15}} -\citation{proba} -\newlabel{prop:stop}{{6.4}{16}} -\newlabel{prop:lambda}{{6.5}{16}} -\newlabel{lm:stopprime}{{6.6}{17}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Experiments}}{17}} -\newlabel{sec:prng}{{7}{17}} -\@writefile{loa}{\contentsline {algocf}{\numberline {2}{\ignorespaces Pseudo Code of the $\chi _{\textit {15Rairo}}$ PRNG\relax }}{18}} -\newlabel{CI Algorithm:2}{{2}{18}} -\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Functions with DSCC Matrix and smallest MT\relax }}{19}} -\newlabel{table:nc}{{1}{19}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Conclusion}}{19}} -\bibstyle{alpha} -\bibdata{biblio} -\bibcite{Banks92}{BBCS92} -\bibcite{bcgr11:ip}{BCGR11} -\bibcite{Nist10}{BR10} -\bibcite{DBLP:journals/combinatorics/BhatS96}{BS96} -\bibcite{Bykov2016}{Byk16} -\bibcite{chgw14oip}{CHG{$^{+}$}14a} -\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces NIST SP 800-22 test results ($\mathbb {P}_T$)\relax }}{20}} -\newlabel{The passing rate}{{2}{20}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{20}} -\bibcite{DBLP:conf/secrypt/CouchotHGWB14}{CHG{$^{+}$}14b} -\bibcite{5376454}{CMZ09} -\bibcite{Devaney}{Dev89} -\bibcite{guyeuxTaiwan10}{GWB10} -\bibcite{LevinPeresWilmer2006}{LPW06} -\bibcite{LEcuyerS07}{LS07} -\bibcite{Marsaglia1996}{Mar96} -\bibcite{proba}{MU05} -\bibcite{Robinson:1981:CS}{RC81} -\bibcite{915385}{SK01} -\bibcite{915396}{SPK01} -\bibcite{ZanSup04}{SZ04} -\bibcite{wbg10ip}{WBGF10} -\newlabel{tocindent-1}{0pt} -\newlabel{tocindent0}{13.28564pt} -\newlabel{tocindent1}{18.39551pt} -\newlabel{tocindent2}{26.57129pt} -\newlabel{tocindent3}{0pt} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Balanced Codes}}{12}} diff --git a/main.blg b/main.blg index b2d7619..01af0c2 100644 --- a/main.blg +++ b/main.blg @@ -3,7 +3,6 @@ Capacity: max_strings=35307, hash_size=35307, hash_prime=30011 The top-level auxiliary file: main.aux The style file: alpha.bst Database file #1: biblio.bib -Warning--I didn't find a database entry for "" You've used 19 entries, 2543 wiz_defined-function locations, 695 strings with 8037 characters, @@ -45,4 +44,3 @@ warning$ -- 0 while$ -- 80 width$ -- 21 write$ -- 259 -(There was 1 warning) diff --git a/main.log b/main.log index 749adc1..8af4a0a 100644 --- a/main.log +++ b/main.log @@ -1,11 +1,11 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2015/dev/Debian) (preloaded format=pdflatex 2015.11.13) 7 APR 2016 10:48 +This is pdfTeX, Version 3.14159265-2.6-1.40.15 (TeX Live 2015/dev/Debian) (preloaded format=pdflatex 2015.4.13) 7 APR 2016 20:22 entering extended mode restricted \write18 enabled. %&-line parsing enabled. **main.tex (./main.tex LaTeX2e <2014/05/01> -Babel <3.9l> and hyphenation patterns for 6 languages loaded. +Babel <3.9l> and hyphenation patterns for 79 languages loaded. 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