2 \citation{915396,915385,5376454}
3 \citation{915396,915385}
5 \citation{Marsaglia1996}
9 \select@language{english}
10 \@writefile{toc}{\select@language{english}}
11 \@writefile{lof}{\select@language{english}}
12 \@writefile{lot}{\select@language{english}}
13 \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}}
14 \citation{guyeuxTaiwan10,bcgr11:ip}
16 \citation{DBLP:conf/secrypt/CouchotHGWB14}
19 \citation{DBLP:conf/secrypt/CouchotHGWB14}
20 \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Preliminaries}}{3}}
21 \newlabel{sec:preliminaries}{{2}{3}}
22 \newlabel{eq:asyn}{{1}{3}}
23 \citation{DBLP:conf/secrypt/CouchotHGWB14}
25 \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Iteration Graph $\Gamma (f^*)$ of the function $f^*$\relax }}{4}}
26 \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
27 \newlabel{fig:iteration:f*}{{1}{4}}
28 \@writefile{loa}{\contentsline {algocf}{\numberline {1}{\ignorespaces Pseudo Code of the $\chi _{\textit {14Secrypt}}$ PRNG\relax }}{4}}
29 \newlabel{CI Algorithm}{{1}{4}}
30 \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Proof Of Chaos}}{4}}
31 \newlabel{sec:proofOfChaos}{{3}{4}}
35 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Devaney's Chaotic Dynamical Systems}}{5}}
36 \newlabel{subsec:Devaney}{{3.1}{5}}
37 \newlabel{sensitivity}{{3.5}{5}}
38 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{A Metric Space for PRNG Iterations}}{5}}
39 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{A metric on $\mathcal {X}_{\mathsf {N},\mathcal {P}}$}}{6}}
40 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.4}{$\Gamma _{\mathcal {P}}(f)$ as an extension of $\Gamma (f)$}}{8}}
41 \newlabel{graphe1}{{2a}{9}}
42 \newlabel{sub@graphe1}{{a}{9}}
43 \newlabel{graphe2}{{2b}{9}}
44 \newlabel{sub@graphe2}{{b}{9}}
45 \@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}}
46 \newlabel{fig:itg}{{2}{9}}
47 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.5}{Proofs of chaos}}{9}}
48 \newlabel{prop:trans}{{3.8}{9}}
50 \citation{DBLP:conf/secrypt/CouchotHGWB14}
52 \citation{Robinson:1981:CS,DBLP:journals/combinatorics/BhatS96,ZanSup04,Bykov2016}
53 \citation{DBLP:journals/combinatorics/BhatS96,ZanSup04}
55 \@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Functions with Strongly Connected $\Gamma _{\{b\}}(f)$}}{11}}
56 \newlabel{sec:SCCfunc}{{4}{11}}
57 \@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Balanced Hamiltonian Cycle}}{11}}
58 \newlabel{sec:hamilton}{{5}{11}}
59 \citation{ZanSup04,DBLP:journals/combinatorics/BhatS96}
62 \citation{Robinson:1981:CS}
63 \citation{DBLP:journals/combinatorics/BhatS96}
65 \citation{Robinson:1981:CS}
66 \citation{DBLP:journals/combinatorics/BhatS96}
69 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.1}{Analysis of the Robinson-Cohn extension algorithm}}{12}}
70 \newlabel{item:nondet}{{1}{12}}
72 \newlabel{item:u'}{{2}{13}}
73 \newlabel{item:VW}{{3}{13}}
74 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{5.2}{Balanced Codes}}{13}}
75 \newlabel{prop:balanced}{{5.1}{13}}
76 \newlabel{eq:sys:zt1}{{3}{14}}
77 \newlabel{eq:sys:zt2}{{4}{14}}
78 \newlabel{eq:TCN:def}{{5}{15}}
79 \@writefile{toc}{\contentsline {section}{\tocsection {}{6}{Stopping Time}}{15}}
80 \newlabel{sec:hypercube}{{6}{15}}
81 \citation{LevinPeresWilmer2006}
82 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.1}{Formalizing the Random Walk}}{16}}
83 \newlabel{sub:stop:formal}{{6.1}{16}}
84 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.2}{Upper bound of Stopping Time}}{17}}
85 \newlabel{sub:stop:bound}{{6.2}{17}}
86 \newlabel{eq:Markov:rairo}{{6}{17}}
87 \newlabel{lm:h}{{6.2}{17}}
88 \newlabel{prop:stop}{{6.4}{18}}
89 \newlabel{prop:lambda}{{6.5}{18}}
91 \newlabel{lm:stopprime}{{6.6}{19}}
92 \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.3}{Practical Evaluation of Stopping Times}}{20}}
93 \newlabel{sub:stop:exp}{{6.3}{20}}
94 \@writefile{loa}{\contentsline {algocf}{\numberline {2}{\ignorespaces Pseudo Code of the stoping time calculus\relax }}{20}}
95 \newlabel{algo:stop}{{2}{20}}
96 \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Average Stopping Time\relax }}{21}}
97 \newlabel{table:stopping:moy}{{1}{21}}
98 \@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Experiments}}{21}}
99 \newlabel{sec:prng}{{7}{21}}
100 \@writefile{loa}{\contentsline {algocf}{\numberline {3}{\ignorespaces Pseudo Code of the $\chi _{\textit {15Rairo}}$ PRNG\relax }}{21}}
101 \newlabel{CI Algorithm:2}{{3}{21}}
102 \@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces Functions with DSCC Matrix and smallest MT\relax }}{22}}
103 \newlabel{table:nc}{{2}{22}}
106 \bibcite{Banks92}{BBCS92}
107 \@writefile{lot}{\contentsline {table}{\numberline {3}{\ignorespaces NIST SP 800-22 test results ($\mathbb {P}_T$)\relax }}{23}}
108 \newlabel{The passing rate}{{3}{23}}
109 \@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Conclusion}}{23}}
110 \bibcite{bcgr11:ip}{BCGR11}
111 \bibcite{Nist10}{BR10}
112 \bibcite{DBLP:journals/combinatorics/BhatS96}{BS96}
113 \bibcite{Bykov2016}{Byk16}
114 \bibcite{DBLP:conf/secrypt/CouchotHGWB14}{CHG{$^{+}$}14}
115 \bibcite{5376454}{CMZ09}
116 \bibcite{Devaney}{Dev89}
117 \bibcite{guyeuxTaiwan10}{GWB10}
118 \bibcite{LevinPeresWilmer2006}{LPW06}
119 \bibcite{LEcuyerS07}{LS07}
120 \bibcite{Marsaglia1996}{Mar96}
121 \bibcite{proba}{MU05}
122 \bibcite{Robinson:1981:CS}{RC81}
123 \bibcite{915385}{SK01}
124 \bibcite{915396}{SPK01}
125 \bibcite{ZanSup04}{SZ04}
126 \bibcite{wbg10ip}{WBGF10}
127 \newlabel{tocindent-1}{0pt}
128 \newlabel{tocindent0}{12.77466pt}
129 \newlabel{tocindent1}{17.77344pt}
130 \newlabel{tocindent2}{25.54932pt}
131 \newlabel{tocindent3}{0pt}
132 \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{24}}
