diff -r 279fa5a06231 -r 6f884231ca57 slides/slides09.tex --- a/slides/slides09.tex Mon Dec 02 10:04:40 2013 +0000 +++ b/slides/slides09.tex Tue Dec 03 03:20:42 2013 +0000 @@ -38,60 +38,8 @@ \renewcommand{\isasymdots}{\isamath{...}} \renewcommand{\isasymbullet}{\act} - - -\definecolor{javared}{rgb}{0.6,0,0} % for strings -\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments -\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords -\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc - -\lstset{language=Java, - basicstyle=\ttfamily, - keywordstyle=\color{javapurple}\bfseries, - stringstyle=\color{javagreen}, - commentstyle=\color{javagreen}, - morecomment=[s][\color{javadocblue}]{/**}{*/}, - numbers=left, - numberstyle=\tiny\color{black}, - stepnumber=1, - numbersep=10pt, - tabsize=2, - showspaces=false, - showstringspaces=false} - -\lstdefinelanguage{scala}{ - morekeywords={abstract,case,catch,class,def,% - do,else,extends,false,final,finally,% - for,if,implicit,import,match,mixin,% - new,null,object,override,package,% - private,protected,requires,return,sealed,% - super,this,throw,trait,true,try,% - type,val,var,while,with,yield}, - otherkeywords={=>,<-,<\%,<:,>:,\#,@}, - sensitive=true, - morecomment=[l]{//}, - morecomment=[n]{/*}{*/}, - morestring=[b]", - morestring=[b]', - morestring=[b]""" -} - -\lstset{language=Scala, - basicstyle=\ttfamily, - keywordstyle=\color{javapurple}\bfseries, - stringstyle=\color{javagreen}, - commentstyle=\color{javagreen}, - morecomment=[s][\color{javadocblue}]{/**}{*/}, - numbers=left, - numberstyle=\tiny\color{black}, - stepnumber=1, - numbersep=10pt, - tabsize=2, - showspaces=false, - showstringspaces=false} - % beamer stuff -\renewcommand{\slidecaption}{APP 09, King's College London, 2 December 2013} +\renewcommand{\slidecaption}{APP 09, King's College London, 3 December 2013} \newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions \newcommand{\bl}[1]{\textcolor{blue}{#1}} @@ -107,15 +55,11 @@ \LARGE Privacy Policies (9)\\[-6mm] \end{tabular}}\bigskip\bigskip\bigskip - %\begin{center} - %\includegraphics[scale=1.3]{pics/barrier.jpg} - %\end{center} - -\normalsize + \normalsize \begin{center} \begin{tabular}{ll} Email: & christian.urban at kcl.ac.uk\\ - Of$\!$fice: & S1.27 (1st floor Strand Building)\\ + Office: & S1.27 (1st floor Strand Building)\\ Slides: & KEATS (also homework is there)\\ \end{tabular} \end{center} @@ -126,31 +70,53 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode{ -\begin{frame}[c] -\frametitle{Last Week} +\begin{frame}[t] +\frametitle{Checking Solutions} + +How can you check somebody's solution without revealing the solution?\pause\bigskip -Recall, the Schroeder-Needham (1978) protocol is vulnerable to replay attacks. +Alice and Bob solve crosswords. Alice knows the answer for 21D (folio) but doesn't +want to tell Bob.\medskip + +You use an English dictionary: -\begin{center} -\begin{tabular}{@{}r@ {\hspace{1mm}}l@{}} -\bl{$A \rightarrow S :$} & \bl{$A, B, N_A$}\\ -\bl{$S \rightarrow A :$} & \bl{$\{N_A, B, K_{AB},\{K_{AB}, A\}_{K_{BS}} \}_{K_{AS}}$}\\ -\bl{$A \rightarrow B :$} & \bl{$\{K_{AB}, A\}_{K_{BS}} $}\\ -\bl{$B \rightarrow A :$} & \bl{$\{N_B\}_{K_{AB}}$}\\ -\bl{$A \rightarrow B :$} & \bl{$\{N_B-1\}_{K_{AB}}$}\\ -\end{tabular} -\end{center}\pause +\begin{itemize} +\item folio \onslide<4->{$\stackrel{1}{\rightarrow}$ individual } + \onslide<5->{$\stackrel{2}{\rightarrow}$ human} + \onslide<6->{$\stackrel{3}{\rightarrow}$ or \ldots} +\only<3>{ +\begin{quote} +``an \alert{individual} leaf of paper or parchment, either loose as one of a series or +forming part of a bound volume, which is numbered on the recto or front side only.'' +\end{quote}} +\only<4>{ +\begin{quote} +``a single \alert{human} being as distinct from a group'' +\end{quote}} +\only<5>{ +\begin{quote} +``relating to \alert{or} characteristic of humankind'' +\end{quote}} +\end{itemize}\bigskip\bigskip -Fix: Replace messages 2 and 3 to include a timestamp:\bigskip +\only<7->{ +hash functions...but Bob can only check once he has also the solution +} + +\end{frame}} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{minipage}{1.1\textwidth} -\begin{center} -\begin{tabular}{@{\hspace{-2mm}}r@ {\hspace{1mm}}l@{}} -\bl{$S \rightarrow A :$} & \bl{$\{B, K_{\!AB}, T_S, \!\{K_{\!AB}, A, T_S\}_{K_{BS}} \}_{K_{AS}}$}\\ -\bl{$A \rightarrow B :$} & \bl{$\{K_{AB}, A, T_S\}_{K_{BS}} $}\\ -\end{tabular} -\end{center} -\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\mode{ +\begin{frame}[c] +\frametitle{Zero-Knowledge Proofs} + +Two remarkable properties:\bigskip + +\begin{itemize} +\item Alice only reveals the fact that she knows a secret.\bigskip +\item Having been convinced, Bob cannot use the evidence in order to convince Carol. +\end{itemize} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -158,391 +124,97 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode{ \begin{frame}[t] -\frametitle{Denning-Sacco Fix} +\frametitle{\begin{tabular}{@{}c@{}}The Idea\end{tabular}} -Denning-Sacco (1981) suggested to add the timestamp, but omit the handshake:\bigskip - -\begin{minipage}{1.1\textwidth} \begin{center} -\begin{tabular}{@{\hspace{-2mm}}r@ {\hspace{1mm}}l@{}} -\bl{$A \rightarrow S :$} & \bl{$A, B$}\\ -\bl{$S \rightarrow A :$} & \bl{$\{B, K_{\!AB}, T_S, \!\{K_{\!AB}, A, T_S\}_{K_{BS}} \}_{K_{AS}}$}\\ -\bl{$A \rightarrow B :$} & \bl{$\{K_{AB}, A, T_S\}_{K_{BS}} $}\\ -\textcolor{lightgray}{$B \rightarrow A :$} & \textcolor{lightgray}{$\{N_B\}_{K_{AB}}$}\\ -\textcolor{lightgray}{$A \rightarrow B :$} & \textcolor{lightgray}{$\{N_B-1\}_{K_{AB}}$}\\ +\begin{tabular}{l@{\hspace{10mm}}r} +\\[-10mm] +\raisebox{10mm}{\large 1.} & \includegraphics[scale=0.1]{pics/alibaba1.png}\\ +\raisebox{10mm}{\large 2.} & \includegraphics[scale=0.1]{pics/alibaba2.png}\\ +\raisebox{10mm}{\large 3.} & \includegraphics[scale=0.1]{pics/alibaba3.png} \end{tabular} \end{center} -\end{minipage}\bigskip - -they argue \bl{$A$} and \bl{$B$} can check that the messages are not replays of earlier -runs, by checking the time difference with when the protocol is last used -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ -\begin{frame}[t] -\frametitle{\begin{tabular}{@{}c@{}}Denning-Sacco-Lowe Fix of Fix\end{tabular}} - -Lowe (1997) disagreed and said the handshake should be kept, -otherwise:\bigskip -\begin{minipage}{1.1\textwidth} -\begin{center} -\begin{tabular}{@{\hspace{-7mm}}r@ {\hspace{1mm}}l@{}} -\bl{$A \rightarrow S :$} & \bl{$A, B$}\\ -\bl{$S \rightarrow A :$} & \bl{$\{B, K_{\!AB}, T_S, \!\{K_{\!AB}, A, T_S\}_{K_{BS}} \}_{K_{AS}}$}\\ -\bl{$A \rightarrow B :$} & \bl{$\{K_{AB}, A, T_S\}_{K_{BS}} $}\\ -\bl{$I(A) \rightarrow B :$} & \bl{$\{K_{AB}, A, T_S\}_{K_{BS}} $}\hspace{5mm}\textcolor{black}{replay}\\ -\end{tabular} -\end{center} -\end{minipage}\bigskip +\small +\only<2>{ +\begin{textblock}{12}(2,13.3) +Even if Bob has a hidden camera, a recording will not be convincing to anyone else +(Alice and Bob could have made it all up). +\end{textblock}} +\only<3>{ +\begin{textblock}{12}(2,13.3) +Even worse, an observer present at the experiment would not be convinced. +\end{textblock}} -When is this a problem?\pause\medskip - -Assume \bl{$B$} is a bank and the message is ``Draw \pounds{1000} from \bl{$A$}'s -account and transfer it to \bl{$I$}.'' \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode{ -\begin{frame}[t] -\frametitle{Privacy} - -\begin{minipage}{1.05\textwidth} -\begin{itemize} -\item we \alert{do} want that government data is made public (free maps for example) -\item we \alert{do not} want that medical data becomes public (similarly tax data, school -records, job offers)\bigskip -\item personal information can potentially lead to fraud -(identity theft) -\end{itemize}\pause - -{\bf ``The reality'':} -\only<2>{\begin{itemize} -\item London Health Programmes lost in June unencrypted details of more than 8 million people -(no names, but postcodes and details such as gender, age and ethnic origin) -\end{itemize}} -\only<3>{\begin{itemize} -\item also in June Sony, got hacked: over 1M users' personal information, including passwords, email addresses, home addresses, dates of birth, and all Sony opt-in data associated with their accounts. -\end{itemize}} -\end{minipage} - -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ \begin{frame}[c] -\frametitle{Privacy and Big Data} - -Selected sources of ``Big Data'':\smallskip{} - -\begin{itemize} -\item Facebook -\begin{itemize} -\item 40+ Billion photos (100 PB) -\item 6 Billion messages daily (5 - 10 TB) -\item 900 Million users -\end{itemize} -\item Common Crawl -\begin{itemize} -\item covers 3.8 Billion webpages (2012 dataset) -\item 50 TB of data -\end{itemize} -\item Google -\begin{itemize} -\item 20 PB daily (2008) -\end{itemize} -\item Twitter -\begin{itemize} -\item 7 Million users in the UK -\item a company called Datasift is allowed to mine all tweets since 2010 -\item they charge 10k per month for other companies to target advertisement -\end{itemize} -\end{itemize}\pause - +\frametitle{Graph Isomorphism} -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ -\begin{frame}[c] -\frametitle{Cookies\ldots} - -``We have published a new cookie policy. It explains what cookies are -and how we use them on our site. To learn more about cookies and -their benefits, please view our cookie policy.\medskip - -If you'd like to disable cookies on this device, please view our information -pages on 'How to manage cookies'. Please be aware that parts of the -site will not function correctly if you disable cookies. \medskip - -By closing this -message, you consent to our use of cookies on this device in accordance -with our cookie policy unless you have disabled them.'' - - -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ -\begin{frame}[c] -\frametitle{Scare Tactics} +\begin{center} +\begin{tabular}{l@{\hspace{10mm}}r} +\includegraphics[scale=0.8]{pics/graphs.png}\\ +\end{tabular} +\end{center} -The actual policy reads:\bigskip - -``As we explain in our Cookie Policy, cookies help you to get the most -out of our websites.\medskip - -If you do disable our cookies you may find that certain sections of our -website do not work. For example, you may have difficulties logging in -or viewing articles.'' - - - - -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ -\begin{frame}[c] -\frametitle{Netflix Prize} - -Anonymity is \alert{necessary} for privacy, but \alert{not} enough!\bigskip - -\begin{itemize} -\item Netflix offered in 2006 (and every year until 2010) a 1 Mio \$ prize for improving their movie rating algorithm -\item dataset contained 10\% of all Netflix users (appr.~500K) -\item names were removed, but included numerical ratings as well as times of rating -\item some information was \alert{perturbed} (i.e., slightly modified) -\end{itemize} - -\hfill{\bf\alert{All OK?}} - +Finding an isomorphism between two graphs is an NP complete problem. \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode{ \begin{frame}[c] -\frametitle{Re-identification Attack} - -Two researchers analysed the data: +\frametitle{Graph Isomorphism Protocol} -\begin{itemize} -\item with 8 ratings (2 of them can be wrong) and corresponding dates that can have a margin 14-day error, 98\% of the -records can be identified -\item for 68\% only two ratings and dates are sufficient (for movie ratings outside the top 500)\bigskip\pause -\item they took 50 samples from IMDb (where people can reveal their identity) -\item 2 of them uniquely identified entries in the Netflix database (either by movie rating or by dates) -\end{itemize} - -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - +Alice starts with knowing an isomorphism between graphs \bl{$G_1$} and \bl{$G_2$}\medskip -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ -\begin{frame}[c] -\frametitle{} - -\begin{itemize} -\item Birth data, postcode and gender (unique for\\ 87\% of the US population) -\item Preferences in movies (99\% of 500K for 8 ratings) -\end{itemize}\bigskip +\begin{enumerate} +\item Alice generates an isomorphic graph \bl{$H$} which she sends to Bob +\item Bob asks either for an isomorphism between \bl{$G_1$} and \bl{$H$}, or +\bl{$G_2$} and \bl{$H$} +\item Alice and Bob repeat this procedure \bl{$n$} times +\end{enumerate}\pause -Therefore best practices / or even law (HIPAA, EU): - -\begin{itemize} -\item only year dates (age group for 90 years or over), -\item no postcodes (sector data is OK, similarly in the US)\\ -\textcolor{gray}{no names, addresses, account numbers, licence plates} -\item disclosure information needs to be retained for 5 years -\end{itemize} - +these are called commitment algorithms \end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode{ \begin{frame}[c] -\frametitle{How to Safely Disclose Information?} - -\only<1>{ -\begin{itemize} -\item Assume you make a survey of 100 randomly chosen people. -\item Say 99\% of the surveyed people in the 10 - 40 age group have seen the -Gangnam video on youtube.\bigskip - -\item What can you infer about the rest of the population? -\end{itemize}} -\only<2>{ -\begin{itemize} -\item Is it possible to re-identify data later, if more data is released. \bigskip\bigskip\pause - -\item Not even releasing only aggregate information prevents re-identification attacks. -(GWAS was a public database of gene-frequency studies linked to diseases; -you only needed partial DNA information in order -to identify whether an individual was part of the study --- DB closed in 2008) -\end{itemize}} - -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ -\begin{frame}[c] -\frametitle{Differential Privacy} - -\begin{center} -User\;\;\;\; -\begin{tabular}{c} -tell me \bl{$f(x)$} $\Rightarrow$\\ -$\Leftarrow$ \bl{$f(x) + \text{noise}$} -\end{tabular} -\;\;\;\;\begin{tabular}{@{}c} -Database\\ -\bl{$x_1, \ldots, x_n$} -\end{tabular} -\end{center} +\frametitle{Non-Interactive ZKPs} -\begin{itemize} -\item \bl{$f(x)$} can be released, if \bl{$f$} is insensitive to -individual entries \bl{$x_1, \ldots, x_n$}\\ -\item Intuition: whatever is learned from the dataset would be learned regardless of whether -\bl{$x_i$} participates\bigskip\pause - -\item Noised needed in order to prevent queries:\\ Christian's salary $=$ -\begin{center} -\bl{\large$\Sigma$} all staff $-$ \bl{\large$\Sigma$} all staff $\backslash$ Christian -\end{center} -\end{itemize} - +\bigskip +This is amazing: Alison can publish some data that contains no data about her secret, +but can be used to convince anyone of the secret's existence. \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode{ \begin{frame}[c] -\frametitle{Adding Noise} - -Adding noise is not as trivial as one would wish: - -\begin{itemize} -\item If I ask how many of three have seen the Gangnam video and get a result -as follows - -\begin{center} -\begin{tabular}{l|c} -Alice & yes\\ -Bob & no\\ -Charlie & yes\\ -\end{tabular} -\end{center} - -then I have to add a noise of \bl{$1$}. So answers would be in the -range of \bl{$1$} to \bl{$3$} - -\bigskip -\item But if I ask five questions for all the dataset (has seen Gangnam video, is male, below 30, \ldots), -then one individual can change the dataset by \bl{$5$} -\end{itemize} - -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - +\frametitle{Problems of ZKPs} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ -\begin{frame}[t] -\frametitle{\begin{tabular}{@{}c@{}}Tor, Anonymous Webbrowsing\end{tabular}} - -\begin{itemize} -\item initially developed by US Navy Labs, but then opened up to the world -\item network of proxy nodes -\item a Tor client establishes a ``random'' path to the destination server (you cannot trace back where the information came from)\bigskip\pause -\end{itemize} - -\only<2>{ -\begin{itemize} -\item malicious exit node attack: someone set up 5 Tor exit nodes and monitored the traffic: -\begin{itemize} -\item a number of logons and passwords used by embassies (Usbekistan `s1e7u0l7c', while -Tunesia `Tunesia' and India `1234') -\end{itemize} -\end{itemize}} -\only<3>{ -\begin{itemize} -\item bad apple attack: if you have one insecure application, your IP can be tracked through Tor -\begin{itemize} -\item background: 40\% of traffic on Tor is generated by BitTorrent -\end{itemize} -\end{itemize}} - - -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ -\begin{frame}[t] -\frametitle{\begin{tabular}{@{}c@{}}Skype Secure Communication\end{tabular}} - -\begin{itemize} -\item Skype used to be known as a secure online communication (encryption cannot be disabled), -but \ldots\medskip - -\item it is impossible to verify whether crypto algorithms are correctly used, or whether there are backdoors.\bigskip - -\item recently someone found out that you can reset the password of somebody else's -account, only knowing their email address (needed to suspended the password reset feature temporarily) -\end{itemize} - - -\end{frame}} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - - - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\mode{ -\begin{frame}[c] -\frametitle{\begin{tabular}{@{}c@{}}Take Home Point\end{tabular}} - -According to Ross Anderson: \bigskip -\begin{itemize} -\item Privacy in a big hospital is just about doable.\medskip -\item How do you enforce privacy in something as big as Google -or complex as Facebook? No body knows.\bigskip - -Similarly, big databases imposed by government -\end{itemize} - - +\bigskip +This is amazing: Alison can publish some data that contains no data about her secret, +but can be used to convince anyone of the secret's existence. \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \mode{ \begin{frame}[c] -\frametitle{\begin{tabular}{@{}c@{}}Next Week\end{tabular}} - -Homework: Which areas should I focus on? +\frametitle{Random Number Generators} \end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - \end{document} %%% Local Variables: