\documentclass[11pt]{article}
\usepackage{latexsym}
\usepackage{graphics}
\usepackage{amsfonts}
\usepackage{epsfig} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%  MODIFY THE TWO LINES BELOW TO MATCH THE LECTURE YOU ARE SCRIBING
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\def\lectureNumber{0}         %enter the index of the lecture
\def\scribe{Ariel Elbaz}      %enter your name
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\setlength{\textheight}{8.5in}          
\setlength{\textwidth}{6.0in}
\setlength{\headheight}{0in}
\addtolength{\topmargin}{-.5in}
\addtolength{\oddsidemargin}{-.5in}


\def\epslion{\varepsilon}
\def\phi{\varphi}
\def\bool{\{0,1\}}

\newenvironment{proof_sketch}{\noindent{\bf Sketch of Proof} (Informal)\hspace*{1em}}{\qed\medskip}
\newenvironment{proof}{\noindent{\bf Proof}\hspace*{1em}}{\qed\medskip}
\newenvironment{claim}{\noindent{\bf Claim}\hspace*{1em}\begin{em}}{\end{em}}
\newcounter{defcounter}
\setcounter{defcounter}{1}
\newenvironment{definition}{\medskip\noindent{\bf Definition \thedefcounter}}{\qed\medskip\stepcounter{defcounter}}
\newcounter{remarkcounter}
\setcounter{remarkcounter}{1}
\newenvironment{remark}{\medskip\noindent{\sc Remark \theremarkcounter.}}{\stepcounter{remarkcounter}}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{fact}{Fact}
\newtheorem{conjecture}{Conjecture}
\newtheorem{assumption}{Assumption}
\newtheorem{example}{Example}
\newcommand{\qed}{\hspace*{\fill}\rule{7pt}{7pt}}
\newcommand{\xor}{\oplus}
\newcommand{\Xor}{\bigoplus}
\newcommand{\ceil}[1]{\left\lceil #1 \right\rceil}
\newcommand{\for}[3]{{\sf for}\hspace{3pt} #1 = #2\hspace{3pt}{\sf to}\hspace{3pt}#3:}
\newcommand{\pr}[1]{{\rm Pr}\left[ #1 \right]}
\newcommand{\ignore}[1]{}
\newcommand{\bydef}{\stackrel{\rm def}{=}}
%\newcommand{\integers}[1]{{\sf Z} \!\!\!\, {\sf Z}_{#1}}
\newcommand{\integers}[1]{{\mathbb Z}_{#1}}
\newcommand{\reals}{{\sf R}}

\begin{document}
\noindent
Columbia University \\
W4995: Introduction To Cryptography, Fall 2003 \\
Instructor: Professor Malkin \\
Lecture \lectureNumber ,   
Notes takes by \scribe \\
\rule{6.0in}{.01in}

\abstract{This document gives a few examples for using \LaTeX. You should use the given file ({\tt ITC0.tex})
    as a template for writing your scribes - change the name to have the correct lecture number that 
    you're scribing, such as {\tt ITC7.tex} if you're scribing lecture 7.
 }

\section{Examples of Theorems, Claims, Lemmas, and Proofs} 

\begin{theorem}
text of a theorem...
\end{theorem}
\begin{proof}
starting a proof...
and ending it
\end{proof}

\begin{claim}
text of a claim
\end{claim}

\begin{proof_sketch}
text of a proof sketch
\end{proof_sketch}

We can also {\em emphasize parts of the text} 

\subsection{A subsection on primes}

\begin{definition}
  $p$ is a prime if $p$ is an integer that has no integral factors but itself and 1. 
\end{definition}

\subsection*{More examples}

\begin{lemma}
text of a lemma
\end{lemma}

\begin{example}
text of an example
\end{example}

\section{Mathematical Equations}


Math equations can either be mixed with text,, such as $y = x + 2$ or $2^{2^{13}}$ or 
$\frac {17!} {2^{34}}$, or else they can be in a sepeate line
$$ Pr[x \geq y] \leq \frac 1 3 $$
or on an automatically numbered equation
\begin{equation}
\forall x | x \leq y \Rightarrow y \geq 0
\end{equation}
and the text continues in the next line.

More advanced equations
\begin{eqnarray*}
V_{s,0}^{\{q_0,q_1\}} &  =  V_{s,0}^{\{q_0\}} \ \cup  & 
V_{s,1}^{\{q_0\}}\circ  
\left(V_{1,1}^{\{q_0\}}\right)^* \circ 
V_{1,0}^{\{q_0\}} 
\\ 
V_{s,2}^{\{q_0,q_1\}} & =  V_{s,2}^{\{q_0\}} \ \cup  &
V_{s,1}^{\{q_0\}}\circ 
\left(V_{1,1}^{\{q_0\}}\right)^* \circ 
V_{1,2}^{\{q_0\}} = 
V_{s,1}^{\{q_0\}}\circ 
\left(V_{1,1}^{\{q_0\}}\right)^* \circ 
V_{1,2}^{\{q_0\}} 
\end{eqnarray*}

\section{Itemizing}
\begin{itemize}
  \item first item
  \item second item
  \item[***] item with my bullet
  \item[\qed] any bullet will go
    \begin{enumerate}
      \item Items can also be numbered
      \item and they can be nested
    \end{enumerate}
\end{itemize}


\section{Writing text}
The \LaTeX  compiler does it's own pagesetting. It converts any number of spaces to a single space, and concatantes sentences to fill the paragraph. You can spread one sentence
over
as 
many 
lines
as 
you 
want, 
it 
will 
continue 
as if it was written in a single line. 
If you want to break a paragraph you need to leave an empty line.


\end{document}
