Contents on Report

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--- a/report/formal.tex
+++ b/report/formal.tex
@@ -1,10 +1,14 @@
 \documentclass[a4paper,10pt]{scrartcl}
-\usepackage[utf8]{inputenc}
+\usepackage[utf8x]{inputenc}
+\usepackage{lmodern}
+\usepackage[T1]{fontenc}
+\usepackage{verbatim}
 \usepackage{amsmath}
 \usepackage{amsthm}
 \usepackage{amssymb}
 \usepackage{listings}
-
+\usepackage{xcolor}
+\usepackage{textcomp}
 %
  \newtheorem{thm}{Theorem}[section]
  \newtheorem{cor}[thm]{Corollary}
@@ -25,9 +29,6 @@
 \date{\today}
 
 \begin{document}
-
-
-
 \maketitle
 
 \begin{abstract}
@@ -44,6 +45,8 @@ developed in RISCAL, and put them into context of formal verification
 systems.
 \end{abstract}
 
+\tableofcontents
+
 \section{Introduction}
 
 \section{Mathematical and Cryptographic Preliminaries}
@@ -173,17 +176,71 @@ $$
 \end{lem}
 
  
- \subsubsection*{The discrete Logarithm problem}
- TODO 
- \subsection{The RSA cryptosystem}
- TODO
+\subsubsection*{The Discrete Logarithm problem}
+Consider an odd prime $p$. Let $b\in\mathbb{Z}_p$ and $0\leq x\leq p-1$. 
+We consider the equality $a\equiv b^x\pmod p$, and call $x$ the \emph{discrete
+logarithm} (or \emph{index}) of $a$ with respect to the basis $b$.\\
+
+The Diffie-Hellmann key exchange algorithm relies on the assumption, that given $a^x$ and
+$a^y$ (without knowing $x$ and $y$), the expression $a^{xy}$ can not be efficiently computed,
+i.e. the discrete logarithm problem is not effectively solvable. Effective solvable in this
+context means that there is an algorithm which proceeds more effective than the brute-force
+method, which proceeds by computing $b^k\pmod p$ until $a\equiv b^x\pmod p$ is reached.\\
 
+TODO: Baby Step Giant Step Algorithm
 
 
+\subsection{The RSA cryptosystem}
+TODO
+
+\subsection{The Diffie-Hellmann cryptosystem}
+TODO
 
 \section{The Formal Verification}
+TODO: Describe algorithm Left-To-Right Exponentation
 
 \appendix
 \section{Listing of the developed Theory}
-
+{\scriptsize \verbatiminput{../numbertheory.txt}}
+
+
+\section{Listing of Implementations}
+\lstset{language=C++,
+    backgroundcolor=\color{black!5}, % set backgroundcolor
+    basicstyle=\ttfamily\tiny,% basic font setting
+ breakatwhitespace=false,         % sets if automatic breaks should only happen at whitespace
+  breaklines=true,                 % sets automatic line breaking
+   captionpos=b,                    % sets the caption-position to bottom
+   deletekeywords={...},            % if you want to delete keywords from the given language
+  escapeinside={\%*}{*)},          % if you want to add LaTeX within your code
+  extendedchars=true,              % lets you use non-ASCII characters; for 8-bits encodings only, does not work with UTF-8
+  frame=single,                           % adds a frame around the code
+  keepspaces=true,                 % keeps spaces in text, useful for keeping indentation of code (possibly needs columns=flexible)
+  language=Octave,                 % the language of the code
+  morekeywords={*,...},           % if you want to add more keywords to the set
+  numbers=left,                    % where to put the line-numbers; possible values are (none, left, right)
+  numbersep=5pt,                   % how far the line-numbers are from the code
+  showspaces=false,                % show spaces everywhere adding particular underscores; it overrides 'showstringspaces'
+  showstringspaces=false,          % underline spaces within strings only
+  showtabs=false,                  % show tabs within strings adding particular underscores
+  stepnumber=1,                  % the step between two line-numbers. If it's 1, each line will be numbered
+  tabsize=2,                      % sets default tabsize to 2 spaces    
+}
+
+{\tiny \lstinputlisting{../src/primefactors.cpp}}
+
+\begin{thebibliography}{9}
+\bibitem{hardylittlewood}
+G.H. Hardy and E.M. Wright. \textit{An Introduction to the Theory of Numbers}.
+Oxford University Press, 6. edition. 2008.
+
+\bibitem{LidlNiederreiter}
+R. Lidl and H. Niederreiter. \textit{Introduction to finite fields and their applications}.
+Cambridge University Press, 1986.
+
+\bibitem{Schreiner}
+W. Schreiner. \textit{The RISC Algorithm Language (RISCAL), Tutorial and Reference}.
+Research Institute for Symbolic Computation (RISC) Linz, 2017.
+
+\end{thebibliography}
 \end{document}