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formalization. \subsection{(Slightly more) Advanced Number Theory} -Describe the algorithm for discrete logarithm here +In this section, we describe a cryptographic algorithm, the \emph{Diffie-Hellmann key exchange} that has its origins +in computational number theory. Within the description of the algorithm, we will pose question \eqref{eq:q2} that we +will answer and formalize in section \ref{sec:formal}. The Diffie-Hellmann key exchange is used, when two parties, say +\textbf{A}lice and \textbf{B}ob want to exchange secret messages. \textbf{A} and \textbf{B} agree on +a (large) prime $p$ and an element $a\in\mathbb{Z}_p$ (of high order, i.e. $a^m\pmod p \equiv 1$ implies that +$m$ is large). The pair $(a,p)$ is called the \emph{public key}. Then, both \textbf{A} and \textbf{B}, choose their private key $x$ and +$y$ respectively, and publish as a public key $a^x\pmod p$ and $a^y\pmod p$ respectively. \textbf{A} takes the published key +$a^y\pmod p$ from Bob and computes the key $M\equiv (a^y)^x \equiv a^{xy}\pmod p$. Similar, \textbf{B} +takes the published key $a^x\pmod p$ from \textbf{A} and computes $M \equiv (a^x)^y \equiv a^{xy}\pmod p$, +to obtain the same key. The key $M$ can now be used by a ``simpler'' crypto-system, such as a +simple displacement chiffre.\\ + +Why is this algorithm useful? The overall concept of the Diffie-Hellmann key exchange rests on the \emph{discrete logarithm problem}. +Let us describe this problem in more detail: Suppose we are given +\begin{itemize} + \item an odd (very large) prime $p$; + \item an element $b\in \mathbb{Z}_p$; +\end{itemize} +If we now have an element $a$ such that $0\leq a\leq p-1$, such that +$$ +a \equiv b^x \pmod p,\qquad 0\leq x\leq p-1, +$$ +then $x$ is called \emph{the discrete logarithm} of $a$ to basis $b$. The assumption of the Diffie-Hellmann algorithm +might now be formulated as follows: Given $a^x\pmod p$ and $a^y\pmod p$ without knowing $x,y$, the expression $a^{xy}\pmod p$ +is not efficiently solvable.\\ + +We are now ready formulate \eqref{eq:q2}: +\begin{itemize} + \item Let $p$ be a large (odd) prime; + \item An element $g$ such that $2\leq g\leq p-1$ of order $p-1$; + \item An element $a\in\mathbb{Z}_p\backslash\{0\}$; +\end{itemize} +\begin{equation}\label{eq:q2} +\text{Find }x\in \mathbb{Z}_p \text{ such that }\, g^x\equiv a\pmod p.\tag{{\rm Q2}} +\end{equation} \section{Analysis of the Algorithms} The first step in formalization is to do the necessary paper-work. We start by analyzing the expected input and output @@ -452,9 +486,9 @@ $2\leq x\leq \lfloor \sqrt{n}\rfloor$, there are finitely numbers, and therefore considered, no further numbers greater $\lfloor \sqrt{n}\rfloor$ need to be considered and hence the algorithm terminates in a finite number of steps. -\subsection{Discrete Logarithm - Algorithm of Tonelli} +\subsection{Discrete Logarithm - The Baby Step Giant Step Algorithm} -\section{The RISCAL Formalization} +\section{The RISCAL Formalization}\label{sec:formal} \section{Outlook} -- 2.1.4