\section{Part 5: Asymmetric Encryption}

\subsection{Generierte Primzahlen}

\begin{verbatim}
openssl prime -generate -bits 8
\end{verbatim}

\begin{verbatim}
prime1 = 211, prime2 = 223, prime3 = 227, e=11
\end{verbatim}

\subsection{Berechnungen}

\begin{align}
    d = e^{-1}\ mod\ ((prime1-1)(prime2-1)) \\
    d = 11^{-1}\ mod\ ((211-1)(223-1)) = 21191
\end{align}

\begin{verbatim}
g = 9, x = 2, y = 3
\end{verbatim}

\begin{align}
    a = g^{x}\ (mod\ prime3) = 9^{2} (mod\ 227) = 81 \\
    b = g^{y}\ (mod\ prime3) = 9^{3} (mod\ 227) = 48 \\
    k_{1} = b^{x}\ (mod\ prime3) = 48^{2}\ (mod\ 227) = 34 \\
    k_{2} = a^{y}\ (mod\ prime3) = 81^{3}\ (mod\ 227) = 34 \\
    k = k_{1} = k_{2} = 34
\end{align}

\subsection{Fragen und Antworten}

1. What attack is the Diffie-Hellman key exchange vulnerable to?

Man in the Middle

2. What measures can be taken to prevent this type of attack?

Encryption, Authentication over QR code or 2Factor-Authentication

3. For the Diffie-Hellman, a generator g is used. Explain what a generator is and how can it be found

4. Show why for the primes 61,23 and the public key e=60 no private key d can be found

\begin{align}
    d = e^{-1}\ mod\ ((p1-1)(p2-1)) \\
    d = 60^{-1}\ mod\ ((61-1)(23-1)) = 60^{-1} mod\ 1320
\end{align}