\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}

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

Man in the Middle attacks

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

RSA Encryption

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