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\begin{document}
\huge{Hill Cipher Exercises (Daily Due Sept 4)}
\large{Katherine E. Stange, CU Boulder}
\section*{Exercise 1}
Find the inverse of the following matrix whose entries are considered modulo $26$:
\[
\begin{pmatrix} 11 & 13 \\ 2 & 3 \end{pmatrix}
\]
Note: you can do this exactly as you would find the inverse of a $2\times 2$ matrix normally, except when you need to divide by something, find the modular inverse using the table.
\section*{Exercise 2}
The matrix given in the last exercise was used as a key to a Hill cipher to encrypt a favourite vegetable of mine, and the resulting ciphertext was YGFI. What is the vegetable?
\section*{Exercise 3}
A $2 \times 2$ Hill cipher encrypted the plaintext SOLVED to give the ciphertext GEZXDS. Find the encryption matrix.
\section*{Exercise 4}
Suppose the matrix
\[
\begin{pmatrix}
1 & 2 \\ 3 & 4 \end{pmatrix}
\]
is used for a $2 \times 2$ Hill cipher.
\begin{enumerate}
\item Compute the determinant. What is bad about this determinant?
\item Find two plaintexts that encrypt to the same ciphertext.
\end{enumerate}
\end{document}