Exercises - The Hill Cipher

1. Suppose $M_1$ and $M_2$ are linear transformations which operate on blocks of two letters, as represented by two integers chosen in the usual way (A=0, B=1, ..., Z=25), and treated as a vector. It is known that $M_1(\textrm{“BA”})=\textrm{“TR”}$, $M_1(\textrm{“AB”})=\textrm{“IL”}$, $M_2(\textrm{“BA”})=\textrm{“BI”}$, and $M_2(\textrm{“AB”})=\textrm{“LL”}$.

1. Find the matrix forms for both $M_1$ and $M_2$

2. Use the matrix form of $M_2 \pmod{26}$ to encrypt the message "NUMBERTHEORYROCKSX"

3. Now use the matrix form of $M_1 \pmod{26}$ to encrypt your answer to (b) (so that the original message has gone through two stages of encryption).

4. Find the matrix form for the composition $M_1$ and $M_2$, $\pmod{26}$

5. Use the composition found in (b) to encrypt the message "NUMBERTHEORYROCKSX". Compare your answer with the answer to part (c).

6. Find the matrix form for the inverse$\pmod{26}$ of the composition found in (d), and show that when it is applied to the answer to (c), the original message "NUMBERTHEORYROCKSX" is revealed. Does encrypting a message twice with this type of encryption make things more difficult for the person trying to break the code?

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