# The Theorem that Won the War: Activities for Part 2.2 (Conjugates)

Author(s):
Jeff Suzuki (Brooklyn College)

1. Find $P$, the permutation that shifts letters one place forward (so $a \rightarrow b$, $b \rightarrow c$, and so on).  Also find $P^{-1}$.
2. Prove:  $(P^{k}NP^{-k})^{-1} = P^{k}N^{-1}P^{-k}$.
3. Consider the following initial setup for an Enigma encryption:

1. Find $Q$, the permutation corresponding to the reflector.
2. Find $N$, the permutation corresponding to the initial setup of the keyboard rotor. Also find $N^{-1}$.
3. Find the permutation $P$ that shifts each letter one place forward (so $a \rightarrow b$, $b \rightarrow c$, and so on).
4. Find the inverse permutation $P^{-1}$.
5. Compute $PNP^{-1}$.
6. Compute $PN^{-1}P^{-1}$.
7. Find $E_{1}$, the permutation that will be used to encrypt the first letter.
8. Verify $E_{1} = PN^{-1}P^{-1} Q PNP^{-1}$.  (Remember, in our paper Enigma we're ignoring the permutation $S$ associated with the plugboard.)
9. Find $E_{2}$ and $E_{3}$.
4. The following explains the significance of the reflector. Using the reflector and keyboard setup shown above, answer the following.
1. Consider an Enigma-type machine that omitted a rotor, so the $k$th letter of a message would use the permutation $H_{k} = P^{k}NP^{-k}$.  Find $H_{1}$ and $H_{1}^{-1}$.
2. Show that by including the reflector $Q$, the $k$th letter of an Enigma message would use the permutation $E_{k} = H_{k}^{-1}QH_{k}$.
3. Find $E_{1}$ and $E_{1}^{-1}$.
4. Why does the reflector make the Enigma encryption easier to use?