The Enigma machine is an electromechanical device for cryptography whose letter enciphering rule changes as its rotors move. At each fixed rotor state, it acts as a permutation of the 26 letters of the alphabet (Rejewski 1980). The photograph above is courtesy of the Central Intelligence Agency (2011).
Let 
be the permutation carrying a keyboard letter through
the plugboard and rotors to the reflector, and let 
be the reflector's permutation.
The reflector pairs the 26 letters, so 
is a permutation involution
with no fixed points. The return path applies the
inverse permutation 
, giving the enciphering permutation
 |
with function composition evaluated from right to left. The permutations 
and 
are conjugate elements,
so 
and 
for every letter 
(Rejewski 1980). No letter enciphers to itself, a restriction
that can rule out proposed alignments of plaintext
and ciphertext. Rotor motion changes 
between keystrokes. Decryption therefore requires reproducing
the same sequence of rotor states, starting from matching initial settings (Grime,
pp. 1-4).
The close-up above shows the keyboard below the lampboard (Central Intelligence Agency 2011).
A plugboard joins letters in pairs, each acting as a transposition, and leaves the remaining letters unchanged. For an alphabet of 
letters, the number of arrangements with exactly 
pairs is
 |
for a positive integer 
and a nonnegative integer 
with 
. This follows by selecting and ordering 
letters, then removing the ordering of the 
pairs and of the two letters in each pair (Grime, pp. 3-4).
With three ordered rotors chosen from five, 26 initial positions per rotor, and ten plugboard pairs, the number of settings is 
(Grime, pp. 3-4).
The reflector and rotor ring settings are held fixed in this count.
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