How does the Enigma machine ensure that no letter is substituted for itself?

Question

In Alan Turing: The Enigma Andrew Hodges describes how the letter encodings performed by a German Enigma machine "would always be swappings" (original emphasis). And goes on to say that

There was a practical advantage to this Enigma property. It meant that the deciphering operation was identical with the enciphering operation. (In group-theory terms, the cipher was self-inverse). ... But it was associated with a grave weakness, in that the substitutions thus performed were always of this very special kind, with the particular feature that no letter could ever be enciphered into itself.

This passage suggests that any encoding that performs swappings, and is thus self-inverse, must have the property that no letter is "enciphered into itself". However this isn't the case, is it? Isn't this only a property of "swappings" in which all letters are swapped for a different letter. That is, doesn't he mean to say that the Enigma substitutions "would always be swappings of every letter for a different one" so that the non-substitution of the same letter is enforced by the specific swappings preformed by the machine?

If so, how is the property of "non-self-substitution" achieved by the machine as a whole? I can see that at the reflector is physically constrained to have this property, but the plugboard clearly lacks it, as does at least one rotor (Rotor III substitutes N for N). And in any case, it isn't clear to me how self-substituted in guaranteed to be avoided in all possible configurations and rotor positions.

What specific properties of the components and configuration of the Enigma machine ensure that no letter is substituted for itself? Does it follow from some basic property of permutations, or is it the result of specific wiring configurations, deliberately made to avoid self-substitution?

Answer 1

At any stage the Enigma machine provided a unique electrical path through the plugboard and rotors from each letter key to the reflector. The reflector then sent the current back down a different path through the rotors and plugboard, so it ended up at a different letter key and that new letter's light. This had two effects:

• at any stage keys were paired (their paths were joined by the reflector) so the same initial set-up would allow encryption and decryption - this was entirely deliberate and an advertised feature
• each letter key had a letter other than itself which it was paired with at each point in time - this was a flaw exploited by Ultra for cryptanalysis, and could have been avoided with a different design of reflector.

Answer 2

I've always found it easier to understand the non-self encoding of a key press by considering the design of the keys on the Enigma keyboard. Each key on the Enigma keyboard is a two-way switch for the key and its associated lamp. When key X is pressed, lamp X is on open circuit. As long as key X is held down lamp X will never light. So, key X will never encode to itself. See diagram

 http://www.matematiksider.dk/enigma/enigma_circuit_big.png