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?

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    $\begingroup$ Why do you believe the passage suggests what you say it suggests? $\endgroup$ – whacka Mar 27 '15 at 22:31
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    $\begingroup$ @whacka: "This" refers to property that whatever substitutions were performed would be swapping, but not (as written) that all letters are swapped. "Associated" is a bit less direct, but suggests that the fact that " no letter could ever be enciphered into itself" is a consequence of "this". $\endgroup$ – orome Mar 27 '15 at 22:34
  • $\begingroup$ I'm not quite sure what the passage is trying to say, but it sounds like if you treat the Enigma machine as a function $f$ taking plaintext and outputting cipertext, then it is an involution - i.e. $f(f(x))=x$. This doesn't imply that $f$ has no fixed points (since the identity function is an involution, for instance) - but it could be that the Enigma machine specified so as an additional requirement. $\endgroup$ – Milo Brandt Mar 27 '15 at 22:41
  • $\begingroup$ @Meelo: Correct, and, I think that highlights the question then: what properties of the Enigma (e.g. physical constraints or deliberate design) ensure that no letter is substituted for itself (i.e. that there are no fixed points). $\endgroup$ – orome Mar 27 '15 at 22:47
  • $\begingroup$ I don't read that text as implying what you say it implies. Obviously, he is implying that, by a swapping, he means a complete swapping - no fixed letters. He never says any self-inverse cipher must not have a fixed point. $\endgroup$ – Thomas Andrews Mar 27 '15 at 22:50

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.
  • $\begingroup$ So (1) unique path (through rotors and plugboard, in each direction) combined (critically) with the fact that (2) the reflector must perform complete swapping of all letters ensures that the machine as a whole does so. Correct? $\endgroup$ – orome Mar 27 '15 at 23:09
  • $\begingroup$ @raxacoricofallapatorius: I am not sure I would use must that way. It did always swap/pair different letters, but it could have been designed differently. It had $26$ keys/letters - if it had had a $27$th and a single reflection back down the incoming route, it would have been more secure. $\endgroup$ – Henry Mar 27 '15 at 23:18
  • $\begingroup$ Good point. But the key thing is: it's that feature of the reflector that drives the whole thing (assuming nothing special about the wiring of the rotors, etc.). $\endgroup$ – orome Mar 27 '15 at 23:20
  • $\begingroup$ Yes: the rotors make cryptanalysis difficult (and an extra rotor even more so in the case of Shark), but it was the the reflector design which led to the no non-substitution property. $\endgroup$ – Henry Mar 27 '15 at 23:35
  • $\begingroup$ What I was missing in thinking about this was the simple fact that (regardless of the details of rotor wiring or position, as long as every letter is connected) there is a unique path that connects to a given letter at the begging/end, so that if, as is the case, the reflector switches between these paths, the letter at the begging and end must be different. $\endgroup$ – orome Mar 28 '15 at 0:34

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