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I recently learned about the Enigma Machine in my cryptography class, but I am a bit confused as to the number of permutations of the wheel settings. According to every article I've read on the matter, the number of different ways the wheels could be set up equals the number of permutations that the wheels can produce. I understand that there are 26*26*26 possible starting settings (assuming three wheels), but what I do not understand is why the settings of each wheel do not come into play. From what I know, each wheel contains a series of jumbled wires that further scramble each incoming letter. Aren't there 26! different ways that these wires can be arranged? For example, when the plaintext is an 'A' and the starting setting of the first wheel is at position n, the first wheel can produce any different letter depending on the wiring inside of the wheel. I just don't get why that is not a factor when determining the number of keys that can be used. Can somebody clear this up please?

EDIT: to clarify, I know that the wiring settings remain constant throughout the encryption procedure, but there are still multiple ways to set it up initially. At this point I'm thinking that each wheel has a specific wiring that is common to all Enigma machines. That would make the most sense to me.

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  • $\begingroup$ The description of your problem is unclear as you are assuming that everyone is familiar with the wheel settings. $\endgroup$ – Carlos Mendoza Dec 15 '15 at 20:35
  • $\begingroup$ I believe the so-called "plugboard settings", the re-wiring you speak of, was also a variable. By the way, only the earliest versions had three rotors and the order of these was also a variable (hence an additional factor of $6$). Later versions had as many as six rotors. Still the encoder only had access to the starting rotor settings and presumably these permutations are what your reference was speaking of. Not $100\%$ sure of my history here but reasonably confident. $\endgroup$ – lulu Dec 15 '15 at 20:56
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The wheels were initially wired in a secret way by the German army, compared to the commercially available Enigma machine (which anyone could buy, so these wirings were never assumed to be secret; the Italian and Spanish governments used these commercial enigma's for their communication (so they posed a much easier target). The Polish mathematicians in the Biuro Szyfrów devised a method to determine those wirings, and communicated them to the French and British allied forces. So the wirings (and the method to determine them, which is as important, if not more) were known to the British in their code breaking efforts.

The way the machines were set up was that for a certain day, 3 out of 5 wheels (initially, IIRC, later more wheels become available, and even later the German Navy went to 4 wheel enigma's) were fixed and their order as well (so $5\times4\times3$ ways to set up the wheels). This was fixed for the day. Then an initial setting was fixed for the wheels ($26^3$ ways) and also there were ring settings that determined the stepping of the wheels. Also there was a "Steckerbrett" which added an extra permutation by wiring 10 up to 13 many pairs to each other (adding extra letter interchanges). So there were a lot of settings, and later in the war that only become more. See wikipedia and many books for more info. Later more wheels with secret wiring became available, but these were reconstructed (and captured!), so they weren't secret for too long.

I think that in analysing the machine we should assume the wheels to be known (but the settings, like what wheels, stepping and initial setting plus Steckers, are assumed to be secret). The commercial one had fewer wheels, all of which known from the start, and no Steckers. So it was a much weaker system (which the Germans recognised, and so they considerably strengthened it.) The wheels could not be rewired, so we must really consider it part of the logic, not part of the keying/setup.

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  • $\begingroup$ Sorry for the necro-post. So what is the algebraic basis, if any, for breaking the code ? $\endgroup$ – gary Jul 22 '17 at 23:58
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    $\begingroup$ @gary None, really. There was no "algebraic" attack that was used by the British. The Poles used some algebra of permutations to reconstruct the wheels, though. Look in issues of "Cryptologia" for stuff like this, there are lots of modern papers on breaking Enigma in there. $\endgroup$ – Henno Brandsma Jul 23 '17 at 5:05

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