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Each Enigma machine setting induces a certain encryption in the sense of a function from the space of plain texts to the set of cipher texts.

The number of different Enigma machine settings can be counted for a given Enigma version (and usage). This is answered here: How many possible Enigma machine settings?.

But do different machine settings necessarily lead to different encryption functions? If not, what is the number of different encryption functions? Or, put differently, what ist the effective key length?

I think that this could be answered using group theory, but I still don't know how exactly this could be done.

Edit: I think I will represent the machine states as elements of a permutation group, provide these representations to a computer algebra system, let the system calculate the order of the group that is generated from all these elements and compare that group order to the total number of machine states. This is not a satisfactory answer because it probably won't give me much insight. It will just be a yes/no - answer.

maya
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    According to the text here the settings are all "mutually exclusive", i.e. represents a different keyed function. – Maarten Bodewes Oct 14 '23 at 18:40
  • "Mutually exclusive" seems to refer to the different plugboard combinations, as well as to a ring containing a single notch or two notches. I also tend to think that the key space is as big as the number of machine settings, but I can't find a rigorous argument... – maya Oct 15 '23 at 09:31
  • Possibly. I thought it would be a good starting point, but if it was well supported in the text then I would have posted it as an answer... – Maarten Bodewes Oct 15 '23 at 11:55
  • I found another text where o.a. the plug board settings were disregarded because they didn't play a major role when trying to analyze the ciphertext. However, that's not the same thing as having identical functions. In a mathematical sense they would count as different functions even if they simply performed a single additional addition of one, as long as the functions created by the keys cannot be rewritten to be identical. – Maarten Bodewes Oct 17 '23 at 23:13

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