# Confused about the number of permutations of the Enigma Machine

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.

• The description of your problem is unclear as you are assuming that everyone is familiar with the wheel settings. – Carlos Mendoza Dec 15 '15 at 20:35
• 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. – lulu Dec 15 '15 at 20:56

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.