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In this article about decryption of Enigma(p.259), author says that

the second set of rotors was stepped three positions beyond the groundsetting

in the Cyclometer Rejewski devised. I would like to know why did he advanced 3 positions for the right rotor in the second set of rotors instead of 1 or 2 positions for the same.

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To understand why it was done so, it's necessary to understand what the cyclometer was. And to understand that, we need to get through a bit of encryption procedures for Enigma, as well as steps used to break that encryption.

The procedure for encrypting with Enigma was (simplified):

  • Set the machine to specific daily setting
  • Generate a random, 3 letter, key (say RFV)
  • Encrypt this key using the initial settings, twice
  • Set the machine to generated key and encrypt the rest
  • Prepend the encrypted key to the message

Because of this, the 1st letter of ciphertext and the 4th one were in fact the same cleartext letter, but encrypted with different settings (as the machine rotated rotors each letter). So were 2nd and 5th as well as 3rd and 6th.

The Rejewski's method worked by gathering a lot of messages from radio interception, and generating permutations from those. E.g. say you intercept 3 messages, with initial 6 letters of:

  1. pkpjxi
  2. jxixtm
  3. xtmrcy

Than you have partial permutations of

  1. p -> j -> x -> r -> ... (by taking 1st and 4th letter)
  2. k -> x -> t -> c -> ... (by taking 2nd and 5th letter)
  3. p -> i -> m -> y -> ... (by taking 3rd and 6th letter)

Those permutations were unique (or at least almost unique) for each possible initial setting of the machine. So, by recovering the permutations from the intercepted messages and looking them up in catalogue, one could recover (with some more tricks that are out of scope of this question) recover the daily keys, and thus all messages.

Now, after you recover the permutations from daily messages, you need the catalogue to look them up. This is where the cyclometer comes up. It wasn't a decryption tool, it was a tool to facilitate creating the catalogue. As you see, the permutations alternate between the same letter encrypted at first and fourth position (let's forget the other sets for brevity). One could encrypt with all possible settings and get the permutations from that, but cyclometer simplified that - it was two rotor sets, interconnected in a way that whatever you pressed was encrypted via first set, fed to the second set for encryption, than back to the first set and so on, until there was a cycle. As the Enigma advances the rotor after encrypting each letter, and the second rotor set in cyclometer mimics machine operation at 4th letter, the rotors in the second set had to be advanced by 3 characters.

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  • $\begingroup$ Could you tell what did you mean by "the second set was to mimic the machine settings after encrypting more letters". $\endgroup$ – justin Nov 13 '17 at 13:52
  • $\begingroup$ @justin edited slightly, is it clear now? $\endgroup$ – Torinthiel Nov 13 '17 at 18:02
  • $\begingroup$ By this phrase "this is what the Enigma do for that letter" did you mean to tell about the double encryption of the random key selected by the operator? $\endgroup$ – justin Nov 14 '17 at 0:36
  • $\begingroup$ The first set set of rotors mimics Enigma's state at start of message. The second one mimics state after encrypting 3 letters. And because the letter encrypted here is the same, it makes sense to calculate cycles for those letters. $\endgroup$ – Torinthiel Nov 14 '17 at 6:25

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