How did the cryptographers of Bletchley Park figured out the chi stream of the Lorenz cipher?

How did the Bletchy Park code breakers figured out the chi stream of the Lorenz cipher, that was obscured in the cipher text, which Britain code breakers eventually decoded. It's written in The Rutherford Journal that John Tiltman deduced the key using a depth found on 30 August 1941. Since he couldn't figure how the key was generated, it was given to W.T. Tutte , who figured out how the key was generated, without ever looking at the Lorenz cipher machine.

Jack Copeland writes in Colossus: The Secrets of Bletchley Park's Codebreaking Computers(p.364)

Here was a method of wheel setting! $\Delta(\chi_1+\chi_2)$ was a periodic sequence, supposed known, with a period of $41\times31=1271$. Lay it against $\Delta(C_1+C_2)$ in each of the 1271 possible relative positions and count the number of agreements for each one.

It's obvious that Britain code breakers have in hand the ciphertexts, from which they can obtain the delta-ed version of the sum of the first and second impulse of a intercepted cipher text i.e $\Delta(C_1+C_2)$. I would like to know how the code breakers figured out the delta-ed version of the chi stream of the Lorenz cipher machine i.e $\Delta(\chi_1+\chi_2)$.

• have you read en.wikipedia.org/wiki/Cryptanalysis_of_the_Lorenz_cipher? – indiscreteLogarithm Dec 27 '17 at 21:34
• Could you tell how codebreakers figured out the start positions of the chi wheel mentioned in the Cryptanalysis of the Lorenz machine. – justin Dec 28 '17 at 1:14
• The first paragraph you link to clearly says: " Initially, this was achieved by two messages having been sent in depth.". This means that they intercepted messages encrypted with the same settings. This is basically key re-use and leaks a lot of information. – Marc Dec 28 '17 at 17:00
• @Marc: To be honest I couldn't get how code breakers figured out the start positions of the chi wheels even though they found a depth. I know that Tutte arranged the cipher text in rows of 41 after many attempts and came to see repetitions. Could you tell how Tutte figured out the start positions of the chi wheels. – justin Dec 28 '17 at 20:24