Editing Colossus computer (section)

==Decryption processes==
{| class="wikitable floatright"
|+ Notation{{sfn|Good|Michie|Timms|1945|loc = 1 Introduction: 12 Cryptographic Aspects, 12A The Problem, (a) Formulae and Notation, p. 16}}
|-
| <math>P</math> || plaintext
|-
| <math>K</math> || key – the sequence of characters used in binary [[XOR]] with <br /> the plaintext to give the ciphertext
|-
| <math>\chi</math> ||''chi'' component of key
|-
| <math>\psi</math> ||''psi'' component of key
|-
| <math>\psi'</math> ||extended ''psi'' – the actual sequence of characters added by <br /> the ''psi'' wheels, including those when they do not advance{{sfn|Good|Michie|Timms|1945|loc = 1 Introduction: 11 German Tunny, 11B The Tunny Cipher Machine, (e) Psi-key, p. 7}}
|-
| <math>Z</math> || ciphertext
|-
| <math>D</math> || de-''chi''—the ciphertext with the ''chi'' component of the key removed{{sfn|Good|Michie|Timms|1945|loc = 1 Introduction: 12 Cryptographic Aspects, 12A The Problem, (a) Formulae and Notation, p. 16}}
|-
| <math>\Delta</math> || any of the above XOR'ed with its successor character or bit{{sfn|Good|Michie|Timms|1945|loc = 1 Introduction: 11 German Tunny, 11C Wheel Patterns, (b) Differenced and Undifferenced Wheels, p. 11}}
|-
| <math>\oplus</math> || the XOR operation{{efn|The [[Truth function|Boolean or "truth" function]] [[Exclusive or|XOR]], also known as [[Exclusive disjunction|Exclusive disjunction and Exclusive or]], is the same as binary [[Modular arithmetic|modulo]] 2 addition and subtraction.}}{{sfn|Good|Michie|Timms|1945|loc = 1 Introduction: 11 German Tunny, 11B The Tunny Cipher Machine, (a) Addition, p. 5}}
|-
| <math>\bullet</math> || Bletchley Park shorthand for telegraphy code ''space'' (zero)
|-
| <math>\mathbf{x}</math> || Bletchley Park shorthand for telegraphy code ''mark'' (one)
|-
|}
By using differencing and knowing that the ''psi'' wheels did not advance with each character, Tutte worked out that trying just two differenced bits (impulses) of the ''chi''-stream against the differenced ciphertext would produce a statistic that was non-random. This became known as [[Cryptanalysis of the Lorenz cipher#The "1+2 break in"|Tutte's "1+2 break in"]].{{sfn|Budiansky|2006|pp=58–59}} It involved calculating the following Boolean function:
:::<math>\Delta Z_1 \oplus \Delta Z_2 \oplus \Delta\chi_1 \oplus \Delta\chi_2 = \bullet</math>
and counting the number of times it yielded "false" (zero). If this number exceeded a pre-defined threshold value known as the "set total", it was printed out. The cryptanalyst would examine the printout to determine which of the putative start positions was most likely to be the correct one for the ''chi''-1 and ''chi''-2 wheels.{{sfn|Carter|2008|pp=18–19}}

This technique would then be applied to other pairs of, or single, impulses to determine the likely start position of all five ''chi'' wheels. From this, the de-''chi'' (D) of a ciphertext could be obtained, from which the ''psi'' component could be removed by manual methods.{{sfn|Small|1944|p=65}} If the frequency distribution of characters in the de-''chi'' version of the ciphertext was within certain bounds, "wheel setting" of the ''chi'' wheels was considered to have been achieved,{{sfn|Small|1944|p=15}} and the message settings and de-''chi'' were passed to the "[[Testery]]". This was the section at Bletchley Park led by Major [[Ralph Tester]] where the bulk of the decrypting work was done by manual and linguistic methods.<ref>{{cite AV media |last=Roberts |first=Jerry |author-link=Jerry Roberts |title= Capt. Jerry Roberts: My Top Secret Codebreaking at Bletchley Park 1941 to 45: Lecture on 11 March 2009 |publisher=University College London |year=2009 |minutes=34 |url=https://www.youtube.com/watch?v=m1bJHG95Xn8 | archive-url=https://ghostarchive.org/varchive/youtube/20211124/m1bJHG95Xn8| archive-date=2021-11-24 | url-status=live|via=YouTube}}{{cbignore}}</ref>

Colossus could also derive the start position of the ''psi'' and motor wheels. The feasibility of utilizing this additional capability regularly was made possible in the last few months of the war when there were plenty of Colossi available and the number of Tunny messages had declined.{{sfn|Copeland|2006|p=77}}