Editing Lorenz cipher (section)

==Operating Principle==
The logical functioning of the Tunny system was worked out well before the Bletchley Park cryptanalysts saw one of the machines—which only happened in 1945, as Germany was surrendering to the Allies.<ref name=Sale>{{ Citation | last = Sale | first = Tony | author-link = Anthony Sale | title = The Lorenz Cipher and how Bletchley Park broke it | url = http://www.codesandciphers.org.uk/lorenz/fish.htm | access-date = 21 October 2010 }}</ref>
[[Image:SZ42-6-wheels-lightened.jpg|right|400px|thumbnail|The Lorenz SZ machines had 12 wheels each with a different number of cams (or "pins").
{|class="wikitable" | border=1 
|-
! [[Cipher Department of the High Command of the Wehrmacht|OKW/''Chi'']]<br>wheel name
|A||B||C||D||E||F||G||H||I||K||L||M
|-
! BP wheel<br>name{{sfn|Good|Michie|Timms|1945|loc = 1 Introduction: 11 German Tunny, 11B The Tunny Cipher Machine, p. 6}}
| align="center" | ψ<sub>1</sub>
| align="center" | ψ<sub>2</sub>
| align="center" | ψ<sub>3</sub>
| align="center" | ψ<sub>4</sub>
| align="center" | ψ<sub>5</sub>
| align="center" | μ<sub>37</sub>
| align="center" | μ<sub>61</sub>
| align="center" | χ<sub>1</sub>
| align="center" | χ<sub>2</sub>
| align="center" | χ<sub>3</sub>
| align="center" | χ<sub>4</sub>
| align="center" | χ<sub>5</sub>
|-
! Number of<br>cams (pins)
|43||47||51||53||59||37||61||41||31||29||26||23
|}
]]

The SZ machine served as an in-line attachment to a standard Lorenz teleprinter. It had a metal base {{convert|19|×|15.5|in|abbr=on|cm|0}} and was {{convert|17|in|abbr=on|cm|0}} high.<ref name = "GRoT11B10"/>  The teleprinter characters consisted of five data [[bit]]s (or "impulses"), encoded in the [[Baudot code#ITA2|International Telegraphy Alphabet No. 2 (ITA2)]]. The SZ machine generated a stream of [[pseudorandom number generator|pseudorandom]] characters as the key that was combined with the plaintext input characters to form the ciphertext output characters. The combination was by means of the XOR (or modulo 2 addition) process.<ref name = "GRoT11B6">{{Harvnb|Good|Michie|Timms|1945|p=6}} of ''German Tunny''</ref>

The key stream consisted of two component parts that were XOR-ed together. These were generated by two sets of five wheels which rotated together. The Bletchley Park cryptanalyst Bill Tutte called these the ''χ'' ("[[Chi (letter)|''chi'']]") wheels, and the ''ψ'' ("[[Psi (letter)|''psi'']]") wheels. Each wheel had a series of cams (or "pins") around their circumference. These cams could be set in a raised (active) or lowered (inactive) position. In the raised position they generated a '1' which reversed the value of a bit, in the lowered position they generated a '0' which left the bit unchanged.{{sfn|Churchhouse|2002|pp=156,157}} The number of cams on each wheel equalled the number of impulses needed to cause them to complete a full rotation. These numbers are all [[Coprime|co-prime]] with each other, giving the longest possible time before the pattern repeated. This is the product of the number of positions of the wheels. For the set of ''χ'' wheels it was 41 × 31 × 29 × 26 × 23 = 22,041,682 and for the ''ψ'' wheels it was 43 × 47 × 51 × 53 × 59 = 322,303,017. The number of different ways that all twelve wheels could be set was {{val|1.603|e=19}} i.e. 16 billion billion.

The set of five ''χ'' wheels all moved on one position after each character had been enciphered. The five ''ψ'' wheels, however, advanced intermittently. Their movement was controlled by the two ''μ'' ("[[Mu (letter)|''mu'']]") or "motor" wheels in series.<ref name = "GRoT11B7">{{Harvnb|Good|Michie|Timms|1945|p=7}} of ''German Tunny''</ref> The SZ40 ''μ''<sub>61</sub> motor wheel stepped every time but the ''μ''<sub>37</sub> motor wheel stepped only if the first motor wheel was a '1'. The ''ψ'' wheels then stepped only if the second motor wheel was a '1'.<ref>{{Citation |last= Roberts |first= Eric |author-link= Eric Roberts |title= The Lorenz Schluesselzusatz SZ40/42 |publisher= Stanford University |url= https://cs.stanford.edu/people/eroberts/courses/soco/projects/2008-09/colossus/lorenzmachine.html }}</ref> The SZ42A and SZ42B models added additional complexity to this mechanism, known at Bletchley Park as ''Limitations''. Two of the four different limitations involved characteristics of the plaintext and so were [[Autokey cipher|autoclaves]].<ref>{{Harvnb|Good|Michie|Timms|1945|p=8}} of ''German Tunny''</ref>

The key stream generated by the SZ machines thus had a ''χ'' component and a ''ψ'' component. Symbolically, the key that was combined with the plaintext for enciphering and with the ciphertext for deciphering, can be represented as follows.<ref name = "GRoT11B7" />

::::key = ''χ''-key ⊕ ''ψ''-key

However to indicate that the ''ψ'' component often did not change from character to character, the term ''extended psi'' was used, symbolised as: ''Ψ'''. So enciphering can be shown symbolically as:

::::plaintext ⊕ ''χ''-stream ⊕ ''ψ'''-stream = ciphertext

and deciphering as:

::::ciphertext ⊕ ''χ''-stream ⊕ ''ψ'''-stream = plaintext.