Review of Lorenz and comments on the work of William Tutte

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Notes

¹Roberts says the ciphers were from the German military police. That is not correct, but the Ordnungspolizei, which was a paramilitary unit, and several other police agencies did use the double Playfair cipher. (See (Hinsley et al. Appendix 5 The German Police Cyphers, pp. 669–673.).

²Roberts began his study at University College London in 1939, but he did not study in London. His college of UCL was evacuated to Aberystwyth on Cardigan Bay in Wales because of the German threat to London. (p. 35).

³Research on the sources of Soviet intelligence for the Battle of Kursk and their impacts somewhat downplays the information that was obtained from Bletchley Park.

“[The] substantive intelligence on German capabilities and intentions at Kursk was painstakingly acquired by conventional Soviet intelligence on the scene and in the German rear, and especially by the intelligence staffs of the Red Army units in the affected areas. Employing the normal intelligence-gathering techniques of a modern army (i.e., aerial reconnaissance, physical observations, interrogations of prisoners, signal intelligence, and agent activity), the Soviets built up an accurate picture of the dispositions, strength, and intentions of their opponents.” (Mulligan 1987, pp. 253–254) (Cf. Hinsley et al. 1981, pp. 624–627.).

⁴William Thomas “Bill” Tutte was born in Newmarket, Suffolk, England on 14 May 1917. In 1935, he entered Trinity College, Cambridge from which he graduated with a degree in chemistry. After graduation, Tutte began to do research in chemistry, but his fascination with mathematics (in particular, with graph theory), which apparently began when as a schoolboy he read Rouse Ball’s Mathematical Recreations and Essays, drew him to mathematics and, indirectly, to Bletchley Park in 1941. After World War II, Tutte returned to Cambridge and studied for a doctorate in mathematics. His advisor was Shaun Wylie. Tutte combined algebra and graph theory in his 1948 thesis. After completing his thesis, Tutte was immediately invited by the geometer H. S. M. Coxeter to take a position at the University of Toronto. In 1962, Tutte moved from Toronto to the University of Waterloo, where he remained until his retirement in 1985. Tutte died in 2002. Tutte was a leading figure in graph theory and matroid theory. (See the MacTutor biography (William Thomas Tutte, 2003)). Two comments might be appropriate here: (1) When Alan Turing was a 17-year-old schoolboy, he received a copy of the 1928 edition of Ball’s Mathematical Recreations and Essays as a school prize, and the last chapter of Ball’s book was Turing’s first introduction to cryptology. (2) Tutte’s advisor Shaun Wylie (1913–2009) served at Bletchley Park from 1941 until 1945. In 1958, Wylie became chief mathematician at the Government Communications Headquarters (GCHQ), from which he retired in 1973.

⁵“Fish referred to encrypted German links from the teleprinter traffic. Bletchley Park gave each link in this network a different fish code name, such as Bream, Herring, Jellyfish and so on, and eventually there were up to twenty-six Fish links.” (p. 59).

⁶See Davies 1995.

⁷See F. H. Hinsley‘s comments (Hinsley and Stripp 1993, p. 141).

⁸Some authors use the term Geheimschreiber in the generic sense to refer to Fish. (See, for example, Hinsley et al. 1981.).

⁹See F. H. Hinsley’s comments (Hinsley and Stripp 1993, p. 142).

¹⁰See Beckman (2003), the essay by Frode Weierud (Joyner 2000, pp. 18–49), and the essay by Lars Ulfving and Frode Weierud (Joyner 2000, pp. 62–100).

¹¹Beurling (1905–1986) was a professor of mathematics at Uppsala University from 1937 until 1954. He then moved to the United States and was a member of the Institute for Advanced Studies at Princeton. Beurling broke Sturgeon during the German attack on Norway when Swedish intelligence was able to tap landlines that were transmitting Sturgeon-enciphered messages.

¹²More likely this would appear as 55M88. Cipher clerks often repeated the figure shift and letter shift commands to ensure that shifts would occur. If the shifts did not occur when required, gibberish would result.

¹³The ordering of the symbols is an example of a binary Gray code; when moving from one symbol to the next in the ordered list, only one impulse is changed. Gray codes were developed by Bell Labs researcher Frank Gray, who applied for a patent on the codes in 1947. The codes were originally designed to be used with electromechanical switches. Such switches did not change position in exact synchrony; therefore, an ordering that required only one switch at a time to change position was desirable. Today, Gray codes are used for error correction.

¹⁴Roberts, at several places in Lorenz, refers to this system as giving three levels of encipherment. It seems better to think of the three sets of wheels combining to produce one level of encipherment (the keystream), which is XORed with the message.

¹⁵The 12 wheels in Figure 2 are arranged in the order .

¹⁶The numbers of pins are relatively prime in pairs.

¹⁷Recall that 9 represents a space.

¹⁸Delta-ing is typically credited to Turing. Tutte (Copeland 2006, p. 363) seems uncertain as to who introduced the method. “I have been asked, ‘When did the study of Δχ and Δψ become usual at Bletchley?’ [In the discussion preceding the quote] I have been much concerned about whether two consecutive symbols in a stream T of dots and crosses were alike or different. Equivalently, but more simply, I could have asked whether ΔT had a dot or cross. It seems that the Δ operation could have been used with advantage in Turing’s method. I do not remember, but find it hard to doubt that it was. … So I was probably doing no new thing when, in our emergency [when the double delta method was developed], I meditated on the possible use of Δ-sequences. But it is from that time that they are prominent in my memory.”

¹⁹In mathematics, a capital delta Δ often denotes “difference.”

²⁰It was also “playfully” called Turingismus.

²¹The process is in some places called “integration.”

²²See (Reed, Diffie, and Fields 2015, pp. 298–299).

²³The new indicator system (the QEP system) was introduced on two Lorenz channels in October 1942.

²⁴This result is based on the facts that German frequently uses double letters and that cipher clerks tended to double the figure shift and letter shift.

²⁵The psi wheels do not move 50% of the time, and when the psi wheels do not move, is •. The psi wheels do move about 50% of the time, and 50% of the time when the psi wheels move, and will be the same and is •. 0.50 + 0.25 = 0.75; therefore, is • 75% of the time.

²⁶See Tutte (2000).

²⁷Thomas “Tommy” Flowers (1905–1998) graduated from the University of London with a degree in electrical engineering. Prior to his work for Bletchley Park, he explored the potential use of electronics in the telephone system at the General Post Office’s Dollis Hill Research Station.

²⁸Of course, the last string need not be 1,271 characters.

²⁹Frank Carter, in his description of rectangling (Copeland 2006, pp. 396–405; see pp. 399–400 for the quote), points out that “the designation just made is arbitrary, and the correct value may be the opposite of the one chosen. That is to say, the final and patterns obtained may have their crosses and dots interchanged. It was easy to discover if this had happened at a later stage of the wheel breaking process, and to make the necessary corrections.”

³⁰See Frank Carter’s “Rectangling,” which is pp. 396–405 of Copeland (2006) for details of rectangling or Tutte’s comments on pages 365–367 of the same reference.

³¹See “Of Men and Machines” by Randell, which is pages 141–149 of Copeland (2006) and also “Turing and the origins of digital computers,” by Randell, which is pages 67–75 of Copeland et al. (2017).

³²Flowers’ paper is introduced by former U.S. Navy codebreaker Howard Campaigne. Campaigne uses the term Geheimschreiber to refer to Lorenz.

³³Donald Michie (1923–2007) was a pioneer in the field of artificial intelligence. He was recruited for Bletchley Park in 1942. Apparently, Michie and Turing often discussed their ideas about machine intelligence while they were serving at Bletchley Park.

³⁴I. J. Good (1916–2009) was Alan Turing’s “statistical assistant” at Bletchley Park. In 1948, he was recruited by GCHQ, where he served until 1959. He moved to the United States in 1967 and was a research professor of statistics at Virginia Tech. Good was a leader in the field of Bayesian statistics.

³⁵MBE is Member of the Most Excellent Order of the British Empire.

³⁶Initially, the MBE recognized only Roberts’ work promoting Bletchley Park. Through his own efforts, Roberts was able to have the statement modified in such a way that his work and, therefore, the work of his codebreaking colleagues at Bletchley Park, was recognized.

³⁷http://billtuttememorial.org.uk/codebreaking/the-testery/.

³⁸In the United Kingdom, blue plaques are installed to commemorate locations associated with famous persons or events.

³⁹For more information about honors for Tutte, see Bill Tutte Memorial Fund.

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Chris Christensen

Chris Christensen teaches mathematics and cryptology at Northern Kentucky University. He feels very fortunate to have been able to visit Bletchley Park several times, see the Colossus rebuild, hear Tony Sale describe its use, and see Colossus determine the chi wheel settings for a message.