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Turing's Treatise on the Enigma (the Prof's Book):

a preface by Andrew Hodges (1998, revised 1999)

For a guide to this website go to the Alan Turing Home Page

The following preface appeared in the volume Mathematical Logic  of the Collected Works of A. M. Turing,  ed. R. O. Gandy and C. E. M. Yates (2001). See the Turing Bibliography page for details of the Collected Works.

Source of the Document

In April 1996 a mass of hitherto secret material was released by the National Security Agency of the United States Government. Amongst them, with reference number NR 964, Box 201, RG 457 was the item described as 'Turing's Treatise on the Enigma.' This became available from the National Archives and Records Administration, Washington DC.

The document as released is a photographic copy of some 140 pages of typescript (some pages being missing or misnumbered). It is not signed or dated but is unquestionably in Turing's own handwriting and (poor) typewriting, with drawings also in his own hand. It is safe to identify it with what was known as 'Prof's Book' at Bletchley Park both according to Joan Murray [12], and also according to a 1945 report, The History of Hut Eight, by A. P. Mahon [10], which was released at the same time.

The unhelpful NSA label is 'ca.1939-1942,' but the date of composition is probably autumn 1940. Joan Murray's article describes it as composed after she joined in June 1940. Mahon's report dates it as written in 1940. (Probably it was not transmitted to the United States until 1943, since another newly released report [1] indicates that the U.S. Army analysts did not know of Turing's early role in Bombe design until that year. Possibly however U.S. Navy analysts knew of it through their work at Bletchley Park in 1942.)

The Background

Variants of the Enigma cipher machine had been used by the German armed services since the 1920s. From September 1938 Turing had given part-time assistance to the British cryptanalytic organisation, the Government Code and Cypher School. Their work was transformed by the transfer of information from Polish mathematical cryptanalysts in July 1939. From 4 September 1939, Turing worked full-time at the GC&CS war-time headquarters, Bletchley Park.

During 1939-40 Turing was foremost in developing logical, statistical and mechanical methods which allowed rapid decryption of some Enigma cipher traffic in 1940. The vital naval ciphers resisted decryption, mainly through an extra complexity (a bigram substitution) in the key-system. Turing took charge of a section (Hut Eight) devoted to this problem. Successful day-to-day decryption (of the main naval cipher) was only achieved in mid-1941. Turing's methods came to be incorporated in an efficient British and American organisations which in the later stages of the war were able to monitor much of the German communications system with momentous consequences [6,7]. But this text only describes the analysis of the Enigma ciphers in 1939-40.

Content of the document

The text appears to have been written as a guide for those starting Enigma work. But as so often, Turing's style is hard to categorise. It is written as narrative history as well as a practical handbook; yet as history it is unsystematic, only sometimes stating the source of ideas. One must recall the total secrecy of the work, and that Turing was typing during the battle of Britain when questions of past credits and future historians must have seemed secondary. The text has the appearance of having been composed while Turing typed it, and never revised.

Pages 1-95 describe hand methods of analysis, including the solving of the wheel wirings. Turing's text does not relate the methods described to methods used by the Polish analysts [9]; it would appear that Turing is here describing methods that he himself (or in collaboration with the other British analysts) worked out in the period before the Polish revelations, but this is not specifically asserted.

Pages 96-123 describe the machine methods which triumphed, principally the Bombe as designed by himself, G. W. Welchman and the British Tabulating Machinery engineer H. Keen in 1939-40. The Bombe was central to the entire Enigma-breaking work conducted at Bletchley Park and (later in the war) in the United States. Again no comment is made relating it to Polish mechanical methods. Turing does however describe in detail the origin of the two original logical principles which gave the Bombe its power:

  1. the idea of using a 'diagonal board' to exploit the self-inverse nature of the plugboard complication, and
  2. 'simultaneous scanning': the following of all possible false implications of a false hypothesis about plugboard settings.

The ideas are independent, as is brought out in Good's discussion [5]. However in another account Good [3] presented Turing's idea for (2) as following 'shortly after' Welchman's idea for (1), and my own 1983 account [8] followed this order of presentation. Welchman's account of (1) in The Hut Six Story [13] is similar. Joan Murray's first-hand account [12] is more ambiguous as to the order of ideas, and so it is noteworthy that Turing's own explanation definitely introduces (2) as preceding (1).

Pages 129-142 describe Turing's early attack on naval Enigma ciphers, which Turing continued to head until late 1942. The problem revolved around the system of bigram substitutions, as described by Good [3,5]. In this section (and only in this section) Turing clearly states the legacy of the Polish analysts.

This final section is however curiously short and inconclusive. At this point I refer to The History of Hut Eight by Mahon [10], which covers the entire period until 1945, and adds a good deal on Turing's work in 1939-1940 for which the principal source must have been Turing himself. There are at least three topics in this Mahon report, conspicuous through absence or vagueness in Turing's own text.

(1) Like Good [3,5], Mahon stresses the centrality of Turing's 'Banburismus' method in the success of Hut Eight after spring 1941. This could produce a partial identification of the wheel order for the day's traffic, thus greatly reducing the time spent on Bombe search. Mahon gives a detailed example of its application, and states (on page 14) that Turing's report gives the theory of it. However Turing's text has only a parenthetic description of Banburismus (on page 134). Possibly, therefore, there is a continuation of Turing's account of naval Enigma methods, yet to be revealed in another version or copy of the report.

(2) Mahon also stresses the frustration of the 1940 period when the Banburismus method could not be brought into use for lack of sufficient captured material, and that the analysts pressed the navy to undertake captures. There is nothing of this is Turing's text.

(3) Mahon also reports on an internal argument persisting throughout 1940 between those guessing probable words in plaintext (in Hut Four), and Turing's group in Hut Eight. According to Mahon, Turing was a 'lamentable explainer' of their side of the argument. Turing's own report does not refer to any such difficulties.

Mahon also illustrates the context within which Turing operated, in particular the 1939-40 atmosphere of 'defeatism' regarding the prospects of Enigma decipherment. Another memoir writer [11] recalls that 'as late as the summer of 1940 I myself heard Commander Denniston, head of GC&CS, saying to the Head of Naval Section, " You know, the Germans don't mean you to read their stuff, and I don't expect you ever will." As regards the naval Enigma, Mahon quotes Turing as saying 'no-one else was doing anything about it and I could have it to myself.' Mahon says, of a moment 'at the end of 1939':

Turing had in fact solved the essential part of the indicator system and that same night he conceived the idea of Banburismus 'though I was not sure it would work in practice, and was not in fact sure until some days had actually broken.'

Turing's own report does describe his guessing of the basic bigram system (page 136), but without a mention of Banburismus, and with a lack of emphasis on the breakthrough that it represented. It is hard to remember on reading his text, that Turing's taking on and persisting with the Enigma variant of such significance in the battle of the Atlantic, and doing so alone against the tide of prevailing wisdom, was perhaps his greatest individual contribution to history. In this respect he was certainly 'a lamentable explainer.'

As regards Turing's mathematical work, the text as we have it presents none. The group-theoretic algebra is described for non-mathematicians; only elementary probability considerations are brought into play and detailed calculations are not given. The text does not mention the log-odds assessment of weight of evidence, as further developed in 1941 by Turing and Good [3,4,5].

The comments above should indicate the many difficulties of evaluating and interpreting Turing's text. Even Mahon, on the spot in 1945, reported finding it hard to collate an accurate history when few records had been kept. But the release of further related documents, and the detailed reconstruction of methods by a number of researchers (including the actual rebuilding of Bombe machinery at the Bletchley Park Museum) should make it possible one day to write a definitive assessment of Turing's role in the Second World War.


I am indebted to Tony Sale, curator of the Bletchley Park Museum, for his assessment of the document and his assistance in the preparation of this description of it. In particular he has made available to me his 1998 work (as yet unpublished) which reconstructs detail lost in the copying process. I am grateful also for comments by Ralph Erskine, Philip C. Marks and Frode Weierud, who at the time of writing (1999) are completing their own annotated edition of Turing's report.

I am also indebted to Lee Gladwin of the National Archives and Records Administration, for finding and copying the released documents. He has published his own survey of the context of the document [2].


[1] William F. Friedman, Report on E Operations of the GC & CS at Bletchley Park, 12 August 1943; released by the National Security Agency, Washington DC, 1997; reference NR 3620, Box 1126, National Records and Archives Adminstration.
[2] Lee A. Gladwin, Alan Turing, Enigma, and the Breaking of German Machine Ciphers in World War II, Quarterly of the National Archives and Records Administration, Washington DC, 29, 202-217 (1997)
[3] I. J. Good, Enigma and Fish, in [7]
[4] I. J. Good, Studies in the History of Probability and Statistics.
XXXVII. A. M. Turing's statistical work in World War II, Biometrika 66, 2, pp. 393-6 (1979), reprinted also in the Collected Works (see [5]).
[5] I. J. Good, Introductory Remarks for the article in Biometrika 66 (1979) in the volume Pure Mathematics (ed. J. R. Britton) of the Collected Works of A. M. Turing (North-Holland, 1992)
[6] F. H. Hinsley et al., British Intelligence in the Second World War, HMSO (1979, 1981, 1984, 1988). Appendix 30 to the final volume adds much to the treatment of the 1939-40 period in the first.
[7] F. H. Hinsley and Alan Stripp (eds.), Codebreakers (Oxford University Press, 1993)
[8] Andrew Hodges, Alan Turing; the Enigma (Burnett, London and Simon & Schuster, New York, 1983; new edition Vintage, London 1992)
[9] W. Kozaczuk (tr. C. Kasparek) Enigma.... (Arms and Armour Press, 1984)
[10] A. P. Mahon, The History of Hut Eight, a report of 1945 released by the National Security Agency, Washington DC, 1996; reference NR 4685, box 1424, RG 457, National Records and Archives Adminstration.
[11] Christopher Morris, Navy Ultra's Poor Relations, in [7]
[12] Joan Murray, Hut 8 and naval Enigma, Part I, in [7]
[13] G. W. Welchman, The Hut Six Story (Allen Lane, London; McGraw Hill, New York, 1982)

More of the Prof's Book on-line:

The Collected Works  edition then continues with the facsimile of four pages from Turing's report. Continue to an online version of this material. For a general introduction to Turing's work on the Enigma see this Alan Turing Internet Scrapbook Page.

The whole of the American text is available on the Turing Digital Archive copy.

A sequence of pages explaining the Enigma can be found on Tony Sale's Second World War Codes and Ciphers site. Tony Sale also offers an explanation of how Turing arrived at the idea of the Bombe, based on Turing's account in the 'Prof's Book.'

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