Prof

by Dermot Turing

  In the 1740s, the Reverend Thomas Bayes had put forward a theorem about the probability of causes. The man of God was not now fashionable, and it was not because he had been dead for 180 years. It was because any mathematician who followed him was at risk of being branded a heretic. By the 1940s, Bayes’s theorem was considered outrageous: the only orthodox way to measure probabilities was to take samples and observe their distribution, in the manner leading to the good old Gaussian Error Function. The inquisition into Bayes’s heresy was led by none other than Professor R.A. Fisher, the Cambridge authority on statistics, he who had said that the subject Alan Turing had chosen for his fellowship dissertation was ‘positively repellent’. As befits an inquisitor, Fisher was powerful, irascible and opinionated, and he believed evangelically in Gauss. The reputation of any academic who might stray from the Gaussian Path was at risk.

  So the name of the man of God could not be uttered at Bletchley Park. Mr Bayes’s theorem was way off the Gaussian Path: applying Bayes’s theory, sampling was not necessary to evaluate probabilities. At Bletchley in 1941 the idea that you might take samples and see what happened was laughable: sampling was never going to work with 159 million million million possible Enigma set-ups. Instead, if the testing of wheel combinations on the Bombe could focus on the combinations most likely to be right, it would save hours of time. If only the codebreakers were allowed to guess – ideally making an informed guess – and measure the reliability of the guess. The theorem of the heretical Mr Bayes allowed you to guess.

  The starting point for guesswork was the codebreaker’s oldest weapon. Frequency analysis has been used by codebreakers for centuries to unravel substitution ciphers, by relying on the unequal reliance of language on particular letters: the same reason that Q in English-language Scrabble is worth ten points but S is only worth one. Frequency analysis ought not to work for Enigma messages, because the scrambling changes with every press of a key on the keyboard – but what if several messages had been enciphered on machines set up the same way? Although the scrambling would change with every new letter of the message, the changes would be the same. If two pieces of plain German were each enciphered on Enigma machines set up in the same way, the intercepts would be more similar to each other than a wholly random allocation of letters. If the same letter appears in the same place in the plain-language texts, the same (enciphered) letter will appear in both cipher-texts if the machines are set up identically. So two pieces of cipher-text lined up one underneath the other will have more coincidences than you would otherwise predict. It’s the difference between a 1:26 chance that any letter would match and about 1:17; not much, but a start.

  Alan Turing’s statistical attack on naval Enigma exploited this difference. Messages were punched as holes into long sheets of paper pre-printed with the alphabet, and pairs of these punched sheets were superimposed and moved sideways from left to right to see where the largest number of overlapping holes appeared. The more overlapping holes, the higher the likelihood that there was a situation of 1:17 correspondence rather than totally random 1:26. This enabled – with a degree of uncertainty – some of the eight possible wheels to be excluded from testing on the Bombe. The process was called ‘Banburismus’, named for the town in Oxfordshire with the cross and the lady on a white horse, where the mundane process of printing long sheets with rows and rows of alphabets was done. In another nod to Banbury, measures of probability were called ‘bans’ or ‘decibans’, using a logarithmic scale to tot up the chances. The process of adjusting guesses based on small observations was distressingly Bayesian. The Cambridge mathematician I.J. Good joined Bletchley Park in April 1940 and was assigned to Hut 8. He is reported to have asked Prof, ‘Aren’t you essentially using Bayes’s theorem?’ and Alan confessed that it was so. Fortunately for Alan, the work at Bletchley Park was secret, so Fisher would not find out about the wartime witchcraft of Banburismus. After the war, Good became a professional defender of Bayesian methodology, writing several papers on the subject; he had to do so from the safety of the United States.

  Banburismus. An original ‘Banbury sheet’ used for early attacks on naval Enigma, found during the restoration of Bletchley Park huts.

  Good also described working with Alan Turing at this time:

  When he attacked a problem he liked to start from first principles, and he was hardly influenced by received opinion. This attitude gave depth and originality to his thinking, and also it helped him to choose important problems. In discussions he was excitable, and his voice would rise to a high pitch, although he was not in the least quarrelsome. Between sentences he had a habit of saying ‘Ah-ah-ah-ah-ah …,’ which made it difficult to interrupt his line of thought, or even to have a line of thought of one’s own!

  Ruthless behaviour

  But still the sinkings continued; even Bayes wasn’t enough to defeat naval Enigma. The Cryptographic History of Work on the German Naval Enigma picks up the story:

  Turing was now faced with the following dilemma. There were only two ways of getting into a key (1) Cribbing (2) Banburismus. Cribbing required some detailed knowledge of the traffic since otherwise one could not predict what a message would say; it therefore seemed necessary to break a few days on Banburismus first. Banburismus needed no knowledge of the content of the traffic but needed at least one known bigram table; it therefore seemed necessary to break a few days on Cribbing first. A further difficulty was that the bombes – essential to complete the break on modern keys – did not start to arrive until the Summer of 1940 and the German Air and Army section working on Enigma (Hut 6) also needed these machines. Thus the testing of even one crib, supposing this to be available, presented a considerable problem. The only really satisfactory solution to the problem was (1) a pinch either of the key sheets for a month or (rather less valuable) of the set of bigram tables, combined with (2) maximum bombe production to enable such a pinch to be exploited. Failing a pinch or a really large number of bombes there was little hope of any progress on up to date material.

  The attempt by the Allies to forestall the German invasion of Norway in April 1940 had not been a military success. Nevertheless, a German patrol boat VP2623 had been captured at Narvik on 26 April with its codebooks and some transmitted messages (valuable as cribs). The captured documents confirmed Alan’s theory about the bigram processes and allowed some messages to be broken. More captures were needed if the breaks were to continue. Fortunately, British Naval Intelligence included an officer of some ingenuity, namely Commander Ian Fleming, who wanted truth to be stranger than his later fiction. He proposed an operation called Ruthless.

  I suggest we obtain the loot by the following means:

  1. Obtain from Air Ministry an air-worthy German bomber.

  2. Pick a tough crew of five, including a pilot, W/T operator and word-perfect German speaker. Dress them in German Air Force uniform, add blood and bandages to suit.

  3. Crash plane in the Channel after making S.O.S. to rescue service in P/L.1

  4. Once aboard rescue boat, shoot German crew, dump overboard, bring rescue boat back to English port.

  F. 12.9.40

  Truth turned out more sober than fiction after all. Although all was prepared, there were no suitable German vessels, and Ruthless was called off. Frank Birch, the head of the German Naval Section, reported to the Admiralty:

  Turing and Twinn2 came to me like undertakers cheated of a nice corpse two days ago, all in a stew about the cancellation of Operation Ruthless. The burden of their song was the importance of a pinch. Did the authorities realise that, since the Germans did the dirt on their machine on June 1st, there was very little hope, if any, of their deciphering current, or even approximately current, Enigma for months and months and months – if ever? Contrariwise, if they got a pinch – even enough to give a clue to one day’s material, they could be pretty sure, after an initial delay, of keeping going from day to day from then on.

  There were other ways to get the material, which were less
cinematic but more secure, and the Navy were keen to oblige. A raid on German-occupied Norway in March 1941 secured some useful material from a trawler called Krebs; in May 1941 a weather ship called München was targeted and her code books and key sheets secured. The Lauenberg, another weather ship, was grabbed in June, just after Germany tore up the Molotov-Ribbentrop pact and invaded Russia. The most extraordinary capture of all happened on 9 May 1941, when U-110, commanded by Kapitänleutnant Lemp, was attacking convoy OB318. Lemp was no novice: he was decorated with the Knight’s Cross and he had sunk 96,314 tons of shipping – including one of the first kills of the war, the passenger liner Athenia sunk on 3 September 1939. But today was not his day. U-110’s periscope was seen by HMS Aubretia which fired a pattern of depth-charges. Then HMS Bulldog and Broadway joined in the hunt. Bulldog set a course to ram, and Lemp, knowing that his boat was a goner, gave the order to abandon ship. But the uncooperative U-boat was a large Type IX-B which refused to sink promptly, and Lemp, realising that the British might capture his books, frantically struck out into the icy water to regain his boat and carry out his duty. He never made it, and was never seen again. A boarding party from Bulldog went over in a whaler and grabbed everything they could, while the men detailed to rescue survivors from U-110 bundled them below to conceal what was going on. The haul included an Enigma machine with its rotors as well as the secret books. The men from Bulldog swore an oath of secrecy, saw to it that U-110 sank properly, and whisked their finds across to Naval Intelligence as soon as they reached land.

  Square root of minus one

  Meanwhile, Hut 8 had done the unthinkable: it had appointed a woman mathematician to join its officer-class codebreakers. The woman was Joan Clarke. Gordon Welchman had supervised Joan when she was studying mathematics at Cambridge, and he recruited her in the spring of 1940. She was allocated to Alan Turing’s team in Hut 8, and her job was to test the results of the Bombe which had been analysing the haul of material from VP2623. She rapidly ‘rose from the ranks of the girls in the big room; but this was obviously because of my degree, and before I had had any chance of proving myself’. Joan was being modest. Her intellect caught Alan Turing’s attention; Commander Travis – Denniston’s first lieutenant – took steps to have her pulled out of the clerical grades to which women were appointed. Nevertheless, the ingrained resistance to women’s achievements remained: pseudo-Germanic terms were usually given to new codebreaking techniques (Turingismus, Yoxallismus and the like, not to mention Banburismus), but when Joan discovered a way to speed up the recovery of Enigma settings she was told it was ‘pure Dillyismus’. For the first time in his life, Alan Turing found in Joan a woman he could talk to, on the same level, despite her sex. They tended to do the late shift together; and after some months of chess-playing and general socialising, Alan Turing turned the unthinkable into the unimaginable.

  I suppose the fact that I was a woman made me different. We did do some things together, perhaps went to the cinema and so on, but certainly it was a surprise to me when he said – I think his words probably were ‘Would you consider marrying me?’ But although it was a surprise, I really didn’t hesitate in saying yes. And then he knelt by my chair and kissed me. We didn’t have very much physical contact. Now the next day, I suppose we went for a bit of a walk together; after lunch, he told me that he had this homosexual tendency, and naturally that worried me a bit, because I did know that that was something that was almost certainly permanent. But we carried on.

  Carrying on involved a round of parental visits. Joan’s father was a clergyman worried about his daughter’s financial security, and Alan’s parents were just sniffy. So was John:

  My parents were pretty well accustomed to my landing them with the young, attractive and lively young women with whom I fell in love at intervals of about six months at a time; they used to come for week-ends and cheered up my father immensely. But Alan’s fiancée was tough going. We parents and elder brother worked like beavers all the week-end on this unpromising female and were exhausted by the exercise (as, no doubt, was she). I have an improbable memory of Alan and his affianced dutifully holding hands in a sandpit, both of them obviously wishing that they could get on with some untried theorem – and that not of the type which would have appealed to me.

  Joan Clarke, who was briefly engaged to Alan Turing, relaxing with other colleagues at Bletchley Park.

  Alan Turing’s reconstructed office in Hut 8. It is said that he chained his tea-mug to the radiator to prevent it being stolen.

  It was more complicated, and more sobering, than John knew. Going through the ritual of parental introductions and scrutiny must have been agonising for Alan. At this stage no one in the Turing household had any ideas about Alan’s sexuality. John himself had been married since 1934 and had two small daughters, living just around the corner from the senior Turings in Guildford, so at this time the succession of girlfriends to cheer up father was a memory, not the present reality. And life in the senior Turing household was not as calm as it might be. Ethel decided at around this time that she would be known by her second name. Julius moved out to live in the Cromwell Hotel in South Kensington, leaving Sara (as she now was) in Guildford, supervising the grandchildren. It wouldn’t have been a fun family visit even in the best of circumstances. After a while reality sank in. Alan and Joan knew it wouldn’t work out and the engagement was off. It had been an imaginary number.

  Golden eggs

  With the aid of the Navy’s ‘pinches’ and a good number of Bombes now in use, decryption of naval Enigma was happening without delay by the summer of 1941. ‘A flood of decrypted and translated signals concerning the operational U-boats began to pour into the Operational Intelligence Centre’ at the Admiralty. During that summer, Alan Turing and Gordon Welchman were summoned to see Sir Stewart Menzies, now head of the Secret Service, who presented each of them with a cheque for £200 for their achievements with Enigma. Better still, Winston Churchill, who loved his secret intelligence, came to Bletchley in person on 6 September to meet the front-line staff, say thank you, and boost morale. Churchill allegedly told the codebreakers that they were the geese that laid the golden eggs. The great man was eager to know how it was done, and Gordon Welchman was told to prepare a short speech.

  When the party turned up, a bit behind schedule, Travis whispered, somewhat loudly, ‘Five minutes, Welchman.’ I started with my prepared opening gambit, which was ‘I would like to make three points,’ and proceeded to make the first two. Travis then said, ‘That’s enough, Welchman,’ whereupon Winston, who was enjoying himself, gave me a grand school-boy wink and said, ‘I think there was a third point, Welchman.’

  It seems credible that Commander Travis wanted to shut him up, for Gordon Welchman was getting frustrated with the bureaucracy. One problem with working for a secret organisation is that nobody can explain why resources are needed, because the persons in charge of resource allocation do not need-to-know. Inspired by Churchill’s pep-talk, he decided to act. Or rather, to write a letter to Churchill, get his colleagues to sign it, and ask Stuart Milner-Barry – Welchman’s deputy in Hut 6 and chess-player extraordinary – to deliver it. Alan Turing was game, the letter was written, and Milner-Barry took the train to London and an expensive taxi to Downing Street. In his hurry and anxiety Milner-Barry left his identity card behind, but the real effrontery was not him brazenly walking into Number Ten (indeed he subsequently became a very senior official in the Civil Service, so evidently had the right skills); it was the breach of etiquette by the signatories in going over the heads of Alastair Denniston and his deputy, Commander E.W. Travis.

  Hut 6 and Hut 8, (Bletchley Park)

  21st October 1941

  Secret and Confidential

  Prime Minister only

  Dear Prime Minister,

  Some weeks ago you paid us the honour of a visit, and we believe that you regard our work as important. You will have seen that, thanks largely to the energy and foresight of Commander Travis
, we have been well supplied with the ‘bombes’ for the breaking of the German Enigma codes. We think, however, that you ought to know that this work is being held up, and in some cases is not being done at all, principally because we cannot get sufficient staff to deal with it. Our reason for writing to you direct is that for months we have done everything that we possibly can through the normal channels, and that we despair of any early improvement without your intervention. […]

  We have written this letter entirely on our own initiative. We do not know who or what is responsible for our difficulties, and most emphatically we do not want to be taken as criticizing Commander Travis who has all along done his utmost to help us in every possible way. But if we are to do our job as well as it could and should be done it is absolutely vital that our wants, small as they are, should be promptly attended to. We have felt that we should be failing in our duty if we did not draw your attention to the facts and to the effects which they are having and must continue to have on our work, unless immediate action is taken.