Alan Turing: The Enigma (44 page)

Nor was there anything heroic about the scheme that Alan devised in 1940 for protecting his savings against imminent disaster. David Champernowne had observed that silver was one thing that had gained in real value during the First World War. Both he and Alan invested accordingly in silver bullion, but while ‘Champ’ prudently kept his in the bank, Alan typically decided to go the whole hog with a Burying Procedure.

Apparently he imagined that by burying the silver ingots, he could recover them after an invasion had been repelled, or that at least he could evade a post-war capital levy. (In 1920, Churchill and the Labour party had both favoured such a policy.) It was an odd idea. It was logical enough to be pessimistic about the outcome of the war, but if there had been an invasion, then surely some transatlantic evacuation of code-breakers would have taken place (just as the Poles had escaped to France), in which case he would have been better off with his savings in a form more suitable for transport. He bought two bars, worth about £250, and wheeled them out in an old pram to some woods near Shenley. One was buried under the forest floor, the other under a bridge in the bed of a stream. He wrote out instructions for the recovery of the buried treasure and enciphered them. At one point the clues were stuck in an old benzedrine inhaler and left under another bridge. He liked talking about ingenious schemes for coping with the war, and once proposed to Peter Twinn an alternative plan of buying a suitcase full of razor blades. It suggested the curious, but not totally impossible, picture of Alan as a street-corner hawker in a reduced Britain.

In August or September 1940 Alan had a week’s holiday, and spent it with Bob, making an effort to give the boy a treat. He had arranged for them to stay at what was for Alan a smart hotel, a renovated castle near Pandy, in Wales. It had been the usual hell for Bob in his first term, but, like Alan, he had survived the year, and at least had not encountered the usual public school anti-semitism. Alan asked a little about the past, and his family, but it was impossible to chat, for Bob had cast out the past as best he could, and Alan had no ability to heal such wounds. In fact, he probably never knew of the scenes which had taken place in Manchester as Bob pleaded unsuccessfully with the H–––– family to rescue his mother from Vienna.

They went fishing and for long walks over the hills. After a day or two Alan made a gentle sexual approach, but Bob rebuffed it. Alan did not ask again. It did not affect the holiday. Bob perceived that the possibility had
been at the back of Alan’s mind from the beginning, but did not feel that Alan had taken advantage of him. He was simply not interested.

None of this was quite what Churchill had in mind when calling the British people to brace themselves to their duties, or speaking of the Empire that might last a thousand years. But duty and empires did not solve ciphers, and Churchill never bargained for an Alan Turing.

If the danger of direct invasion diminished, the attack on shipping was itself an invasion of the British metabolism. In the first year of war the sinkings by U-boats had not been the dominating problem. More significant were the disposition of the merchant fleets of newly occupied and neutral countries, the closure to trade of both the Channel and the Mediterranean, and the reduced capacity of British ports and inland transport to absorb whatever arrived.

From late 1940, however, the position began to clarify. The British-controlled merchant fleet had to supply an island separated by only twenty miles from an enemy continent, and to do so from bases thousands of miles across submarine-infested seas. Britain also had to continue the economic system on which vast populations around the globe depended, and to remain at war at all, had to attack Italy in a Middle East which was now as distant from Britain as was New Zealand. The lessons of 1917 had been applied, and a convoy system introduced since the outbreak of war, but the hard-pressed Navy could not escort convoys far into the Atlantic. And this time, Germany had achieved in a few weeks what four years of machine-guns and mustard-gas had sought to prevent. There were U-boat bases on the French Atlantic coast.

One factor alone weighed against the probability of German victory in the naval war. The U-boat force, so phenomenally successful in 1917, had not been built up in time for 1939. The bluffing over Danzig had meant that Hitler blundered into war while Dönitz commanded less than sixty submarines. Short-sighted strategy would keep the numbers at this level until late 1941. Although the sudden increase in U-boat successes after the collapse of France was alarming, it was not in itself a British disaster.

To remain capable of a belligerent policy Britain required imports of thirty million tons a year. A capital stock of thirteen million tons of shipping was available for this purpose. During the year after June 1940, U-boat sinkings were to deplete that stock by an average of 200,000 tons a month. This loss in capacity could just about be replaced. But anyone could see that a U-boat force just three times larger, enjoying a corresponding degree of success, would have a crippling effect both on the level of current supply, and on the total stock of shipping. Each U-boat was sinking more than twenty ships in its life-time, and there was no counter-strategy while the
U-boat remained invisible. It was the logical, rather than the physical, power of the U-boat that was its strength. It was the German failure to follow up this tremendous advantage against its only remaining enemy that allowed a period of reprieve in which to counter this logical power with new weapons of information and communication. Radio direction finding and radar had already joined sonar in taking the Admiralty a short way beyond the resources of Nelson. The work of Hut 8 was still far behind.

Alan had begun his investigation of the naval Enigma messages on his own, but then was joined (for a time) by Peter Twinn and Kendrick. Clerical work was done by women who would be called ‘big room girls’. Then in June 1940 there was a new mathematical recruit: Joan Clarke, who was one of several ‘men of the Professor type’ to be a woman. The principle of equal pay and rank being stoutly resisted by the civil service, she had to be promoted to the humble rank of ‘linguist’ that the pre-war establishment reserved for women, and there was talk by Travis of her being made a WRNS officer so that she could be better paid. But in the Hut itself, a more progressive Cambridge atmosphere prevailed. She had just been reading for Part III, and was recruited for Bletchley by Gordon Welchman, who had supervised her for projective geometry in Part II. Her brother being a Fellow of King’s, she had once met Alan in Cambridge.

So in the summer of 1940, Alan Turing found himself in the position of telling other people what to do, for the first time since school. It was like school inasmuch as the WRNS and the ‘big room girls’ played the role of ‘fags’, and because it meant meeting, or avoiding, members of the armed services. Alan’s methods for dealing with the clerical assistance, and other administrative problems, which gradually grew in scale, were like those of a shy school ‘brain’ who had been made a prefect by virtue of winning a scholarship. On the other hand, one notable difference from school was that it brought him for the first time into contact with women.

The rest of 1940 saw little progress with naval Enigma. The April U-boat capture, although largely wasted, had given them something to work on
¹⁹
– and it was for this reason that Joan Clarke had been directed to Hut 8.

It had enabled GC and CS to read during May 1940 the naval Enigma traffic for six days of the previous month, and thus to add considerably to its knowledge of the German Navy’s [radio] and cypher organisation. GC and CS was able to confirm that, though the Germans resorted to fairly simple hand codes and cyphers for such things as light-ships, dockyards and merchant shipping, their naval units, down to the smallest, relied entirely on the Enigma machine. More important still, it established that they used only two Enigma keys – the Home and the Foreign – and that U-boats and surface units shared the same keys, transferring to the Foreign key only for operations in distant waters.

But only a further five days’ traffic, for days in April and May, were broken
in the rest of 1940, and ‘the advance of knowledge had also confirmed GC and CS’s worst fears about the difficulty of breaking even the Home key, in which 95 per cent of German naval traffic was encyphered.’ Alan’s work showed that they could not hope for progress without further captures. But while they waited, he was not idle. He developed the mathematical theory that would be required to exploit them. There was far, far more to it than the building of Bombes.

Looking at the cipher traffic, an experienced hand might say that such and such a thing ‘seemed likely’, but now that mass production was the objective, it was necessary to make vague, intuitive judgments into something explicit and mechanical. Much of the mental apparatus required for this had already been constructed in the eighteenth century, although it was new to GC and CS. The English mathematician Thomas Bayes had seen how to formalise the concept of ‘inverse probability’ – this being the technical term for the likely cause of an effect, rather than the probable effect of a cause.

The basic idea was nothing but the common sense calculation of ‘likeliness’ of a cause, such as people would use all the time without thinking. The classical presentation of it was like this: suppose there to be two identical boxes, one containing two white balls and one black ball, the other containing one white ball and two black balls. Someone then has to guess which box is which, and is allowed to make an experiment, that of taking just one ball out of either box (without, of course, looking inside.) If it turns out to be white, the common sense judgment would be that it is
twice as likely
that it has come from the box containing two white balls, as from the other. Bayes’s theory gave an exact account of this idea.

One feature of such a theory was that it referred not to the happening of events, but to the changes in a state of mind. In fact, it was very important to bear in mind that experiments could only produce relative changes in ‘likeliness’, and never an absolute value. The conclusion drawn would always depend upon the
α priori
likeliness which the experimenter had had in mind at the beginning.

To give a concrete feel to the theory, Alan liked to think in terms of a perfectly rational person obliged to make bets upon hypotheses. He liked the idea of betting, and put the theory into the form of odds. So in the example, the effect of the experiment would be to double the odds, one way or the other. If further experiments were allowed, the odds would eventually increase to very large numbers although in principle, certainty would never be attained. Alternatively, the process could be thought of as one of accumulating more and more evidence. From this point of view, it would be more natural to think of
adding
something each time an experiment was made, rather than of
multiplying
the current odds. This could be achieved by using logarithms. The American philosopher C.S.
Peirce had described a related idea in 1878, giving it the name ‘weight of evidence’. The principle was that a scientific experiment would give a numerical ‘weight of evidence’ to be added to, or subtracted from, the likeliness of a hypothesis. In the example, the discovery of a white ball would add a weight of log 2 to the hypothesis that the box it came from was the one with two white balls. It was not a new idea, but
²⁰

Turing was the first to recognise one value of naming the units in terms of which weight of evidence is measured. When the base of logarithms was
e
he called the unit a natural ban, and simply a ban when the base was 10 Turing introduced the name deciban in the self-explanatory sense of one-tenth of a ban, by analogy with the decibel. The reason for the name ban was that tens of thousands of sheets were printed in the town of Banbury on which weights of evidence were entered in decibans for carrying out an important process called Banburismus.

So a ‘ban’ of evidence was something that would make a hypothesis ten times as likely as it had been before. Rather like a decibel, a deciban would be ‘about the smallest change in weight of evidence that is directly perceptible to human intuition’. He had mechanised guessing, and was ready to put it on machines which would add up decibans to arrive at a rational decision.

Alan developed the theory in several ways. The crucially important application lay in a new procedure for making experiments, later to be called ‘sequential analysis’. His idea was to set a target for the weight of evidence required one way or the other, and to continue making observations until that target was attained. This would be a far more efficient method than deciding in advance how many experiments to make.

But he also introduced the principle of judging the value of an experiment, by the amount of weight of evidence that it would, on average, produce; and he even went on to consider the ‘variance’ of the weight of evidence produced by an experiment, a measure of how erratic it was likely to be. In bringing these ideas together, he brought the art of guessing, as employed in cryptanalysis, into the 1940s. Typically, he had worked it out all for himself, either not knowing of earlier developments (as in the case of ‘weight of evidence’ defined by Peirce) or preferring his own theory to the statistical methods pioneered by R.A. Fisher in the 1930s.

Now, therefore, when they thought that a crib was ‘probably’ right, or that one message had ‘probably’ been transmitted twice, or that the same setting had ‘probably’ been used twice, or that one particular rotor was ‘probably’ the outermost one, there lay the possibility of adding up the weight of evidence from faint clues in a systematic, rational way, and of designing their procedures so as to make the most of what they had. To save an hour thereby was to gain an hour in which a U-boat gained six miles upon a convoy.