Alan Turing: The Enigma The Centenary Edition

by Andrew Hodges

  Both were introduced to the large ‘laboratory’ hut where the research projects were in progress, and found Alan at work there. If civilians from Cambridge were apt to find him unusually careless in appearance, his divergences from respectability were very much more noticeable at military Hanslope. With holes in his sports jacket, shiny grey flannel trousers held up with an ancient tie, and hair sticking out at the back, he became the cartoonist’s ‘boffin’ – an impression accentuated by his manner of practical work, in which he would grunt and swear as solder failed to stick, scratch his head and make a strange squelching noise as he thought to himself, and yelp when shocked by the current that he forgot to turn off before soldering the joints in his ‘bird’s nest’ – so they called it – of electronic valves.

  But Robin Gandy was struck in another way on about the first day that he set to work investigating the effectiveness of high-permeability cores in the transformers of the radio receivers. There were two engineers in his section, who started the tedious task of testing the things, when Alan pottered in, and decided that it should all be solved from theoretical principles – in this case, it being an electromagnetic problem, from Maxwell’s equations. These he wrote down at the top of his paper, just as though it were some contrived Tripos question instead of one from real life, and eventually performed a tour de force of partial differential equations to get an answer.

  Donald Bayley was impressed in a similar way by the speech encipherment project, which at Hanslope became known as the Delilah. Alan had offered a prize for the best name, and awarded it to Robin for his suggestion of Delilah, the biblical ‘deceiver of men’. It made full use of his experience in cryptanalysis, and as Alan would explain, was designed to meet the basic condition that even if the equipment were compromised, it would still provide complete security. Yet the system he had conceived on board the Empress of Scotland a year earlier was essentially very simple.10 It was a mathematician’s design, and one which had depended upon Alan asking ‘But why not?’

  What he had done was to consider the roomful of equipment which made up the X-system, and to ask what were the crucial features which made it into a secure speech cipher. The Vocoder was not essential, although it had been the starting point of the project. Nor was the business of quantising the output amplitudes into a number of discrete levels. By jettisoning these he reduced the number of ideas involved to two: the fact that it sampled the speech at a succession of moments in time, and the fact that it used modular addition, like a one-time pad.

  The Delilah was based on these two ideas from the beginning, while in the X-system they had arrived by a back-door route. The point about sampling was that it removed the redundancy of the continuous sound wave. Any sound signal could be represented by a curve such as:

  The point was that it would be unnecessary to transmit the whole curve. It would be sufficient to communicate the knowledge of certain points on the curve, provided that the recipient could thereupon perform the exercise of ‘joining the dots’ to reconsitute the curve. This could be done, at least in principle, provided that it was known how sharply the curve was allowed to wiggle in between the points. Since sharp wiggles would correspond to high frequencies, it followed that provided there was a limit on the frequencies contained in the signal, then a sequence of discrete points, or samples, of the curve, taken at regular intervals, would contain all the information of the signal. Since telephone channels did in any case cut off high frequencies, the restriction on allowed ‘wiggling’ of the curve was no real restriction at all, and in fact a rather small number of samples could be shown to suffice to convey the signal.

  The idea was well-known to communication engineers. In the X-system, it was the practice to sample each of the twelve 25 Hz channels fifty times a second. These figures were illustration of a general result, that it was necessary to sample at a rate of twice the maximum variation in frequency of the sound, or bandwidth. There was an exact mathematical result to this effect, proved as early as 1915 but which Shannon had re-stated11 and discussed with Alan at Bell Labs. If, for instance, the sound signal were restricted to frequencies less than 2000 Hz, then a sample taken 4000 times a second would be exactly enough to reconstitute the signal. There would be precisely one curve of the stated frequency restriction that passed through all the sampled points. Alan described and proved this result to Don Bayley as the ‘Bandwidth theorem’. His ‘Why not?’ had come in asking why this elegant fact could not be made the pivot on which to turn the whole encipherment process.

  The figure of 2000 Hz was in fact the one he intended to use, and his encipherment process would start with the speech signal being sampled 4000 times a second. The Delilah would then have to effect the addition of these sampled speech amplitudes to another stream of key amplitudes. The addition would be done in modular fashion, meaning that while speech sample amplitude of 0.256 units and key amplitude 0.567 units would be added to give 0.823 units, the addition of 0.768 and 0.845 would give 0.613, not 1.613. The result of all this would be a train of sharp ‘spikes’, of heights varying between zero and one unit*:

  The next problem was that of how to transmit the information of these ‘spike’ heights to the receiver. In contrast to the X-system, Alan planned no quantisation of amplitudes here. He wanted to transmit them as directly as possible. In principle the ‘spikes’ themselves could be transmitted, but being of such short duration, a few microseconds in fact, they would require a channel which could carry very high frequencies. No telephone circuit could do this. To use a telephone channel the information of the ‘spikes’ would have to be encoded into an audio-frequency signal. Alan’s proposal was to feed each ‘spike’ into a specially devised electronic circuit with an ‘orthogonal’ property. This meant that its response to a spike of unit amplitude would be a wave with unit height after one time interval, and with zero height at every other unit time interval:

  Assuming the circuit to be ‘linear’, meaning that the input of a ‘spike’ of say half a unit would produce exactly half of this response, the effect of feeding into it the train of ‘spikes’ would be to ‘join the dots’ in a very precise way. The information of each ‘spike’ would go exactly into the amplitude of the response of this circuit one unit of time later, and nowhere else.

  The transmission would then be straightforward, and could be achieved by perfectly standard means; the decipherment process did not require any further new ideas.* Apart from the question of supplying a key-system, this was all that was required for the Delilah to effect an ‘adding-on’ encipherment of speech, the analogy of what agents like Muggeridge, or the machines which produced the Fish signals, or the Rockex, were all alike doing for telegraph or teleprinter. If the key were truly ‘random’, or without any discernible pattern, such a speech cipher system would be as secure as the Vernam one-time cipher for telegraph tape, and on exactly the same argument. From the enemy’s point of view, if all keys were equally probable, then all messages would be equally likely. There would be nothing to go on.†

  The disadvantage of the simple Delilah system, as compared with the X-system, was that its output signal would be one of bandwidth 2000 Hz, rather than a stream of digits, and that it would have to be communicated perfectly, or all would be lost. In particular, any variation in time delay, or distortion in amplitudes, would ruin the decipherment process. Sender and receiver would have to keep in time to the microsecond for the same reason. This was why it could never be used for long-range short-wave transmissions. But it could be used for local short-wave, for VHF and for telephone communication. For tactical or domestic purposes, therefore, it held considerable potential.

  Don Bayley was very eager to work on the Delilah, but it was not at first allowed. He was assigned to other tasks, and it only gradually became possible for him to spend time on Alan’s project. It was several months before formal permission was granted for his participation, and even then it was only on the understanding that he would have to do other jobs from time to time
.

  Alan’s waiting for help coincided with the time when everyone was waiting for the rather more important question of the Second Front. And it was, after all, the enterprise for which all his efforts, fascinating and depressing alike, had helped to secure the conditions. But back at Bletchley Park there was quite a different reason for excitement in Newman’s section. They had shown that even in the time of intricate planning and coordination, there was room for initiative. In fact there had been a scramble at the last minute in the latest development of the treasure hunt. Again it was the fresh generation that had done it, disproving an assumption that something could not be done. It was something they could be proud to tell Alan Turing about.

  Using the new electronic Colossus, installed since December, Jack Good and Donald Michie had made the marvellous discovery that by making manual changes while it was in operation, they could do work that hitherto it had been assumed would have to be done by hand methods in the Testery. The discovery meant that in March 1944 an order had been placed with Dollis Hill for six more Colossi by 1 June. This demand could not possibly be met, but with desperate efforts one Mark II Colossus was finished on the night of 31 May, and others followed. The Mark II included technical improvements, was five times faster, and also incorporated 2400 valves. But the essential point was that it incorporated the means for performing automatically the manual changes that Jack Good and Donald Michie had made. The original Colossus, by recognising and counting, was able to produce the best match of a given piece of pattern with the text. The new Colossus, by automating the process of varying the piece of pattern, was able to work out which was the best one to try out. This meant that it performed simple acts of decision which went much further than the ‘yes or no’ of a Bombe. The result of one counting process would determine what the Colossus was to do next. The Bombe was merely supplied with a ‘menu’; the Colossus was provided with a set of instructions.

  This greatly extended the role of the machine in bringing Fish to a state of ‘cornucopian abundance’. As with the Bombe, it was not that the Colossus did everything. It was at the centre of an extremely sophisticated and complex theory, in which far from being ‘dull and elementary’, the mathematics involved was by now at the frontiers of research. There were in fact many ways in which the Colossus could be used, exploiting the flexibility offered by its variable instruction table. It took the analysts’ work into a quite new realm of enchantment. In one of the main uses, the human and the machine would work together:12

  …The analyst would sit at the typewriter output and call out instructions for a Wren to make changes in the programs. Some of the other uses were eventually reduced to decision trees and were handed over to the machine operators.

  These ‘decision trees’ were like the ‘trees’ of the mechanical chess-playing schemes. It meant that some of the work of the intelligent analyst had been replaced by the electronic hardware of the Colossi; some went into the devising of instructions for them; some into the ‘decision trees’ that could be left to uncomprehending ‘slaves’; and some retained for the human mind. When off duty, they had talked about the machines playing chess, taking intelligent decisions automatically. In their work, in this new extraordinary phase, the arbitrary dispensations of the German cryptographic system had brought something like this into being – and even more uncanny for those who did it, a sense of dialogue with the machine. The line between the ‘mechanical’ and the ‘intelligent’ was very, very slightly blurred. Whatever its application to the great surprise that awaited the Germans, they were having a wonderful time in seeing the history of the future.

  No one at Hanslope, seeing the strange civilian boffin cycling across with a handkerchief around his nose (it was his hay fever period) could have connected him with the success of the assault on Normandy. And by now his part in the necessary conditions for making it was something that lay in the past; the success he wanted was of something truly and more wholly his own. As ten years before, it was his privilege to continue in his own way, with the least waste of energy, the civilisation which demanded harsher sacrifice from others. And it was another kind of invasion that he had in mind, one not yet ready for announcement.

  The successful passage of 6 June 1944 roughly coincided with the point at which it became possible for Alan and Don Bayley to get down to work on constructing the Delilah equipment, clearing up the rather messy efforts that the Prof had made on his own. The main task was that of building the circuit to produce the highly accurate ‘orthogonal’ response. It was the design of this circuit that had absorbed most of Alan’s earlier thought and experiment. He had realised that it could be synthesised out of standard components. This was an entirely new idea to Don Bayley, as was the mathematics of Fourier theory* that had been used to attack it. It was a tough problem, which Alan said had involved spending a whole month in working out the roots of a seventh degree equation. Although he was an amateur and self-taught electronic engineer, he was able to tell his new assistant a good deal about the mathematics of circuit design, and for that matter was by now able to show most of them in the laboratory hut a thing or two about electronics. But it needed Don to bring his practical experience to bear on the problem, and to tame the straggly bird’s nest. He also kept beautifully neat notes of their experiments, and generally kept Alan in order.

  Alan came in most mornings on his bicycle – sometimes in pouring rain, of which he appeared oblivious. He had been offered the use of an official car to take him to and fro, but declined it, preferring the use of his own motive power. Once, very unusually, he was late in, and even more dishevelled than usual. As explanation he produced a grubby wad of £200 in notes, explaining that he had dug them up out of their hiding-place in a wood and that there were two silver bars still to be recovered.

  But late in the summer, as the bridgehead was finally established and the Allied armies swept their way across France, he gave up his lodgings with Mrs Ramshaw at the Crown Inn, and moved to the Officers’ Mess at Hanslope Park. At first he occupied a room on the top floor of the mansion (he had one to himself, enjoying a rather more privileged status than the junior officers), and then later moved into a cottage in the walled kitchen garden, which he shared with Robin Gandy and a large tabby cat. The cat was called Timothy, and had been brought to Hanslope by Robin on his return from staying with friends in London. Alan was well-disposed towards Timothy, even though (or perhaps because) it had a way of launching playful swats at the typewriter keys when he was at work.

  As a place to hibernate while they waited for the war to end, Hanslope enjoyed one special advantage. The mess officer was Bernard Walsh, the proprietor of Wheeler’s, the smart oyster restaurant in Soho. As if by magic, fresh eggs and partridges found their way on to the Hanslope dining table, while the rest of Britain chomped its way through Woolton pie and reconstituted egg. These might be supplemented by a rabbit from the copse, or a duck egg from the pond which lay at the bottom of the meadow surrounding the house, and Alan was also able to have the apple that as a rule he would always eat before going to bed. He would go out for walks or runs round the fields, and might be seen thoughtfully chewing on leaves of grass as he loped along, or perhaps burrowing around for mushrooms. During the year, a timely Penguin guide13 to edible and inedible fungi had appeared, of which he made use to bring back amazing specimens for Mrs Lee (who organised the day-to-day catering) to cook for him. He particularly relished the name of the most poisonous of all, the Death Cap or Amanita phalloides. He would roll the name off his tongue with glee, and they would all search for it, but they never found a specimen.

  Once of an evening he went out for a run and managed to break his ankle by slipping on a slime-covered brick which formed part of the garden path, just as he was getting into his stride. He had to be sent off in an ambulance to have it set in hospital. But at other times the Prof delighted everyone by winning the sports day race, and by defeating the young Alan Wesley (another of the March intake) who rashly challenged h
im to a circuit of the large field. He was accepted (with due respect to his differences) as one of a crowd of junior officers. At lunchtimes, they would gather in the mess, and look at the newspapers: the Daily Mirror first, for the comic strip Jane. Don Bayley, who was very keen on military matters, might tell them about the developing strategy of the eastward moving armies, while Alan might hold forth on some topic of a scientific or technical flavour, such as why water was opaque to electromagnetic radiation of the radar wavelength, or how a rocket could accelerate its own fuel quickly enough. They would sometimes go for lunchtime walks together, Timothy the cat accompanying them. Robin Gandy was learning Russian – not because of his earlier membership of the Communist party (which had lapsed in 1940), but because of his admiration for the Russian classics. Robin still considered himself a fellow-traveller, and in this respect 1941 had not changed Alan’s view that he was quite misguided. But there was little discussion of politics at Hanslope, where the prevailing attitude was that of doing the job without asking questions.

  Every month or so there was a Mess Night at which dress uniform, or in Alan’s case a dinner jacket, would be required, and at which pheasant might be on the menu. This he would enjoy, for although generally so austere he liked to live it up occasionally, dancing vigorously with the ladies of the ATS afterwards. There was plenty of social gossip and intrigue, which Alan rather liked hearing about, and discussing with Mrs Lee and Mary Wilson. Sometimes his own rather glamorous position as the mysterious Prof, coupled with his unthreatening friendly way with the women members of staff, aroused a mild jealousy. In this respect, he kept his own secret.

  It was the first time in his life that he had mixed with ordinary people for any length of time, people picked out neither by social class nor by a special kind of intellect. It was a typical Turing irony that this should happen at an establishment working for the secret service. He liked its unpretentiousness, and perhaps the escape from the intellectual pressure at Bletchley. He certainly had the pleasure of being a large fish in a small pond. This liking was reciprocated. There was an occasion when he was invited to a drinking-party organised by the Other Ranks. For some reason it did not come off, but he was still very pleased: partly at breaching social class barriers, but surely also because of the allure, felt almost inescapably by a homosexual of his background, of that vast unknown England of working-class men.