Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game

by Andrew Hodges

  Welchman not only saw the possibility of improvement, but quickly solved the problem of how to incorporate the further implications into a mechanical process. It required only a simple piece of electrical circuitry – soon to be called the ‘diagonal board’. The name referred to its array of 676 electrical terminals in a 26 × 26 square, each corresponding to an assertion such as (KG), and with wires attached diagonally in such a way that (KG) was permanently connected to (GK). The diagonal board could be attached to the Bombe in such a way that it had precisely the required effect. No switching operations were required for this; the following of implications could still be achieved by the virtually instantaneous flow of electricity into a connected circuit.

  Welchman could hardly believe that he had solved the problem, but drew a rough wiring diagram and convinced himself that it would work. Hurrying to the Cottage, he showed it to Alan, who was also incredulous at first, but rapidly became equally excited about the possibilities it opened up. It was a spectacular improvement; they no longer needed to look for loops, and so could make do with fewer and shorter ‘cribs’.

  With the addition of a diagonal board, the Bombe would enjoy an almost uncanny elegance and power. Any assertion reached, say (BL), would feed back into every B and every L appearing in either plain-text or cipher-text. With this four-fold proliferation of implications at every stage it became possible to use the Bombe on any ‘crib’ of three or four words. The analyst could select a ‘menu’ of some ten or so letters from the ‘crib’ sequence – not necessarily including a ‘loop’, but still as rich as possible in letters bound to lead to implications for other letters. And this would provide a very severe consistency condition, sweeping away billions of false hypotheses with the speed of light.

  The principle was startlingly like that of mathematical logic, in which one might seek to draw as many conclusions as possible from a set of interesting axioms. There was also a particularly logical subtlety about the process of deduction. As so far described, the operation would require the trying out of one plugboard hypothesis at a time. If (AA) were brought down by its own contradictions, then (AB) would be tried, and so on, until all 26 possibilities had been exhausted. Only then would the rotors move on by one step and the next position be examined in the same way. But Alan had seen that this was unnecessary.

  If (AA) were inconsistent then it would generally lead to (AB), (AC), and so forth, in the process of following all the implications. This would mean that all these were also self-contradictory, and there was no need to try them out. An exception would occur when the rotor position was actually correct. In this case either the plugboard hypothesis would also be correct, and would lead to no contradictions; or would be incorrect and lead to every plugboard statement except the correct one. This meant that the Bombe was to stop when the electric current had reached either only one, or just twenty-five, of twenty-six terminals. It was this rather complicated condition that the relay switches had to test. This was not at all an obvious point, but seeing it made the process twenty-six times faster. Alan would comment on the likeness to mathematical logic, in which a single contradiction would imply any proposition. Wittgenstein, discussing this point, had said that contradictions never got anyone into trouble. But these contradictions would make something go very wrong for Germany, and lead to bridges falling down.

  Thus the logical principle of the Bombe was the wonderfully simple one of following the proliferation of implications to the bitter end. But there was nothing simple about the construction of such a machine. To be of practical use, a Bombe would have to work through an average of half a million rotor positions in hours rather than days, which meant that the logical process would have to be applied to at least twenty positions every second. This was within the range of automatic telephone exchange equipment, which could perform switching operations in a thousandth of a second. But unlike the relays of telephone exchanges, the Bombe components would have to work continuously and in concert, for hours at a stretch, with the rotors moving in perfect synchrony. Without the solution of these engineering problems, in a time that would normally see no more than a rough blueprint prepared, all the logical ideas would have been idle dreams.

  Even with Bombes designed and in the pipeline, the problem of the Enigma was far from solved. A Bombe would not take all the work out of the probable word method. One very important point was that when the consistency conditions were met, and a Bombe came to a halt, this did not necessarily mean that the correct rotor position had been arrived at. Such a ‘stop’, as it would be called, could arise by chance. (The calculation of how often such chance ‘stops’ were to be expected was a nice application of probability theory.) Each ‘stop’ would have to be tested out on an Enigma to see whether it turned the rest of the cipher-text into German, until the correct rotor position was discovered.

  Nor was it a trivial matter to guess the probable word, nor to match it against the cipher-text. A good cipher clerk, indeed, could make these operations impossible. The right way to use the Enigma, like any ciphering machine, was to guard against the probable word attack by such obvious devices as prefacing the message with a variable amount of random nonsense, inserting X’s in long words, using a ‘burying procedure’ for stereotyped or repetitious parts of the transmission, and generally making the system as unpredictable, as un-mechanical, as was possible without loss of comprehensibility to the legitimate receiver. If this were done thoroughly the accurate ‘cribs’ required for the Bombe could never be found. But perhaps it was too easy for the Enigma user to imagine that the clever machine would take care of itself, and there were often regularities for the British cryptanalysts to exploit.

  Even when they had overcome subtleties of this kind, and learnt how to guess words with perfect accuracy, the story was far from over. Deciphering one message would not help to fight the war. The problem was to solve every message, of which there would be thousands every day in each network. The solution of this problem would depend upon the cipher system as a whole. In a system as simple as the pre-war use of repeated triplets of indicator letters, a single solved message could be used to undo the whole process, find the ‘ground-setting’, and thus reveal the entire traffic. But the enemy would not always be so obliging. Moreover, there was a sort of double bind, since the process of guessing a word with virtual certainty would only become possible when there was a good acquaintance with the traffic as a whole. The Bombe would be of little use unless a break into that traffic were first made in some other way.

  With the Luftwaffe signals they did have another way – the method with perforated sheets which worked for the nine-letter indicator system. During the autumn of 1939, the construction of the sixty sets of sheets was completed, and a copy taken to the French cryptanalysts at Vignolles. This was an act of hope. No Enigma messages had been solved since December 1938, so they had no assurance that by the time the sheets were completed they would be of any use. But the hope was justified, for10

  ‘At the end of the year’, GC and CS records, ‘our emissary returned with the great news that a key had been broken (October 28, Green)* on the … sheets he had taken with him. Immediately we got to work on a key (October 25, Green) …; this, the first wartime Enigma key to come out in this country, was broken at the beginning of January 1940’. The GC and CS account continues: ‘Had the Germans made a change in the machine at the New Year? While we waited … several other 1939 keys were broken. At last a favourable day arrived … The sheets were laid … and [the] Red of 6 January was out. Other keys soon followed ….’

  Their luck had held, and the perforated sheets gave the first entry into the system. It was like the Princeton treasure hunt, in that each success would give the clue to the next goal, that of speedier and more comprehensive decryption. Special methods like those of the sheets – and there were many other algebraic, linguistic, and psychological tricks – could open up the way for something better. But it was never simple, for the rules were changing all the time
, and they had to run as fast as they could to keep up. They were only just in time and, had they fallen behind by a few months, might never have caught up. In the spring of 1940 it was particularly precarious, as they held on with a mixture of ingenuity and intuition – or as the military were likely to call it, sheer bloody guesswork.

  Guessing and hoping were entirely characteristic of current British operations. The government had little more idea of how a war could be won, or even of what was happening, than did the public. It seemed that the British and German armed forces had, after all, agreed to have a battle again, but the British Tweedledee was decidedly reluctant to be the one to start, and the German Tweedledum expected it all to be over by six o’clock. Tweedledee’s weaponry was still concealed behind Chamberlain’s umbrella. The Red King was snoring on the square to the east, and no one (not even at Bletchley) knew what he was dreaming of. The blockade was supposed to bring an already ‘stretched’ Germany to crack from within, if only Britain could ‘hold out’. Half desired, half dreaded by the British rulers, was the reappearance of the Monstrous Crow, currently flapping ambiguously on the other side of the Atlantic.

  Appropriately, the Luftwaffe messages so laboriously and expensively deciphered at Bletchley in March 1940 turned out to consist mostly of nursery rhymes sent as practice transmissions. Even there, where at least they were busy with very exciting work, a sense of unreality and anticlimax was often felt. It was the same at Cambridge. Alan would return there occasionally for leave days, to work on mathematics and to see friends. At King’s they had all dutifully trooped down into air-raid shelters (all except Pigou, who refused to compromise with the Luftwaffe), but the promised bombardment had not come. Three quarters of the children evacuated to Cambridge had returned home by mid-1940.

  Yet the war had not been over by Christmas; Alan had exercised his option to suspend his fellowship for the duration of the war on 2 October 1939, and although his course on the Foundations of Mathematics had been advertised in the lecture list, it was not to be given. And there was Finland. Once during this period there was a party in Patrick Wilkinson’s room, where Alan met a third-year undergraduate, Robin Gandy, who was reading mathematics and also rather conscientiously trying to defend the Communist party line. ‘Hands off Finland’ was double-talk such as Alan despised, but he liked Robin Gandy, and instead of marching off in disgust, led him on gently with Socratic questioning to arrive at a contradiction.

  And one thing that was real, even in the phoney war, was the conflict at sea. Just as in the First World War, it was the strength and the weakness of the offshore European island that war with Britain was an attack upon the world trading economy. One third of all merchant shipping was British, and apart from coal and bricks there was scarcely a commodity in which the island was self-sufficient. Despite the blockade Germany could survive by pressing the resources and labour of Europe into its service. But British survival depended upon the ocean lanes. There lay the critical, paramount asymmetry.

  It was the sea war that would become Alan’s particular province. In early 1940 the different Enigma systems were divided among the chief cryptanalysts, who were allocated huts outside the Bletchley mansion. Welchman took over the army and air force Enigma systems, in Hut 6, joined by a number of new recruits. Dillwyn Knox took the Italian Enigma* and that used by the German SD, again with new recruits. These systems, which did not use plugboards, suited his psychological methods. And Alan was allocated Hut 8 in which to head the work on the naval Enigma signals. Other huts housed sections translating and interpreting the output; thus Hut 3 dealt with the army and air force material issuing from Hut 6, while the naval signals, if and when any were produced, would be interpreted by Hut 4, which was headed by Frank Birch.

  Alan probably knew little of the context in which he was working, apart from the general air of urgency that issued from Hut 4. This was probably just as well, for the context was not exactly an encouraging one. He was working for the Admiralty, which only grudgingly had relinquished naval cryptanalysis to GC and CS. Traditionally, the Royal Navy expected autonomy. As possessor of the world’s largest fleet, the Admiralty might be supposed capable of organising warfare for itself. Yet it had signally failed to learn the lesson that navies depended not only upon force but upon information, for guns and torpedoes were impotent unless in the right place at the right time. Like the giant Cyclops, ‘Our Fighting Navy’ was decidedly one-eyed. Naval Intelligence was embodied in an organisation that anyone of the new generation would find absurdly Victorian, if not criminally incompetent.

  Only in the First World War had any Naval Intelligence Division been set up, and this had declined in peace-time into Kafkaesque fantasy. In 1937, the NID was11 ‘… neither interested in nor equipped to collect or disseminate information about the organization, dispositions, and movements of foreign fleets … the situation was very little better than it had been … in 1892…. Large old-fashioned ledgers were used in which to enter in longhand the last known whereabouts of Japanese, Italian and German warships …. These reports were often months old, and only once a quarter [were] the supposed dispositions of foreign navies … issued to the Fleet.’ The Movements Section of the NID (consisting of a single part-time officer) ‘did not even subscribe to Lloyd’s list, which would at least have provided a daily and highly accurate record of all the world’s merchant ships. Reports of the movement of warships from the Secret Service were virtually non-existent … The possibility of locating ships at sea was … even more remote than that of obtaining up-to-date information about them when they were in port.’ The admirals did not really want to know.

  By September 1939 a new man, Norman Denning, had somewhat improved the position. There was a card-index instead of ledgers, a direct telephone link to Lloyd’s, and a Tracking Room on which a plot of merchant ships’ positions could be up-dated. Links with GC and CS were not so successful. Indeed, the cryptanalytic organisation, captured by the Foreign Office after the First World War, tended to be treated as the enemy. Denning continued to plot its reconquest by the Admiralty until February 1941.

  But the forward-looking Denning had also been able to establish the principle that a new sub-section of the NID, the Operational Intelligence Centre, which replaced the old Movements Section, should receive and coordinate information from all sources. This had been impossible in the First World War and represented a revolutionary advance. On the eve of war the OIC stood by with a staff of thirty-six. They had many problems to overcome, but the main problem of 1939 was that they had virtually no information to coordinate. Like Tweedledee, the Admiralty could hit out bravely at anything it could see, but it could see very little.

  Occasionally, Coastal Command aircraft would catch sight of U-boats, and the RAF had been persuaded to inform the Admiralty when this happened. Aerial reconnaissance was limited to the hiring of a commercial pilot to take shots of the German coastline. Information from agents in Europe was ‘scanty. The best … came from a black market dealer in silk stockings with a contact in the German Naval Post Office, who from time to time was able to give the address of mail for certain ships, thus providing some fragmentary clues to their whereabouts.’ When the Rawalpindi was sunk in November 1939, the Admiralty were unable to discover even the class of ship responsible. And as for signals, not only were the Enigma-enciphered messages indecipherable, but the German Navy12

  went over to war-time wireless procedure shortly before the attack on Poland, putting an end to the possibility of following its movements by correlating call-signs with the results of direction-finding, and it was to be months before work on the German naval signals system at GC and CS and in the [OIC] … made it possible to produce even tentative deductions on the basis of Traffic Analysis. The first step was to distinguish U-boat from other German naval communications, and it is some indication of the extent of the black-out that this elementary advance was not made until the end of 1939.

  Until the outbreak of war, ‘the naval sub-section of
the German Section’ of GC and CS ‘which was started with one officer and a clerk as late as May 1938, still had no cryptanalysts.’ It was just one aspect of the failure even to try to meet the German challenge. The prospects were better now, with the help of the Poles and with Bombes on the way, but the overall picture was dire:13

  Since the outbreak of war GC and CS had continued to give work on the G[erman] A[ir] F[orce] variant of the Enigma priority over its attack on the naval traffic. It had done so for two good reasons. The GAF traffic was more voluminous. Over and above that, those who worked on the naval Enigma had been held up first by the fact that the German Navy used the machine more carefully than the GAF, so that by the beginning of 1940 GC and CS had been able to break the settings for only 5 days of 1938, and then by the discovery that, sometime about the outbreak of war, the naval machine had undergone more radical modification than had the GAF’s. During 1940 small amounts of captured naval cypher material had confirmed that, while both still used only three wheels at a time, the naval Enigma’s wheels were selected from … 8 instead of from 5.

  To make any headway, Alan would need something more to go on. ‘From December 1939 GC and CS had left the Admiralty in no doubt about the urgency of this … requirement, but the Admiralty had had little opportunity to meet it.’ But there was (at least at sea) a war on, which meant that the German authorities had to work on the assumption that the Enigma machine itself would be captured. Indeed this was so; the Polish revelations had only given GC and CS seven months’ start in this respect, for ‘Three Enigma wheels had been recovered from the crew of U-33 in February 1940.’ But this ‘had not provided a sufficient basis for a further advance.’ Possession of the naval machine, while necessary, would have been far from sufficient. If the German navy used the machine ‘more carefully’, then its key systems were perhaps much less transparent than the foolish repeated triplets exploited by the Poles. And a few days of sparse peacetime traffic would provide a slender basis upon which to mount an attack.