Intelligence in War: The Value--And Limitations--Of What the Military Can Learn About the Enemy

by John Keegan

  Disc positions (three discs): 26 3 26 3 26 = 17,576

  Disc sequence (ABC, ACB, BCA, BAC, CAB, CBA) = 6

  Plugboard connections = over 100 billion

  Total = 10,000 billion.12

  That number does not allow for rotating the outer rims on the three discs, which multiplies it by 17,576.

  The task faced by an interceptor of an Enigma-encrypted text may be represented in this way. If he were able to check “one setting every minute [he] would need longer than the age of the universe to check every setting.”13 Even if he had got possession of an Enigma machine, and so had only to proceed through the initial settings of the discs (17,576) to see if the encrypt rendered a plain text, he would still, working day and night, need two weeks to check all the settings, allowing one minute for each.14 No wonder Scherbius advertised his machine as generating “unbreakable” ciphers and that the Germans believed theirs to be so.

  BREAKING ENIGMA

  Yet Enigma was to be broken and not long after it had been put into use. Those who achieved the solution were cryptanalysts of the Polish army which, as the defender of the Versailles state most resented by post-war Germany, took a keen and necessary interest in German military encrypted transmissions. What is extraordinary, positively intellectually heroic, about the Polish effort is that it was done initially by the exercise of pure mathematics. As Peter Calvocoressi, an initiate of the British cryptanalytic centre at Bletchley Park, has succinctly put it, “in order to break [a machine] cipher, two things are needed: mathematical theory and mechanical aids.”15 The Poles eventually designed a whole array of mechanical aids—some of which they passed to the British, some of which the British replicated independently, besides inventing others themselves—but their original attack, which allowed them to understand the logic of Enigma, was a work of pure mathematical reasoning. As it was done without any modern computing machinery, but simply by pencil and paper, it must be regarded as one of the most remarkable mathematical exercises known to history.

  To do the work the Polish army recruited in the late 1920s a number of young civilian mathematicians from university mathematics faculties, including Henryk Zygalski, Jerzy Rozycki and Marian Rejewski. Marian Rejewski was to prove the most creative; like the others, he came from western, formerly German Poland, and spoke German fluently. In 1932, soon after the German army had adopted, on 1 June, the Enigma machine as its principal encryption instrument, and his own return from postgraduate study at Göttingen, he began to work on intercepted German encrypts in the Polish general staff building in Warsaw. The Poles had already learnt how to break German super-enciphered codes. From 1928 onwards, however, they had been defeated by strange messages which were clearly enciphered and probably, they concluded, the product of a machine system. The young cryptanalysts were set to learn its secrets.

  What the Poles were intercepting were five-letter groups which betrayed no frequency. In technical terms, the message was itself the key, a continuous one which did not repeat unless at very long mathematical intervals (once in many millions of times, as we have seen). Yet it must, as Rejewski knew, obey a mathematical rule. He set out to construct the cipher’s mathematical basis.

  The messages he was given were, we now know, produced in the following manner. After setting up his machine by printed instruction, which prescribed the disc (or rotor) order, the position of the rim and the plugging, the operator chose his own preliminary rotor setting and typed in a three-letter group, which he then repeated; this instructed the recipient how to set up his own machine for that particular transmission (and was to reveal clues to decipherment that were to be of great use, particularly to Bletchley Park). He then typed in the message with his left hand, writing down with his right hand the letters as they appeared illuminated one by one on the lamp board. Next, he passed what he had written to a radio operator, who transmitted it to the receiving station; it was this process which denied Enigma the status of an on-line system, though it would have been easy to achieve had it been linked directly to a transmitter. At the receiving end, the recipient typed in the letters he received and took down those illuminated on his lamp board, which disclosed the decrypted meaning.

  Rejewski got only the encrypt. Quite quickly, however, he recognised that the first three letters were separate from the body of the message, and that the second three letters were an encryption of the first three. These two three-letter groups provided, in short, a key to the very much larger key which was the message itself. If the two preliminary three-letter groups could be broken, two results would follow: first, the electro-mechanics of Enigma itself could be reconstructed, in part at least; second, some intercepted messages could be decrypted.

  Rejewski devised a set of equations which would allow him to allot real alphabetical values to the first six encrypted letters. He was able to deduce that, in the groups, say, ABC followed by DEF, D would be an encryption of A (via electromechanical permutation), E would be an encryption of B and F would be an encryption of C. He decided to designate the permutations produced by the first (fixed) disc as S, those produced by the rotors as L, M, N and that produced by the reflector as R. As a result he wrote three equations, the first of which he expressed as:

  AD = SPNP-1 MLRL-1M-1PN-1p-3NP MLRL-1 M-1 p N-1 p- S-1

  The other two were equally complex and, he writes, “the first part of our task [was], essentially, to solve this set of equations in which the left sides, and on the right side only the permutation P and its powers are known, while the permutations S, L, M, N, R are unknown. In this form, the set is certainly insoluble.”16

  “Therefore,” Rejewski goes on, “we seek to simplify it. The first step is purely formal and consists in replacing the repeated product MLRL-1 ? M-1 . . . with the single letter Q. We have thereby temporarily reduced the number of unknowns to three, namely S, N, Q.”

  Non-mathematicians will be unable to follow Rejewski’s subsequent pages of equations. They conclude, however, as follows: “the method described above for [recovering] N could be applied by turns to each rotor, and thus the complete inner structure of the Enigma machine could be reconstructed.”17

  That was the Polish triumph: the penetration of the Enigma secret by pure mathematical reasoning. During the thirties, the Poles also managed to keep abreast of successive German refinements of Enigma, both electromechanical and procedural, and they succeeded in manufacturing duplicates of the Enigma machine. As its transmissions became more difficult to break, they also devised an electromechanical device (the “bombe,” apparently so-called after its ticking, which was thought to resemble that of an infernal machine) which tested solutions of encrypts faster than was possible by paper methods. Meanwhile they shared their knowledge with the French cryptanalytic service, France being Poland’s principal ally. The French themselves, through a financially corrupt German informant, known as Asché (French pronunciation of HE, the initials of his cover name), were acquiring documents which revealed many of Enigma’s operating secrets; Asché, the brother of a general, appears eventually to have been unmasked and to have been shot for treason in 1943.18 The Poles and the French certainly worked together closely on German ciphers throughout the thirties: latterly the French were also co-operating with the British Government Code and Cipher School (GCCS) located at Bletchley Park. During the period 24–25 July 1939, just before Germany’s invasion of Poland, French and British officials visited Warsaw, where the Poles passed them each a reconstructed model of the Enigma machine.

  BREAKING ENIGMA AGAIN

  By then the Poles were no longer able to read Enigma intercepts, because of mechanical complications—particularly the introduction of two extra discs, increasing the possible number of disc orders from six to sixty—and procedural changes. Nevertheless, they were able to pass to the British reconstructed machines which reproduced the internal wiring of the discs, which to their annoyed consternation was foolishly simple, A being wired to B and so on. They also introduced to the British—who had hit
upon the idea themselves—the concept of subjecting intercepts to treatment by punched sheets. Rejewski, besides being a pure mathematician, also had a practical bent and had grasped, from his theoretical understanding of how Enigma worked, that there would be repetitions in the permutations and that those could be identified by representing encrypted letters as perforations in large sheets of paper. Given enough intercepts, and overlaid sheets, their arrangement on a light-table would reveal repetitions, when they occurred, by light shining through. Repetitions would support disc-settings though not prove them; those would have to be established by subsequent work.

  The British decryption operation, though eventually far larger than the Polish, proceeded on the whole by a different method: to use Calvocoressi’s distinction, it depended more upon mechanical aids than mathematical theory, though many mathematicians worked at Bletchley and it owed its start to the Polish mathematical endeavour. Gordon Welchman, one of the most gifted of the Bletchley mathematicians, who came from a Cambridge fellowship to Bletchley Park right at the beginning of the war, distinguished four periods in its early history: (1) the preparatory period, ending with the making of complete sets of the perforated sheets in early 1940; (2) the period of dependence on the sheets, ending on 10 May 1940, when the Germans ceased to encrypt the second three-letter group which communicated the setting; (3) a subsequent period when the cryptanalysts were largely dependent on exploiting German operators’ carelessness in procedure; and (4) from September 1940 when Bletchley began to acquire its own bombes, similar in principle to those devised in the thirties by the Poles.19

  Welchman divides the development of his own thinking about how to decrypt the Enigma intercepts into ten steps, spread over several months. It was not officially his concern, since he had been set to study German radio call signs. That was necessary but routine work and Welchman’s acute mathematical mind began almost involuntarily to engage with the letter groups on the intercepts he was passed. The first three steps he describes had to do with speculation about whether the two three-letter groups in the preliminaries always contained pairs of encrypts of the same letter three positions apart (as the rotor turned). When he decided that they did, he moved to a calculation of probabilities of how often paired letters would appear, establishing a number which he thought manageable (Step 4). He next concluded that the Germans’ military Enigma was much less complex than they—and the British—thought, because the plugboard did not in practice increase the number of permutations to be tested. “With only the 60 wheel [rotor] orders and 17,576 ring [rim] settings to worry about, we are down to a million possibilities. In fact we have reduced the odds against us by a factor of around 200 trillion. This was Step 5 and quite a gain!”20 Step 6 was a further calculation of probabilities while Step 7 was his independent perception of how perforated sheets could eliminate many unfruitful possibilities. Steps 8, 9 and 10 led him to see how the sheets should be used; “if we could find twelve females [fruitful pairings] on the Red [war] and Blue [training] key for a particular day, we could confidently expect to discover that key after an average of 780 stackings [superimposing the perforated sheets on the light-table] . . . so in great excitement I hurried to tell . . . Dilly about it. Dilly was furious.”21

  Dilly Knox, son of the Bishop of Manchester and brother of E.V. (Evoë), editor of Punch, and Ronnie, a famous Roman Catholic convert priest, had been a fellow of King’s College, Cambridge, was a veteran of Room 40 and had spent all his subsequent life as a government cryptanalyst. On the establishment of the Government Code and Cipher School at Bletchley Park in August 1939 he became principal assistant to Commander Alastair Denniston, another Room 40 veteran who was now GCCS’s head.22 Eccentric and solitary, he was quite unsuited to the task. “Neither an organisation man nor a technical man,” in Welchman’s words, Knox belonged to an earlier age of code-breaking when puzzles were dissolved by flashes of inspiration rather than rigorous analysis. He had tried his hand at Enigma but had concluded that “there were simply too many unknown factors that had to be solved simultaneously. Although Knox had worked out a mathematical procedure for recovering the daily settings, it depended on first knowing the internal rotor wirings, and there just seemed to be no way of isolating that part of the equation.”23 In short, what Rejewski had achieved, Knox could not. He was simply not a good enough mathematician. No wonder that, with his introverted temperament, he flew into a rage when his clever young subordinate Welchman arrived to claim that he could see a way through the thickets that had defeated him.

  Had Welchman been easily put upon, things might have rested there, Enigma might have taken months longer to crack and the Battle of Britain and the Battle of the Atlantic proved even harder to survive. Fortunately Welchman was not to be browbeaten. Though told to go back and get on with his compilation of call signs, he went to the Deputy Director, Commander Edward Travis, but, sensibly, not simply to complain but to present a plan of organisation and action. Welchman had perceived the first lesson in winning bureaucratic battles: present an alternative scheme. He expressed his fear that, once the Phoney War went hot, Bletchley would be overwhelmed by a volume of vital radio traffic it would be unable to read. To cope with the oncoming rush, he proposed dividing Bletchley Park’s growing staff into five sections working in shifts twenty-four hours a day: a Registration Room to do traffic analysis; an intercept Control Room to direct the listening stations to the most promising senders; a Machine Room to co-ordinate the work of the first two; a Sheet-stacking Room, under the Machine Room’s control; a Decoding Room to deal with any messages that yielded to decryption. Welchman also proposed increasing the number of listening centres, to include one operated by the air force which would listen out for Luftwaffe messages; the principal listening station, in an old fort at Chatham, was, though very efficient, operated by the army.24

  Travis not only accepted Welchman’s scheme but persuaded Denniston to instigate it, so that Bletchley, just in time, was already operating effectively when the storm broke on 10 May 1940. There was another fortuitous event. Bletchley already knew about the bombe, from the Poles. It now acquired bombes of its own. The original design was the work of Alan Turing, another Cambridge mathematics don who had been recruited at the same time as Welchman. Turing was Welchman’s intellectual superior. Indeed, he was one of the foremost mathematicians in the world, who, as a visiting fellow at Princeton in 1936, had written the theory of the digital computer, a universal calculating machine which did not yet exist; computers bear the alternative name of “Turing machines.”25 Turing’s design for a bombe was being developed by the British Tabulating Machine Company, whose products were largely punched-card devices. Turing’s was electromechanical, of much greater speed and power, but Welchman proposed an alteration in the design which allowed possible, but wrong, Enigma settings to be eliminated much faster.

  The bombes could not, of course, test every possible Enigma setting, which would have required the calculating speed of a large modern computer. Welchman, but also Turing and others, had realised that parts of many Enigma intercepts were formal and repeated: the full name and rank of the addressee, for example, or the title of the originating headquarters. These might be guessed and came to be called “cribs” (an English public-school term for an illegal guide to a Latin or Greek translation). The bombe method depended upon guessing a crib and testing letter substitutions, by repetitive mathematical process, within the cycle of 17,576 positions in which the rotors could be set. It proved very fruitful.

  The crib method, however, would not have worked unless the German operators had revealed clues to the setting by carelessness, laziness or error. “The machine would have been impregnable if it had been used properly,” in Welchman’s opinion.26 It was used properly by the operators of a number of branches of the German armed forces and government. Three naval Enigma keys, including the important key code-named Barracuda by Bletchley and used for high-level signals during fleet operations, were never broken; Pink, the high-level
Luftwaffe key, was broken only after a year of use and thereafter only rarely; Green, the German army home administration key, was broken only thirteen times during the war and then with some prisoner of war help (“such was the security of Enigma when properly used”); Shark, the Atlantic U-boat key, proved unbreakable between February and December 1942, a crucial period in the Battle of the Atlantic; the Gestapo key, used from 1939 to 1945, was not broken at all.27

  The pattern of breaks was not random. The Gestapo seems, not unnaturally, to have taken great care; the German army and navy, which had long-established signal branches, made use of well-trained and experienced operators; the weakness lay most obviously with the Luftwaffe, a new service founded only in 1935. Its operators were probably younger and less experienced. A Luftwaffe key was the first to be broken by Bletchley, which thereafter broke almost all Luftwaffe keys intercepted, sometimes on the first day they were identified.

  Bletchley called two forms of mistake made by German operators—each was the product of laziness—the “Herivel tip” and “Sillies” (or “Cillies”). John Herivel’s tip was the result of a brainwave. He guessed that an operator, after setting the rotor rims, would place them in their slots with the selected letters uppermost. These letters might then form the first three letters of the encrypt, thus revealing the rotor setting which, for the first few years of the war, remained unchanged throughout the day. The “Herivel tip” often yielded a result, which greatly shortened decryption.28