Alan Turing: The Enigma (37 page)

^*
10
³⁴
is 10,000,000,000,000,000,000,000,000,000,000,000 – a number comparable with the number of elementary particles in a large building. But 10
^10
34
is far bigger: as 1 followed by 10
³⁴
zeroes it would require books with the mass of Jupiter to print it in decimal notation. It could be thought of as the number of possible man-made objects. Skewes’ number was much bigger again, as 1 followed by 10
^10
34
zeroes! In actual fact mathematicians had certainly thought about numbers far larger than these, here we have only gone through three stages of growth, but it is not difficult to make up a new notation to express the idea of going through ten such stages, or 10
¹⁰
, or 10
¹⁰
; or of regarding even these as just the first step in a process of super-growth, and rhen defining super-super-growth, and then.... Such definitions, indeed,had already played a role in the theory of ‘recursive functions’, one of the other approaches to the idea of ‘definite method’ which had been found equivalent to thet of the Turing machine. But Skewes’ number was certainly remarkably large for a problem which could be expressed in such elementary terms.

^*
Certainly one attraction to Alan of the
New Statesman
would have been its exceptionally demanding puzzle column. In January 1937 he was delighted when his friend David Champernowne defeated such runners-up as M.H.A. Newman and J.D. Bermal in giving a witty solution, phrased in Carrollian language, to a problem set by Eddington called ‘Looking Glass Zoo’. (It depended upon a knowledge of the matrices used by Dirac in his theory of the electron.) But Alan’s comments on the Abdication, naive in idealism perhaps but certainly not ill-informed, indicate very clearly that his interest in the journal would not have been confined to this feature.

^*
Ulam writes further that ‘von Neumann had great admiration for him and mentioned his name and “brilliant ideas” to me already, I belive, in early 1939,. ... At any rate von Neumann mantioned to me Turings’s name several times in 1939 in conversations, concerning mechanical ways to develop formal mathematical systems.’

^*
In what follows, code refers to any conventional system of communicating text, whether secret or not. Cipher is used for communications designed to be incomprehensible to third parties. Cryptography is the art of writing in cipher; cryptanalysis that of deciphering what has been concealed in cipher. Cryptology covers both the devising and breaking of ciphers. At the time, these distinctions were not made, and Alan Turing himself referred to cryptanalysis as ‘cryptography’.

^†
The British spying organisation, variously entitled SIS, M16. Apart from this top-level administrative overlap it was and remained essentially distinct from the cryptanalytic department.

^*
David Champernowne also discussed the principle of the chain reaction with Alan after reading an article about it by J.B.S. Haldane in the
Daily Worker
.

^*
that is, for looking at even more zeroes of the zeta-function.

^*
Alan was wrong.

4

The Relay Race

Gliding o’er all, through all,

Through Nature, Time, and Space,

As a ship on the waters advancing,

The voyage of the soul – not life alone,

Death, many deaths I’ll sing.

Alan
reported next day, 4 September, to the Government Code and Cypher School, which had been evacuated in August to a Victorian country mansion, Bletchley Park. Bletchley itself was a small town of ordinary dullness, a brick-built Urban District in the brickfields of Buckinghamshire. But it lay at the geometric centre of intellectual England, where the main railway from London to the north bisected the branch line from Oxford to Cambridge. Just to the north-west of the railway junction, on a slight hill graced by an ancient church, and overlooking the clay pits of the valley, stood Bletchley Park.

The trains were busy with the evacuation of 17,000 London children into Buckinghamshire, swelling Bletchley’s population by twenty-five per cent. ‘The few who returned,’ said an urban district councillor, ‘no one on earth would have billeted, and they did the wisest thing eventually to return to the hovels from whence they came.’ In these circumstances, the arrival of a few select gentlemen for the Government Code and Cypher School would have caused little stir, although it was said that when Professor Adcock first arrived at the station, a little boy shouted ‘I’ll read your secret writing, mister!’ in a most disconcerting way. Later on there were complaints by local residents about the do-nothings at Bletchley Park, and it was said that the MP had to be prevented from asking a question in Parliament. They had the pick of accommodation: the few hostelries of mid-Buckinghamshire. Alan was billeted at the Crown Inn at Shenley Brook End, a tiny hamlet three miles north of Bletchley Park, whither he cycled each day. His landlady, Mrs Ramshaw, was one of those who lamented that an able-bodied young man was not doing his bit. Sometimes he helped out in the bar.

The early days at Bletchley resembled the arrangements of a displaced senior common room, obliged through domestic catastrophe to dine with another college, but nobly doing its best not to complain. In particular there was a strong King’s flavour, with old-timers Knox, Adcock and Birch, and the younger Frank Lucas and Patrick Wilkinson as well as Alan. The shared background in Keynesian Cambridge was probably helpful for Alan. In particular it offered a link with Dillwyn Knox, a figure not generally noted for geniality or acessibility by Alan’s contemporaries. GC and CS was by no means a vast establishment. On 3 September, Denniston wrote
¹
to the Treasury:

Dear Wilson,

For some days now we have been obliged to recruit from our emergency list men of the Professor type who the Treasury agreed to pay at the rate of £600 a year. I attach herewith a list of these gentlemen already called up together with the dates of their joining.

Alan
was not quite the first, for according to Denniston’s list there were nine of these ‘men of the Professor type’ at Bletchley by the time that he arrived with seven others the next day. Over the following year, about sixty more outsiders were brought in.

The ‘emergency in-take quadrupled the cryptanalytic staff of the Service sections and nearly doubled the total cryptanalytic staff.’ But only three of these first recruits came from the science side. Besides Alan, there were only W.G. Welchman and John Jeffries.
^*
Gordon Welchman was the senior figure, lecturer in mathematics at Cambridge since 1929 and six years older than Alan. His field was algebraic geometry, a branch of mathematics then strongly represented at Cambridge, but one which never attracted Alan; their paths had not crossed before.

Welchman had not been involved with GC and CS before the outbreak of war as Alan had, and thus found himself, as a newcomer, relegated by Knox to the task of analysing the pattern of German call-signs, frequencies, and so forth. As it transpired, this was a job of immense significance, and his work rapidly brought such ‘traffic analysis’ to a quite new standard. It made possible the identification of the different Enigma key-systems, as was soon to prove so important, and opened GC and CS eyes to a much wider vision of what could be done. But no one could decipher the messages themselves. There was just a ‘small group which, headed by civilians and working on behalf of all three Services, struggled with the Enigma.’ This group consisted first of Knox, Jeffries, Peter Twinn and Alan. They established themselves in the stables building of the mansion, soon dubbed ‘the Cottage’, and developed the ideas that the Poles had supplied at the eleventh hour.

There was no glamour about ciphers. In 1939 the job of any cipher clerk, although not without skill, was dull and monotonous. But ciphering was the necessary consequence of radio
^*
communication. Radio had to be used for aerial, naval, and mobile land warfare, and a radio message to one was a message to all. So messages had to be disguised, and not just this or that ‘secret message’, as with spies and smugglers, but the whole communication system. It meant mistakes, restrictions, and hours of laborious work on each message, but there was no choice.

The
ciphers used in the 1930s did not depend on any great mathematical sophistication, but on the simple ideas of
adding on
and
substituting
. The ‘adding on’ idea was hardly new; Julius Caesar had concealed his communications from the Gauls by a process of adding on three to each letter, so that an A became D, a B became an E, and so on. More precisely, this kind of adding was what mathematicians called ‘modular’ addition, or addition without carrying, because it meant Y becoming B, Z becoming C, as though the letters were arranged around a circle.

Two thousand years later, the idea of modular addition by a
fixed
number would hardly be adequate, but there was nothing out-of-date about the general idea. One important type of cipher used the idea of ‘modular addition’, but instead of fixed number, it would be a varying sequence of numbers, forming a
key
, that would be added to the message.

In practice, the words of the message would first be encoded into numerals by means of a standard code-book. The job of the cipher clerk would then be to take this ‘plain-text’, say

For this to be of any use, however, the legitimate receiver had to know what the key was, so that it could be subtracted and the ‘plain-text’ retrieved. There had to be some
system
, by which the ‘key’ was agreed in advance between sender and receiver.

One way of doing this was by means of the
one-time
principle. This was one of the few sound ideas of 1930s cryptography, as well as the simplest. It required the key to be written out explicitly, twice over, and one copy given to the sender, one to the receiver of the transmission. The argument for the security of this system was that provided the key were constructed by some genuinely completely random process, such as shuffling cards or throwing dice, there could be nothing for the enemy cryptanalyst to go on. Given cipher text ‘5673’, for instance, the analyst might guess that the plain-text was in fact ‘6743’ and the key therefore ‘9930’, or might guess that the plaintext was ‘8442’ and the key ‘7231’, but there would be no way of verifying such a guess, nor reason to prefer one guess to another. The argument depended upon the key being absolutely patternless, and spread evenly over the possible digits, for otherwise the analyst
would
have reason to prefer one guess to another. Indeed, discerning a pattern in the apparently patternless was essentially the work of the cryptanalyst, as of the scientist.

In
the British system, one-time pads were produced, to be used up one at a time. Provided the key was random, no page was used twice, and the pads were never compromised, the system was fool-proof. But it would involve the manufacture of a colossal quantity of key, equal in volume to the maximum that the particular communication link might require. Presumably this thankless task was undertaken by the ladies of the Construction Section of GC and CS, which on the outbreak of war was evacuated not to Bletchley but to Mansfield College, Oxford. As for the system in use, that was no joy either. Malcolm Muggeridge, who was employed in the secret service, found it
²

a laborious business, and the kind of thing I have always been bad at. First, one had to subtract from the groups of numbers in the telegram corresponding groups from a so-called one-time pad; then to look up what the resultant groups signified in the code book. Any mistake in the subtraction, or, even worse, in the groups subtracted, threw the whole thing out. I toiled away at it, getting into terrible muddles and having to begin again…