Enigma

hands and sat down.
    He was back.

3

    Whenever anyone asked Jericho why he was a mathematician—some friend of his mother, perhaps, or an inquisitive colleague with no interest in science—he would shake his head and smile and claim he had no idea. If they persisted, he might, with some diffidence, direct them to the definition offered by G. H. Hardy in his famous Apology: 'a mathematician, like a painter or a poet, is a maker of patterns'. If that didn't satisfy them, he would try to explain by quoting the most basic illustration he could think of: pi—3.14—the ratio of a circle's circumference to its diameter. Calculate pi to a thousand decimal places, he would say, or a million or more, and you will discover no pattern to its unending sequence of digits. It appears random, chaotic, ugly. Yet Leibnitz and Gregory can take the same number and tease from it a pattern of crystalline elegance:
    pi/4=1-1/3+1/5-1/7+1/9--
    and so on to infinity. Such a pattern had no practical usefulness, it was merely beautiful—as sublime, to Jericho, as a line in a fugue by Bach—and if his questioner still couldn't see what he was driving at, then, sadly, he would give up on them as a waste of time.
    On the same principle, Jericho thought the Enigma machine was beautiful—a masterpiece of human ingenuity that created both chaos and a tiny ribbon of meaning. In the early days at Bletchley he used to fantasise that some day, when the war was over, he would track down its German inventor, Herr Arthur Scherbius, and buy him a glass of beer. But then he'd heard that Scherbius had died in 1929, killed—of all ludicrously illogical things—by a runaway horse, and hadn't lived to see the success of his patent.
    If he had, he would have been a rich man. By the end of 1942 Bletchley estimated that the German had manufactured at least a hundred thousand Enigmas. Every Army headquarters had one, every Luftwaffe base, every warship, every submarine, every port, every big railway station, every SS brigade and Gestapo HQ.
    Never before had a nation entrusted so much of its secret communications to a single device.
    In the mansion at Bletchley the cryptanalysts had a roomful of captured Enigmas and Jericho had played with them for hours. They were small (little more than a foot square by six inches deep), portable (they weighed just twenty-six pounds) and simple to operate. You set up your machine, typed in your message, and the ciphertext was spelled out, letter by letter, on a panel of small electric bulbs. Whoever received the enciphered message merely had to set up his machine in exactly the same way, type in the cryptogram, and there, spelled out on the bulbs, would be the original plaintext.
    The genius lay in the vast number of different permutations the Enigma could generate. Electric current on a standard Enigma flowed from keyboard to lamps via a set of three wired rotors (at least one of which turned a notch every time a key was struck) and a plugboard with twenty-six jacks. The circuits changed constantly; their potential number was astronomical, but calculable. There were five different rotors to choose from (two were kept spare) which meant they could be arranged in any one of sixty possible orders. Each rotor was slotted on to a spindle and had twenty-six possible starting positions. Twenty-six to the power of three was 17,576. Multiply that by the sixty potential rotor-orders and you got 1,054,560. Multiply that by the possible number of plugboard connections—about 150 million million—and you were looking at a machine that had around 150 million million million different starting positions. It didn't matter how many Enigma machines you captured or how long you played with them. They were useless unless you knew the rotor order, the rotor starting positions and the plugboard connections. And the Germans changed these daily, sometimes twice a day.
    The machine had only one tiny—but, as it turned out, crucial—flaw. It could never encipher a letter as itself: an A would never emerge from it as an A, or a B as a B, or a C as a C. . . Nothing is ever itself: that was the great guiding principle in the breaking of Enigma, the infinitesimal weakness that the bombes exploited.
    Suppose one had a cryptogram that began:
    IGWH BSTU XNTX EYLK PEAZ ZNSK UFJR CADV _
    And suppose one knew that this message originated from the Kriegsmarine's weather station in the Bay of Biscay, a particular friend of the Hut 8 cribsters, which