Dreamers and Deceivers (25 page)

Alan had been fiddling with codes more complicated than that since he was a boy in St. Leonards-by-Sea. He knew that code-writing had come a long way since Caesar’s campaigns.

“Well,” continued Denniston, “the German code is trillions of times more complicated than Caesar’s. Imagine that someone in Berlin wants to send a message to a U-boat in the North Atlantic. The person in Berlin writes a message by hand in plain German. It might say, ‘das wetter ist trübe,’ which means, ‘the weather is cloudy.’ He then types the message on an Enigma machine, which looks a little like a typewriter. Take a look.”

A picture of an Enigma machine flashed on the projection screen.

“And here’s where the magic begins,” continued Denniston. “The Enigma machine changes each letter in the message. It might change ‘das wetter ist trube’ into ‘b-t-t o-o-f-d-s-e e-i-c g-t-t-a-i.’ The message is now encoded, and that’s what is sent to the U-boat via the wireless telegraph. Is everyone following me so far?”

Alan already had about a thousand questions, but he chose not to interrupt.

“Now,” said Denniston, “the person in the U-boat receives the encoded message, and he types it into his own Enigma machine, which changes every encoded letter back into its original plain German letter.”

“I suppose,” asked Gordon Welchman, who taught math at Cambridge with Alan and had come with him to Bletchley Park, “we have no idea how the Enigma does what it does?”

“Actually,” replied Denniston, “that’s not correct. Thanks to our friends in Poland, we know
exactly
how it works. While England was burying its head in the sand for the past decade, Poland was preparing for war. Just before the first German tanks crossed their border, our Polish friends explained to us that they had not only figured out the Enigma’s wiring, they’d even built one of their own.”

“So why do any of us even need to be here?” asked John Jeffreys, another mathematician from Cambridge. “It sounds like the Poles did our work for us.”

“Hardly,” said Denniston. “There’s more to the Enigma than its
wires. Each letter typed into the machine first goes to a plug board that sits between the keyboard and the user. It works like a telephone switchboard, and an operator randomly selects new settings for it every day. So, on Monday, it might turn the letter ‘a’ into ‘b,’ and ‘b’ into ‘a’; on Tuesday, it might turn ‘a’ into ‘t,’ and ‘t’ into ‘a’; and on Wednesday it might keep ‘a’ as ‘a.’ Even though we know how the wiring works, we don’t know which setting the Germans have chosen for any particular day.”

Denniston now flashed a picture of three wheels, each labeled with letters, onto the projection screen.

“These are the Enigma’s rotors,” he explained. “The signal for each individual letter is carried by electric current to a rotor sitting above the right side of the keyboard. The right rotor changes the letter into a new letter, and then sends the letter to an adjacent rotor in the middle. The middle rotor changes the letter again and sends it to the left rotor, which changes the letter yet again and sends it to a reflector. The reflector changes the letter for the third time and sends it back to the left rotor, which changes the letter and sends it back to the middle rotor, which changes the letter and sends it back to the right rotor, which sends the letter back to the plug board for one last switcheroo.

“Here’s an illustration of the process, but for the sake of simplicity, it leaves off the plug board.” Denniston flipped a switch and projected his final slide:

“By the time the plug board and the rotors have done their work, the original letter could be anything! The final, encoded letter appears on a board of little lamps, each labeled with a different letter. The lamp board looks like a keyboard, but with little lamps instead of buttons on a keyboard. The user simply writes down each letter on a piece of paper after the Enigma produces it.”

“How long does all this letter-switching take?” asked Jeffreys.

“It all happens in less than a second,” said Denniston. “And the trickiest thing is that the rotors are always moving. Once one letter goes through this process, one of the rotors turns a notch, which means that if ‘q’ became ‘u’ the first time you pressed ‘u’ on the Enigma—as in the example on the screen—then ‘u’ might become ‘z’
the next time you press ‘u,’ even if it’s just a moment later. Even though certain settings only change once a day, the alignment of the rotors is constantly changing. In fact, the alignment changes after every letter of every message, which means the code changes after every letter of every message.”

“Bloody hell!” said Welchman, looking stunned. “We’re supposed to crack this code? By my estimate, there are over 17,000 possible combinations produced by the rotors. That isn’t so bad, but there are 150 trillion ways of connecting ten pairs of letters on that blasted plug board. And I assume the Krauts randomly change the starting position of each rotor every day, just like the plug board? So every day is a new day of trillions of new possible combinations.”

“Quite right, Mr. Welchman,” replied Denniston. “It’s actually 17,576—but, believe it or not, it’s much worse than that. The Germans choose three rotors every day out of a stock of five possible rotors. That means there are sixty possible rotor combinations. In other words, the number of potential combinations is 60 times 17,576 times 150 trillion.

“The U-boat version of Enigma is even more complicated. Its user selects three rotors from a stockpile of eight possible ones. That means there are 336 possible rotor combinations, making it 336 times 17,576 times 150 trillion.”

“So, that’s almost a sextillion combinations,” Alan said, speaking up for the first time. “885,376,800,000,000,000,000 by my calculations. If we tested a possible combination of rotors and plug board settings every second of every day for 28 trillion years, we’d still have 55 billion combinations untested. And even if we were somehow able to decipher a single letter in a single word, after 28 trillion years, we’d have to start all over again for the very next letter, because the Enigma’s rotors spin after each one.”

Alan looked around the table and saw stunned expressions on his colleagues’ faces.

“Yes,” said Denniston, “that’s quite right.”

“I have only have one question, Alastair,” said Alan, who was
dismissive of protocol even when his mind
wasn’t
consumed with mathematical puzzles; it never occurred to him to refer to his superior by rank.

“Yes, Turing,” said Denniston, clearly irked that a subordinate was addressing him by his first name, “what’s your question?”

Alan knew he’d never be the life of a cocktail party or set a marathon record, although he had almost qualified for Britain’s Olympic team. He knew he’d never write like Keats or speak like Churchill. But he understood more about mathematics and probability than anyone in England. From the moment he had heard about the Enigma, he never doubted his ability to build and program a machine—an electronic brain—that could outguess any machine the Nazis could come up with.

It would be like tackling the decision problem.

It would be fun.

“My question is,” said Alan Turing, “when can I start?”

Bletchley Park

October 26, 1940

Alan sat alone in his office, his fingers stained with fountain pen ink, a week’s worth of dirty clothes flung haphazardly around the spartan room, and a notebook before him filled with almost illegible scribblings. Hundreds of pages were covered with algorithms, German phrases, pages-long equations, and dozens of diagrams with arrows pointing in a seemingly infinite number of directions, connecting a seemingly infinite number of signs and symbols.

Alan had slept until noon, and now, nine hours later, he still looked as if he had just woken up. Despite the three-mile bicycle ride to Bletchley Park from his room in Shenley Brook End, he hadn’t bothered to change out of his pajamas. Alan had met his professorial comrades’ informality and raised it, throwing in a few Turingesque eccentricities for good measure.

To prevent the symptoms of hay fever, he biked to work wearing a gas mask. To keep his oversize trousers up, he used a string of yarn around his waist instead of a belt. To protect his finances, he had
bought and buried bars of silver that would likely hold their value even in the worst-case scenario of a successful German invasion, although in his absentmindedness, he had already forgotten the exact spot in the woods where he’d buried them.

Perhaps oddest of all was Alan’s decision to chain his coffee mug to a radiator in his office. He didn’t require much, but he needed his black coffee. And coffee cups, which were difficult to come by in war-torn England, were prime targets for petty thieves.

“Turing!” shouted Alastair Denniston, barging into Alan’s office uninvited and unwelcomed. “Why weren’t you in your office this morning?”

“I was.”

“What do you mean you were? I was here at nine o’clock, and you were nowhere to be found.”

“I’m afraid you just missed me,” Alan said, smiling. “I left my office at three o’clock in the morning.”

“Very funny,” said an unamused Denniston. “Look, I’m not here to talk about your work hours or your unprofessional appearance for that matter. Heaven knows you put in as much time as anyone else here, even if you have unorthodox ways of doing it. My concern, rather, is what you’re working on.”

“The U-boat Enigma?” asked Alan.

“Quite right,” said Denniston. “In case you haven’t noticed, the Krauts are bombing London to smithereens. They’ve already marched through Poland, Denmark, Norway, Holland, Belgium, Luxembourg, and France. We might bloody well be next. Seven thousand civilian fatalities in London last month alone.”

“Why,” Alan asked calmly, trying and failing to mask his annoyance with this seemingly pointless interruption, “are you telling me things I already know?”

“Because,” Denniston bellowed, “I’m not sure why you’re spending so much time on the U-boat Enigma, which hardly anyone believes can be broken, when you could be spending more time on the version of the Enigma being used by the German army and Luftwaffe.

“You know, Turing, no one ever surrendered to a ship. But if the Luftwaffe pummels us into submission and the German army gets a
foothold in Britain, it’s Katy-bar-the-door. Then you’ll really need that silver bullion you buried and lost!”

“Yes,” Alan replied, still trying to keep a lid on his rising temper, “but we’ve already made great strides against the army and air force Enigma. Thanks to the tireless work of the few mathematicians you finally hired last September we’re deciphering the army’s and the Luftwaffe’s encrypted messages faster than ever.”

“I’m not trying to detract from the great work you’ve done,” said Denniston, trying to offer an olive branch to his star code-breaker. “Everyone knows that you’re the most technically gifted cryptographer at Bletchley, and without you, our deciphering mechanisms would be a hundred times slower than they are. I’m only asking why you wouldn’t keep working to make them
another
hundred times faster. The intelligence loses its value with every hour that passes between our receiving it and our deciphering it. Shouldn’t you be working on
that
? Everyone who isn’t named ‘Alan Turing’ thinks there’s only a one-in-a-million chance of anyone being able to break the U-boat Enigma. You’re tilting at windmills!”