Cryptonomicon

Suppose that Turing's bicycle were capable of generating a different alphabet for each one of its different states. So the state ([theta] = 0, C = 0) would correspond to, say, this substitution alphabet:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

Q G U W B I Y T F K V N D O H E P X L Z R C A S J M

but the state ([theta] = 180, C = 15) would correspond to this (different) one:

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

B O R I X V G Y P F J M T C Q N H A Z U K L D S E W

No two letters would be enciphered using the same substitution alphabet--until, that is, the bicycle worked its way back around to the initial state ([theta] = 0, C= 0) and began to repeat the cycle. This means that it is a periodicpolyalphabetic system. Now, if this machine had a short period, it would repeat itself frequently, and would therefore be useful, as an encryption system, only against kids in treehouses. The longer its period (the more relative primeness is built into it) the less frequently it cycles back to the same substitution alphabet, and the more secure it is.

The three-wheel Enigma is just that type of system (i.e., periodic polyalphabetic). Its wheels, like the drive train of Turing's bicycle, embody cycles within cycles. Its period is 17,576, which means that the substitution alphabet that enciphers the first letter of a message will not be used again until the 17,577th letter is reached. But with Shark the Germans have added a fourth wheel, bumping the period up to 456,976. The wheels are set in a different, randomly chosen starting position at the beginning of each message. Since the Germans' messages are never as long as 450,000 characters, the Enigma never reuses the same substitution alphabet in the course of a given message, which is why the Germans think it's so good.

A flight of transport planes goes over them, probably headed for the aerodrome at Bedford. The planes make a weirdly musical diatonic hum, like bagpipes playing two drones at once. This reminds Lawrence of yet another phenomenon related to the bicycle wheel and the Enigma machine. "Do you know why airplanes sound the way they do?" he says.

"No, come to think of it." Turing pulls his gas mask off again. His jaw has gone a bit slack and his eyes are darting from side to side. Lawrence has caught him out.

"I noticed it at Pearl. Airplane engines are rotary," Lawrence says. "Consequently they must have an odd number of cylinders."

"How does that follow?"

"If the number were even, the cylinders would be directly opposed, a hundred and eighty degrees apart, and it wouldn't work out mechanically."

"Why not?"

"I forgot. It just wouldn't work out."

Alan raises his eyebrows, clearly not convinced.

"Something to do with cranks," Waterhouse ventures, feeling a little defensive.

"I don't know that I agree," Alan says.

"Just stipulate it--think of it as a boundary condition," Waterhouse says. But Alan is already hard at work, he suspects, mentally designing a rotary aircraft engine with an even number of cylinders.

"Anyway, if you look at them, they all have an odd number of cylinders," Lawrence continues. "So the exhaust noise combines with the propeller noise to produce that two-tone sound."

Alan climbs back onto his bicycle and they ride into the woods for some distance without any more talking. Actually, they have not been talking so much as mentioning certain ideas and then leaving the other to work through the implications. This is a highly efficient way to communicate; it eliminates much of the redundancy that Alan was complaining about in the case of FDR and Churchill.

Waterhouse is thinking about cycles within cycles. He's already made up his mind that human society is one of these cycles-within-cycles things (8) and now he's trying to figure out whether it is like Turing's bicycle (works fine for a while, then suddenly the chain falls off, hence the occasional world war) or like an Enigma machine (grinds away incomprehensibly for a long time, then suddenly the wheels line up like a slot machine and everything is made plain in some sort of global epiphany or, if you prefer, apocalypse) or just like a rotary airplane engine (runs and runs and runs; nothing special happens; it just makes a lot of noise).

"It's somewhere around . . . here!" Alan says, and violently brakes to a stop, just to chaff Lawrence, who has to turn his bicycle around, a chancy trick on such a narrow lane, and loop back.

They lean their bicycles against trees and remove pieces of equipment from the baskets: dry cells, electronic breadboards, poles, a trenching tool, loops of wire. Alan looks about somewhat uncertainly and then strikes off into the woods.

"I'm off to America soon, to work on this voice encryption problem at Bell Labs," Alan says.

Lawrence laughs ruefully. "We're ships passing in the night, you and I."