After all these years, it still brought tears to Anna's eyes.
Kaz responded with tales of how he was training Tiber. Anna thought it was sad, to train horses for the battlefield. Because of their size, they would be the first to be killed. Kaz wanted to agree, but thought it might be unsoldierly to do so. He offered a consolation: within the coming decade, most horses would be replaced by tanks as the main weapon for thrusts through the enemy's lines.
Anna was surprised to find herself asking about tanks, a topic in which she had absolutely no interest. The Germans and French had the best new ones, Kaz said. The difficulty was that Poland did not have the heavy industry to produce tanks, and were unable to buy new models from the French or even the British. Those two countries had begun a feverish buildup of their own. He then turned the conversation to a more interesting subject, Anna.
“I understand you're a student at Poznan University.”
“Well, yes and no.”
Kaz raised an eyebrow.
“I
was
a student. I gave up my course work when I became involved in a research project—on weather forecasting.”
“Ah, then that would explain how you could get away to Warsaw at this time of the year.”
“Yes, we've had a very intense summer. I needed a break.”
“Intense? Weather forecasting? I would have thought it was a timeless topic—to be approached in a methodical, unhurried way.” Kaz wanted to bite his tongue. He wished it hadn't come out quite that way; he hoped she wouldn't be offended.
“Actually, it is urgent. The Air Force needs better forecasts. Short-range planes can get stuck up in the air, unable to land because of thunderstorms. Running out of gas.”
“I see your point. That would be dangerous.”
“Not to speak of embarrassing.” Anna smirked.
“And you've been working on...?”
“Studying weather balloons. We fill them with hydrogen so they rise rapidly. Above 4,000 meters, they often change direction as they encounter upper-atmosphere winds. Those currents help us predict the movements of weather fronts.”
“You've been...?”
Anna took a deep breath; she normally hid her interest in calculus when talking to young men. Or men of any age, for that matter.
“I've been working on a mathematical model of upper-atmosphere air currents.”
“Really?”
Anna was even more surprised; he seemed interested in the details. Now, she appreciated Jerzy's enthusiasm for his hobby. She also was grateful for the equations she had scribbled in her notebook, following Marian's recent instructions to disguise any notes she might make on Enigma.
She reached over the coffee table for a napkin, and scratched down an equation of the vertical velocity of balloons, how they slowed down as the air got thinner, and how the upward velocity depended on the elasticity of the balloon.
He still seemed interested. Apparently, he had at least a smattering of calculus. She wrote down a second equation—the horizontal acceleration of balloons as they hit upper-air currents.
“Impressive. You're really comfortable with calculus.”
“And I'm pleased—a cavalry officer is interested.”
He put his hand on the sofa. Near Anna's, but not quite touching.
“Well, actually, my interest doesn't come from the cavalry.”
Anna waited, hoping he would elaborate.
“When I was in high school, I loved math. In fact, I wanted to go to university to study engineering.... But things didn't work out.”
Again she was curious, hoping he would provide details. He did.
“My family—my father—wouldn't hear of it. I come from a line of military officers. My grandfather was a general, my father a colonel. His heart was set on my continuing the family tradition.... I'm afraid we had a few, ah, mmm... disagreements. But then we compromised....” His voice trailed off.
“Compromised?” Anna didn't want to state the obvious: some compromise, off to the military academy. But then, of course, the father did have a hold over his son. He was the one who would pay for university.
“I would go to the academy for one year. If my heart were still set on math, I could transfer to the university. He would no longer object.”
“So you liked the academy when you got there?”
“Well, yes and no. I started out with a concentration on artillery; that seemed the closest to mathematics. In a sense, it was. But if you have a knack for math, you quickly get bored with parabolic trajectories of shells. And in combat, the artillery works from tables; they don't do their own calculations.”
“Sounds as though you were disappointed. Why didn't you switch to the university after the first year?”
“I had a lot of friends. In the military, the sense of comradeship is remarkable. You know that some day your life may depend on your friends. And I became very interested in the cavalry; I've always loved horses.”
“So we have the same two interests—mathematics and horses. You picked horses; I picked math.”
“Put that way, horses seem pretty trivial.” Kaz spoke gently, without the hint of offense.
“Not at all. Not at all. With the grim international situation, it's obvious, how important our armed forces are.” Anna waved her hand in front of her face, attempting to shoo away a fly. She put her hand back on the sofa. Close once more, but not quite touching.
“You found your niche; you seem unusually good at mathematics.”
“It's in the genes.... My grandfather was a cousin of Marie Curie.” Again, he seemed impressed.... “I'm also radioactive,” she added with a smile. “I glow in the dark.”
“My little glowworm.”
Anna suddenly realized that she and Kaz were alone. She hadn't noticed; twenty minutes earlier, Krystyna and Pawel had slipped away. She felt a warm glow. Kaz was different—so unlike the boys she dated in high school, who were so... so
shallow
. Even Ryk, the fun-loving Ryk, always seemed to be skating along the surface. She'd known him since she was fourteen, but somehow she wondered if she really knew him at all. She was so much more comfortable with Kaz. Now, after only one evening, she felt as though she had known him for years.
6
Steckered
When you have eliminated the impossible, whatever remains, however improbable, must be the truth.
Sherlock Holmes,
The Sign of Four
T
he next afternoon, Anna curled up in a well stuffed chair and began to read
Anna Karenina
. This was, she realized, the first time in several years she had enjoyed the luxury of sitting down with a novel. When Krystyna dropped by to pick her up for the evening's festivities, Anna demurred; enough parties for the time being. Krystyna smiled quizzically. Anna responded: I'll be going out with Kaz on Saturday. That's the first time he can get away.
For the next three weeks, Krystyna got the same answer: Anna would be seeing Kaz. Then, on Tuesday of the following week, Anna received a message: a telegram from the “Special Meteorology Project” for her to return—at once. At this time of day, she couldn't contact Kaz by phone to break their date. She wrote, and then rewrote, a telegram to him; and rewrote a follow-up letter at least half a dozen times, saying how much she hoped to get back to Warsaw.
There was no need to be concerned. Three days earlier, Kaz had put in for a transfer to a base near Poznan. It would come through quickly; many officers were eager to transfer the other way, from Poznan to Warsaw.
W
hen Anna got back to the “Special Meteorology Project,” she was surprised at the upbeat mood. Substantial progress had been made with Rejewski's machine, although undeciphered messages were still piling up. The following day, Henryk reappeared, carrying a large case. He called the group of four together and opened the case with a flourish. Inside was an Enigma machine with an attachment—something like a small telephone switchboard, dotted with round holes or sockets, each labeled with a letter. A tangle of cables connected the sockets in no apparent order.
“That,” announced Henryk, pointing to the switchboard, “is a plugboard, or, as the Germans call it, a Steckerboard. It comes to us, compliments of Army intelligence.
“Its purpose is just what Jerzy suggested—a second encoding. It's an integral part of the machine, and that makes it easy to use. Each of the short cables can be plugged into two sockets connecting any two letters—say C and K; then C and K are steckered together. When a key is pushed, and C comes out of the three-wheeled scrambler, the current is routed through the steckerboard, changing the C to a K; the K appears in the lighted board at the top. As the connections run both ways, the steckerboard will also switch every K to the letter C. So the machine works like the original: at any initial setting, it can be used either to code or decode a message.”
Jerzy was leaning over the machine, obviously fascinated—like an eight-year old preparing to take a watch apart.
“Yes, Jerzy, you may play with it; the present settings don't mean anything in particular.”
Jerzy began to unplug the cables, resetting them in a different order. Each time, he set the wheels to AAA and pushed one of the keys, scribbling the results on a pad.
“The steckerboard seems like a very simple second coding,” Henryk continued, “and it is. C always becomes K, and K always becomes C, as long as the cable settings remain unchanged. If that were the only coding, it would be child's play to decipher it. But it's not. We have two codings in tandem: first the scrambling from the three wheels, with the results then passed through the steckerboard.
“Unfortunately, this simple system has spectacular results. Consider. The first letter, A, may be steckered to any of the other 25 letters, obviously giving 25 possibilities.”
Marian squirmed in his seat; he had been suffering most acutely from the frustration of preceding months. His mind was now racing ahead; he interrupted Henryk.
“Once A is steckered, there are 24 remaining unsteckered letters. The next one, B, can be steckered to any of the other 23 open plugs, and so on. Thus, the number of possible combinations is 25 x 23 x 21.... That comes to a total of...”
Henryk paused to let Marian demonstrate his impressive ability to do calculations in his head. He waited expectantly, head slightly forward, brows raised, his ten long fingers pressed against the table.
“What? Almost 10 trillion?” Marian reported after a few seconds.
“Actually,
only
8 trillion.” Henryk paused to let the number sink in. “For each of these possibilities, there are the 100,000-plus combinations coming through the three-wheeled scrambler. In very rough terms, that makes almost a quintillion possibilities—or, if you like it straight and simple, a billion billion.”
Without realizing it, Anna sighed.
“I think,” Henryk said in a record understatement, “that this will require the best from both our gorillas and idiots. Indeed, the next round goes to the idiots. Brute force, by itself,
simply won't work any more.
We're to the end of the line with that approach. We expect a lot from Marian's brute of a machine, but not magic.”
The responses of his small audience varied. Marian smiled slightly; it was the first time that Henryk had allowed himself to use the terms gorillas and idiots. Jerzy and Anna, as the chief idiots, were now on the spot; they looked decidedly glum at the huge odds against them. In the past two years, the possible encodings had escalated from a “mere” 17 thousand to a billion billion. To read a message by brute force would take many trillion times as long. They should live so long.
“I also think,” concluded Henryk, “that we might all turn the problem over in our minds before meeting again—next Wednesday morning, I would suggest.”
They didn't get anywhere the next Wednesday. Nor the following Wednesday, nor the next, nor any of the Wednesdays until mid November.
Then they collectively found the key that would be used, not only by the Poles, but also by the French and especially the British all the way up to the time of the invasion of France in May 1940, when the Germans would have another particularly nasty surprise.
When they sat down for the regular Wednesday meeting, Henryk asked Anna, as the raporteur, to sum up the current situation as she saw it; this would give them a starting point.
Unconsciously, Anna sat a bit straighter in her chair, giving a formality to her report. “There are obviously two parts of the puzzle: the wheels from the original Enigma—the scrambler, in other words—plus the steckerboard. Our basic problem is: how can we solve the scrambler settings without first knowing the steckerboard settings, and vice versa? In other words, we have to solve both sets of settings—the wheels and the steckerboard—
simultaneously
.”
Anna noticed that Marian was stroking his beard. She continued.
“What do we know that might be helpful?
“First, some of the messages start with 'From,' or VON in German. It's not clear how much help that is. After it goes through the steckerboard, VON may be transmogrified into
any
three other letters. In spite of it all, there might be something here,” she said, trying to be upbeat. “We've also collected a list of other similar cribs.”
Marian was now stroking his beard even more slowly, more thoughtfully. Anna hoped he was still listening. She took up her story.
“Second, the pattern of messages has remained the same: a preamble followed by two sets of three letters, followed by a string, followed by groups of five letters. Thus, it is reasonable to assume that the two sets of three letters—the inscrutable six—still give the wheel settings. But how can we use that information? Again, it will go through the steckerboard, which will change all the letters from the scrambler. Again, we're driven back to our fundamental problem: it's hard to see how we can crack any corner of the problem without figuring out everything—wheels and steckerboard—at once. We will have to...”
Her voice trailed off because it looked as if Marian was ready to say something.