interesting things for the last ten years or so, and Waterhouse had stumbled upon a rare find. 'Tell me about it,' he insists.
'It's a cryptanalysis problem,' Waterhouse says. 'Non-Enigma.' He goes on to tell the story about the messages from U-553. 'When I got to Bletchley Park this morning,' he concludes, 'I asked around. They said that they had been butting their heads against the problem as long as I had, without any success.'
Suddenly, Alan looks disappointed and bored. 'It must be a one-time pad,' he says. He sounds reproachful.
'It can't be. The ciphertext is not devoid of patterns,' Waterhouse says.
'Ah,' replies Alan, perking up again.
'I looked for patterns with the usual
'As in Baudot code,' Alan says[15]. He looks guardedly interested again.
'So I converted each letter into a number between one and thirty-two, using the Baudot code. That gave me a long series of small numbers. But I wanted some way to convert all of the numbers in the series into one large number, just to see if it would contain any interesting patterns. But this was easy as pie! If the first letter is R, and its Baudot code is 01011, and the second letter is F, and its code is 10111, then I can simply combine the two into a ten-digit binary number, 0101110111. And then I can take the next letter's code and stick that onto the end and get a fifteen-digit number. And so on. The letters come in groups of five-that's twenty-five binary digits per group. With six groups on each line of the page, that's a hundred and fifty binary digits per line. And with twenty lines on the page, that's three thousand binary digits.
So each page of the message could be thought of not as a series of six hundred letters, but as an encoded representation of a single number with a magnitude of around two raised to the three thousandth power, which works out to around ten to the nine hundredth power.'
'All right,' Alan says, 'I agree that the use of thirty-two-letter alphabet suggests a binary coding scheme. And I agree that the binary coding scheme, in turn, lends itself to a sort of treatment in which individual groups of five binary digits are mooshed together to make larger numbers, and that you could even take it to the point of mooshing together all of the data on a whole page that way, to make one extremely large number. But what does that accomplish?'
'I don't really know,' Waterhouse admits. 'I just have an intuition that what we are dealing with here is a new encryption scheme based upon a purely mathematical algorithm. Otherwise, there would be no point in using the thirty-two-letter alphabet! If you think about it, Alan, thirty-two letters are all well and good-as a matter of fact, they are essential-for a teletype scheme, because you have to have special characters like line feed and carriage return.'
'You're right,' Alan says, 'it is extremely odd that they would use thirty-two letters in a scheme that is apparently worked out using pencil and paper.'
'I've been over it a thousand times,' Waterhouse says, 'and the only explanation I can think of is that they are converting their messages into large binary numbers and then combining them with other large binary numbers-one-time pads, most likely-to produce the ciphertext.'
'In which case your project is doomed,' Alan says, 'because you can't break a one-time pad.'
'That is only true,' Waterhouse says, 'if the one-time pad is truly random. If you built up that three- thousand digit number by flipping a coin three thousand times and writing down a one for heads and a zero for tails, then it would be truly random and unbreakable. But I do not think that this is the case here.'
'Why not? You think there were patterns in their one-time pads?'
'Maybe. Just traces.'
'Then what makes you think it is other than random?'
'Otherwise it makes no sense to develop a new scheme,' Waterhouse says. 'Everyone in the world has been using one-time pads forever. There are established procedures for doing it. There's no reason to switch over to this new, extremely odd system right now, in the middle of a war.'
'So what do you suppose is the rationale for this new scheme?' asks Alan, clearly enjoying himself a great deal.
'The problem with one-time pads is that you have to make two copies of each pad and get them to the sender and the recipient. I mean, suppose you're in Berlin and you want to send a message to someone in the Far East! This U-boat that we found had cargo on board-gold and other stuff-from Japan! Can you imagine how cumbersome this must be for the Axis?'
'Ahh,' Alan says. He gets it now. But Waterhouse finishes the explanation anyway:
'Suppose that you came up with a mathematical algorithm for generating very large numbers that were random, or at least random-looking.'
'Pseudo-random.'
'Yeah. You'd have to keep the algorithm secret, of course. But if you could get it-the algorithm, that is- around the world to your intended recipient, then they could, from that day forward, do the calculation themselves and figure out the one-time pad for that particular day, or whatever.'
A shadow passes over Alan's otherwise beaming countenance. 'But the Germans already have Enigma machines all over the place,' he says. 'Why should they bother to come up with a new scheme?'
'Maybe,' Waterhouse says, 'maybe there are some Germans who don't want the entire German Navy to be able to decipher their messages.'
'Ah,' Alan says. This seems to eliminate his last objection. Suddenly he is all determination. 'Show me the messages!'
Waterhouse opens up his attache case, splotched and streaked with salt from his voyages to and from Qwghlm, and draws out two manila envelopes. 'These are the copies I made before I sent the originals down to Bletchley Park,' he says, patting one of them. 'They are much more legible than the originals-' he pats the other envelope '-which they were kind enough to lend me this morning, so that I could study them again.'
'Show the originals!' Alan says. Waterhouse slides the second envelope, encrusted with TOP SECRET stamps, across the table.
Alan opens the envelope so hastily that he tears it, and jerks out the pages. He spreads them out on the table. His mouth drops open in purest astonishment.
For a moment, Waterhouse is fooled; the expression on Alan's face makes him think that his friend has, in some Olympian burst of genius, deciphered the messages in an instant, just by looking at them.
But that's not it at all. Thunderstruck, he finally says, 'I recognize this handwriting.'
'You do?' Waterhouse says.
'Yes. I've seen it a thousand times. These pages were written out by our old bicycling friend, Rudolf von Hacklheber. Rudy wrote those pages.'
Waterhouse spends much of the next week commuting to London for meetings at the Broadway Buildings. Whenever civilian authorities are going to be present at a meeting-especially civilians with expensive sounding accents-Colonel Chattan always shows up, and before the meeting starts, always finds some frightfully cheerful and oblique way to tell Waterhouse to keep his trap shut unless someone asks a math question. Waterhouse is not offended. He prefers it, actually, because it leaves his mind free to work on important things. During their last meeting at the Broadway Buildings, Waterhouse proved a theorem.
It takes Waterhouse about three days to figure that the meetings themselves make no sense-he reckons that there is no imaginable goal that could be furthered by what they are discussing. He even makes a few stabs at
