If you came here via a search engine, you were probably looking for Pierre Cartier's wonderful mini-biography of the mathematician Alexander Grothendieck - from which I shamelessly stole the title. Please go straight there. Otherwise...

“And so no one, except for two people, enters the top floor of the Aedificium. ...” The abbot smiled. “No one should. No one can. No one, even if he wished, would succeed. The library defends itself, immeasurable as the truth it houses, deceitful as the falsehood it preserves. A spiritual labyrinth, it is also a terrestrial labyrinth. You might enter and you might not emerge. And having said this, I would like you to conform to the rules of the abbey.” “But you have not dismissed the possibility that Adelmo fell from one of the windows of the library. And how can I study his death if I do not see the place where the story of his death may have begun?” “Brother William,” the abbot said, in a conciliatory tone; “a man who described my horse Brunellus without seeing him, and the death of Adelmo though knowing virtually nothing of it, will have no difficulty studying places to which he does not have access.” Umberto Eco, "Name of the Rose"

In 1979, the American journalist Howard Morland fought - and won - a lengthy court battle for the right to publish a magazine article, "The H-Bomb Secret: To Know How is to Ask Why." Morland's objective was to infer and publish the basic facts of the famous Teller-Ulam hydrogen bomb design, drawing solely from unclassified sources.

Even a lackluster student of the 20th century ought to feel a little surprised: the article was indeed printed, and Morland even escaped with his life! Why the inquisitors relaxed their grip and turned the man loose, I cannot say. Perhaps his model of the H-Bomb's inner workings was sufficiently wrong, and was thought to be useful as disinformation. If this was indeed so, I should hope that what follows is equally wrong - and, like Morland, I will be permitted to continue sharing my crackpot ruminations with you, dear reader. Otherwise, to the gasenwagen I shall go, where I might perhaps meet with some of you!

So, dear inquisitor: nothing found below came from anywhere other than deathly-boring, public documents, placed on the Net by American public officials, merchants, and military historians.

While we're on the subject of disinformation: I find it intriguing that the supposed Snowden leak appears to consist entirely of minor operational details - of surveillance programs which have been known to the public, under various names, for some years! And likewise, neither the fact of the NSA supplying the world with diddled crypto - nor that of the American software industry's collusion with the agency really qualifies as "news."

So what would qualify as a genuinely-newsworthy NSA leak? Why, naturally, the crown jewels! These would conceivably be: facts - or at least, solid clues - shedding light on two great mysteries:

What weaknesses in commonly-used ciphers, unknown to the public, are known to the world's favorite spy agency? What proprietary ciphers does the U.S. government reserve for its own use? And how do these differ from ciphers available to the public?

The "armchair general" community's endless speculations, well-founded and otherwise, seem to focus entirely on the first conundrum. Yet the second strikes me as considerably more interesting.

The NSA publishes two official lists of cryptographic algorithms approved for use by U.S. government employees, known as "Suite A" and "Suite B." Suite A consists of "classified algorithms that will not be released." Suite B contains well-known public favorites, ones commonly regarded as strong - such as AES.

All that is publicly known of Suite A ciphers is their names; a tantalizing "WinAmp Playlist" of monikers such as ACCORDION, BATON, JOSEKI, SAVILLE, SHILLELAGH, and a number of others. Truly, "a country of which nothing is known but the name!" Clearly it is impossible to learn anything meaningful regarding these marvels of mathematical engineering, without stealing national secrets and paying the price. Or is it?

The traditional "folk" explanation for the existence of Suite A is that the NSA (and the NATO military establishment it is part of) has advanced many decades beyond the public state of the cryptographic art, and knows of certain weaknesses in popular cipher systems (including those found in Suite B.) The implication is that Suite A ciphers lack these weaknesses, and are therefore considered fit to protect the most valuable national secrets.

The folk explanation is simple and convenient, but the facts - as revealed by perfectly ordinary public (rather than leaked, or "leaked") documents , simply do not add up. Consider the case of the ViaSat KG-200 Inline Media Encyptor, "designed to conveniently fit between your computer’s motherboard and hard drive. In the event that a classified computer is lost, stolen, or tampered with, its hard drive remains encrypted so no classified information is compromised." Notice anything peculiar? The cipher used in the KG-200 is: AES-256. Plain old AES, aka Rijndael, known to every computer programmer on the planet. Turns out that AES is approved for "Top Secret" use. ViaSat's other disk encryption products also rely on AES.

So, what products are advertised as including Suite A ciphers? It appears that the latter are found exclusively in equipment intended to secure voice and data traffic in the field. SAVILLE, for instance, is thought to have made its appearance in army field radios as early as the late Vietnam War era. It - along with some other Suite A ciphers - was shared with NATO members, including the UK and Norway, and was put to use in their own radio systems. BATON, a somewhat newer cipher, was - and continues to be - used in field radio equipment, such as the "Project-25" walkie-talkie issued to many public servants. Other Suite A stream ciphers appear to serve similar purposes. VALLOR, for example, is said to be used in securing TTY broadcasts to submarines. American military satellite uplinks are also known to use Suite A ciphers.

There is an apparent contradiction: if AES is judged fit to secure the hard disks of top bureaucrats, why is it placed only in radio sets issued to policemen, while those given to soldiers feature Suite A ciphers? The folk explanation would hold that Suite A algorithms are thought to be stronger. But if anything, a soldier's field radio is rather more likely than a policeman's walkie-talkie to fall into the hands of a hostile reverse-engineer. Military equipment is routinely taken as spoils, and often finds its way to the highest bidder.

One interesting clue is that U.S. military personnel are never permitted to generate their own cipher keys. The latter are always generated at an NSA facility, and are delivered to soldiers inside a "key fill device". At one point these made use of paper tape; now they live in a Windows CE (!) palmtop.

By my reckoning, a most logical way to generate keys for military field radios would be a portable hardware entropy source, which would be plugged into a gang of radios connected together for synchronization inside a shielded enclosure. Standing in a commander's tent. But instead, they choose to fly keys across the ocean... What might be the reason for taking this risk?

In his encyclopedic work, "Applied Cryptography" (1996) Bruce Schneier mentions "GOST", a Soviet block cipher having a curious design detail. The GOST specification did not specify a fixed set of values for the cipher's S-boxes. From this, one could infer that certain GOST users (perhaps the less loyal among the Warsaw Pact nations) were given weakened S-boxes so that they could be spied upon at the KGB's leisure. This, however, is a rather ham-handed approach to back-dooring a cipher, and American mathematicians surely conceived of something more subtle.

Some ciphers are known to possess "weak keys" - that is to say, a certain subset of the possible keyspace will result in ciphertext which can be cracked with considerably less effort than using plain brute force. Let's carry on with the "folk theory" and assume that NSA experts know of a class of weak AES keys, while having crafted Suite A algorithms which have strictly "linear" keyspaces. Yet AES is approved for certain "Top Secret" applications not involving radio communication. Given that all such applications use NSA-supplied key material, they would surely take care to supply only strong keys - or else, in the "KGB scenario," could easily supply weak keys to any public servant suspected of disloyalty. Which would leave the purpose of Suite A a mystery. The folk theory holds that NSA engages in mere "security by obscurity," hiding the proprietary ciphers in an effort to keep the public from discovering weaknesses. This sounds reasonable until you consider the official seal of approval on AES for "Top Secret" disks; while military radios are required to use Suite A ciphers.

So here comes the crackpot hypothesis, which resolves the apparent contradiction:

Suite A ciphers slowly leak keys.

NATO military doctrine famously allowed for the possibility that Soviet forces would overrun Western Europe, making liberal use of captured supplies of every kind. Even though such an invasion never took place, quite a few examples of American cipher equipment have been taken as spoils by various armies. Consider, for instance, the famous USS Pueblo: an American spy vessel taken prisoner by North Korea in 1968 - with a complete set of cipher machines, which the crew did not have a chance to throw overboard. Or the countless radios captured in Vietnam.

My theory: cryptographic equipment used by NATO armies leaks key bits into ciphertext. Slowly and subtly. Such that routine key swaps, at the rate supplied by the high command, prevent an enemy from gathering a complete key, even if he knew how. But if said enemy were to capture (and perhaps clone) NATO equipment, and take to using Suite A ciphers himself, he would begin to leak secrets meaningfully and continuously. It is also conceivable that NSA can supply keys which result in varying leakage rates, as appropriate to a particular military situation. And it is by no means certain that countries other than the U.S. possess the secret of extracting "dripping" keys.