This wild menagerie makes our blunt-toothed modern housekeys look rather tame. And indeed, Matt Blaze, a colleague in computer security, has shown that locks and keys today are badly under-engineered.

Grand Vizier Sissa Ben Dahir’s wheat order in fulfillment

One imagines that the king agreed happily. The vizier, though, was laughing up his sleeve. Concealed in his humble request was the explosive phenomenon of exponential growth. On the tenth square of the board would have been deposited more than one thousand grains, on the twentieth, over a million. The number of grains on the last square would have been 9,223,372,036,854,775,808. (That is, 2⁶³.) At 7,000 grains per pound, this quantity of wheat would have exceed 650,000,000,000 tons–about one thousand times the annual worldwide production today!

What does this have to do with secret codes? Exponential growth is the phenomenon that keeps code-breakers at bay and makes encryption practical.

The most popular encryption system until recently was something called DES (Data Encryption Standard). It uses secret keys consisting of 64 bits. In binary, a DES key looks like this:

1001001001010111101011110101000101010001101010001010101001010110.

How many possible DES keys are there, i.e., how many possible ways to write out 64 bits? There are 2⁶⁴, that is, the number 2 doubled 64 times. Now, the straightforward way to break an encryption system with 64-bit keys is to try every key until the correct one is found. On average, this requires 2⁶³ tries. (It suffices to try half of all possible keys on average to hit the right one, and 2⁶³ = 2⁶⁴ / 2). Recall that 2⁶³ is exactly the number of grains of wheat on that last square of Ben Dahir’s chessboard.

The astonishing thing is that computers today are capable of handling computations of this size. In fact, in the most prominent public key-cracking contest (run by RSA Laboratories), the largest “cracked” key was 64 bits in length. It took a large network of computers nearly five years. Today it would require substantially less expensive computing resources.

Today, the key length of secret-key encryption schemes-such as the Advanced Encryption Standard (AES), a successor to DES-runs up to 256 bits. That’s far beyond the reach of all of any computer today. Cracking a 256-bit key isn’t four times as hard as cracking a 64-bit key, it’s 2¹⁹² times as hard. To put this in perspective, 2²⁵⁶ far exceeds the number of atoms in the Sun–not to mention all of the planets in the Solar System. That’s why cryptographic keys can provide so much power in such a small compass.

As for Sissa Ben Dahir, the tale does not relate whether King Shirham’s humor operated on the cosmic scale required to pardon his Grand Vizier’s joke with good grace.

• For technical reasons, the effective key length of DES is only 56 bits. A special-purpose computer capable of cracking DES keys in a matter of days was demonstrated in 1998. The inadequacy of DES for long-term cryptographic protection is now universally acknowledged, and security architects instead employ AES, typically with 128-bit keys.
• Encryption systems like RSA-so called “public-key” cryptosystems-require rather longer keys, due to the keys’ special algebraic structure.  For present needs, the most commonly recommended length for RSA keys is 2048 bits.

(Many thanks to Emerald Bay Books and to Lee Nagan, my publisher’s excellent, punctilious printer / typesetter.)

Here’s a 1620 edition of the works of Silius Italicus, a Roman poet. (A piquant aside: It may have belonged to a the Bishop of Metz, a bastard son of kind Henry IV, as I mention here.)  This is dense reading. An entire epic poem in less volume than a deck of cards. I’m content just to admire the binding.

Silius Italicus, On the Second Punic War, 1620 Dutch edition

Few people besides Juels could have written such a book, especially with such authenticity. Playing to his strengths as scientist, cryptography expert, and classically trained thinker, he delivers a tale that is compelling, intellectually stimulating, and destined to resonate with the technical community.

During my commute to work by train, I keep my head well buried in a book or technical article. While I imagine I have a forbidding look of concentration, and reticence is the rule on Boston trains, someone did once buttonhole me. I was penning notes in an article, when a woman leaned over and asked, “Is that a fountain pen?”

My fountain pen also instantly attracts other enthusiasts in business meetings. There’s fast camaraderie in that tiny, half-empty little boat in which users of archaic writing instruments bear against the tide of modernity. There’s no headway to be made against the flood of cheap pens, themselves being whelmed beneath a wave of typing, with video and other threats to text in general curling somewhere over the horizon.

In Tetraktys, Ambrose not only uses a fountain pen, but writes in the distinctive style of Oxbridge classicists (with touches like the occasional substitution of an epsilon for an `e’). When he comes upon similar handwriting, he knows that he’s found another member of his own, nearly extinct clan…

Each item in this menagerie is interesting enough to be the centerpiece of its own blog post. But there’s one brooch that I particularly like because within its contours lies an elegant little puzzle. This puzzle gives a flavor of how Ambrose reasons in Tetraktys.

The brooch, as you see, is shaped like a serpent; the handsome green is a patina that formed over the bronze. Richard Hattatt, a major collector, and writer of four books on ancient brooches, once owned it. On the right is the hand-drawn sketch he published in his Iron Age and Roman Brooches. Hattatt called the brooch a “freak.” It was originally circular or “penannular,” a common shape in antiquity—as with the brooches you see below, which Hattatt sketched on the same page as the “S-brooch.”

Many centuries ago, however, someone bent this brooch from its original, circular shape into a serpentine one. Who reworked it and why this person did so, we’ll never know.

But what we do know is that the S-brooch was refashioned in the ancient world, and not after it was unearthed by someone in the south of England. So here’s the puzzle: How do we know? Here’s one more picture, to help reveal the clue…

To learn the answer, click here!

Tetraktys mentions such a vase in the Boston Museum of Fine Arts. It depicts the Homeric heroes Achilles and Ajax taking a break from slaughtering Trojans to play a board game.

Bilingual vase--black-figure side (Lysippides)

Bilingual vase--red-figure side (Andokides)

Ambrose sees this vase as an emblem of the Greek awakening to philosophical and anatomical self-awareness. The Black Figure warriors are clumsy—half-iconic, half-representational. In Red Figure, these same two warriors become fluid, animate—a still frame from a story we feel as though we’re on the cusp of grasping or remembering. (Compare the eyes especially. In Black Figure, they’re hieroglyphics. In Red Figure, they’re windows on the soul.)

Why the difference? The main reason is technical. The Black Figure technique required that details be etched. The Red Figure technique allowed them instead to be painted, with vastly more delicate results.

The red-figure technique evolved around the year 530 B.C.—also the time of Pythagoras’ emigration to Kroton, where he would found his cult. It was just several years after Thespis invented tragedy as a dramatic form. This was the period in which the Greeks found the languages of self-expression and self-awareness—intellectual, visual, dramatic—that established them not just as progenitors of Western thought, but intelligible as teachers and masters even today.

"Pythagoras," the Musical Instrument

Between the tracks was a row of metal tubes, like a suspended battery of organ pipes, with hammers in between. The whole apparatus swayed, the hammers knocking gently against the tubes, producing an eerily tempered music…

Tetraktys,  p. 186

While not named explicitly in Tetraktys, these chimes in the Kendall / MIT subway station are a musical instrument called “Pythagoras.” They’re operated by a handle beside the passenger platform. At the moment that  Ambrose was passing, it seems that someone was swinging the handle…

]]> http://www.tetraktysnovel.com/?feed=rss2&p=815 0 http://www.tetraktysnovel.com/?p=1175 http://www.tetraktysnovel.com/?p=1175#comments Tue, 29 Sep 2009 22:35:53 +0000 Ari http://www.tetraktysnovel.com/?p=1175

Distinguishing authentic artifacts from forged or misattributed ones is an essential challenge for archaeologists and art historians. Scientific tests, though, often prove incomplete or inconclusive (or simply misleading). In such cases, the best line of defense against forged artifacts is an expert with a finely honed intuitive sense for authenticity—sometimes called a fakebuster. (In Tetraktys, Dr. Jerusalem is a consummate fakebuster.)

Can you use your fakebusting skills to figure out which image is real and which is fake?

Attic vase depicting Athena and Herakles, circa 520 B.C., Andokides painter

Click here for the answer!

Creative Problem Solving, Cousin to Brainstorming

But nothing of substance is shaken free from the bindweed of hierarchy and corporate oneupsmanship, the sterility of the room to eye and mind, the burden of respectful consensus. “Brainstorming” is cousin to “thinking outside the box”—another aspiration of boxed-in cubicle dwellers. And it’s handmaiden to “innovation,” the process by which a company or nation undergoes wholesale transformation.

It’s all well and good for me to rail against clichés.  So where do Promethean sparks of imagination come from—at least in my small world of computer-science research?

One colleague tells me that she can only think in contact with water. Her muse sings in the shower. Another can only think with his shoes off. (Perhaps his habit ensures that he’s surrounded only by devoted colleagues.) I myself think better amid the impersonal noise of a café, with the pressure to seal my mind off, than around comfortable and familiar distractions.

I’ve seen the best ideas flutter down at unpredictable times. One of my lab’s best known papers began with a colleague’s joke. We were discussing rumored plans by the European Union to embed microchips in Euro banknotes. “Wouldn’t it be funny,” this colleague quipped, “if Switzerland put chips in its banknotes that jammed Euros?” We figured out how to make this mischief work—and voila! A new privacy-enhancing chip design emerged.

Usually inventive ideas are like cats. They only come unbidden. For instance, there’s the famous story about Kekulé’s discovery of the atomic structure of benzene. A vision came to him spontaneously of snakes forming a circle by biting one another’s tail, revealing the molecule’s ringlike structure.

The predictive power of crowds and markets over lone individuals is well documented. Groups filter outliers: A boon for forecasting, but a bane for invention. I recently attended an event in which two groups “brainstormed” independently. Instead of doubling creative power, the presence of two groups halved it. Each group drew up a list of ideas. They decided that the valuable ore lay at the intersection of their lists. One person sensibly urged them to throw away precisely this fool’s gold—the ideas that were clearly in the air and therefore stale—and keep the outliers instead. But he was an outlier. They threw away his advice.

In Tetraktys, the NSA (National Security Agency) has the boldness to bring in a strange young man—Ambrose Jerusalem, the hero of the novel—to tackle an insoluable problem. The agency recognizes that it needs not a meeting of minds, not groupthink, but eccentricity.

In the midst of an economic recession in which governments, companies, and the media are all trumpeting invention and innovation as economic lifesavers, where should they turn? There are no formulas or sage pronouncements. Sometimes invention is a joke. Sometimes it’s a unbidden vision. It’s generally undemocratic (strangely, not unlike a corporation). If there’s a universal rule, as best I can gather, it’s that invention shuns crowds. And another:  If a spark descends from the heavens, the word “brainstorm” will almost certainly blow it out.

This blog is also posted on the EMC web  site.

Message in a Sheepskin

This book—of which you see a bit of the cover—is a sixteenth-century copy of Castiglione’s Il Cortegiano. It’s an imaginary dialogue set in the court of the Duke of Urbino that aims to outline the qualities of the ideal Renaissance courtier. Paper was once so expensive that it was regularly recycled into new books. Embedded in the thin vellum (sheepskin) cover of this book is a manuscript fragment. You can make out the beautiful calligraphic quality of the handwriting—but not the words.

I suppose I could try to decipher it. But sometimes the richest mystery is the one so fragmentary that the very questions it poses are inscrutable. Then it’s truly the stuff that dreams are made on.

Distinguishing authentic artifacts from forged or misattributed ones is an essential challenge for archaeologists and art historians. Scientific tests, though, often prove incomplete or inconclusive (or simply misleading). In such cases, the best line of defense against forged artifacts is an expert with a finely honed intuitive sense for authenticity—sometimes called a fakebuster. (In Tetraktys, Dr. Jerusalem is a consummate fakebuster.)

In homage to an important theme in Tetraktys, here is the first of what I plan to make a regular series of fakebusting puzzles.

Can you use your fakebusting skills to figure out which image is real and which is fake?

Greek diadem, circa 300 B.C.

Click here for the answer!