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Enigma code setting, for example, or the location of a submarine. He felt that something here needed measuring, mathematically. It was not the probability, which would traditionally be expressed as an odds ratio (such as three to two) or a number from zero to one (such as 0.6, or 60 percent). Rather, Turing cared about the data that changed the probability: a probability factor, something like the weight of evidence. He invented a unit he named a “ban.” He found it convenient to use a logarithmic scale, so that bans would be added rather than multiplied. With a base of ten, a ban was the weight of evidence needed to make a fact ten times as likely. For more fine-grained measurement there were “decibans” and “centibans.”

Shannon had a notion along similar lines.

Working in the old West Village headquarters, he developed theoretical ideas about cryptography that helped him focus the dream he had intimated to Vannevar Bush: his “analysis of some of the fundamental properties of general systems for the transmission of intelligence.” He followed parallel tracks all during the war, showing his supervisors the cryptography work and concealing the rest. Concealment was the order of the day. In the realm of pure mathematics, Shannon treated some of the same ciphering systems that Turing was attacking with real intercepts and brute hardware—for example, the specific question of the safety of Vigenère cryptograms when “the enemy knows the system being used.”♦ (The Germans were using just such cryptograms, and the British were the enemy who knew the system.) Shannon was looking at the most general cases, all involving, as he put it, “discrete information.” That meant sequences of symbols, chosen from a finite set, mainly letters of the alphabet but also words of a language and even “quantized speech,” voice signals broken into packets with different amplitude levels. To conceal these meant substituting wrong symbols for the right ones, according to some systematic procedure in which a key is known to the receiver of the message, who can use it to reverse the substitutions. A secure system works even when the enemy knows the procedure, as long as the key remains secret.

The code breakers see a stream of data that looks like junk. They want to find the real signal. “From the point of view of the cryptanalyst,” Shannon noted, “a secrecy system is almost identical with a noisy communication system.”♦ (He completed his report, “A Mathematical Theory of Cryptography,” in 1945; it was immediately classified.) The data stream is meant to look stochastic, or random, but of course it is not: if it were truly random the signal would be lost. The cipher must transform a patterned thing, ordinary language, into something apparently without pattern. But pattern is surprisingly persistent. To analyze and categorize the transformations of ciphering, Shannon had to understand the patterns of language in a way that scholars—linguists, for example—had never done before. Linguists had, however, begun to focus their discipline on structure in language—system to be found amid the vague billowing shapes and sounds. The linguist Edward Sapir wrote of “symbolic atoms” formed by a language’s underlying phonetic patterns. “The mere sounds of speech,” he wrote in 1921, “are not the essential fact of language, which lies rather in the classification, in the formal patterning.… Language, as a structure, is on its inner face the mold of thought.”♦ Mold of thought was exquisite. Shannon, however, needed to view language in terms more tangible and countable.

Pattern, as he saw it, equals redundancy. In ordinary language, redundancy serves as an aid to understanding. In cryptanalysis, that same redundancy is the Achilles’ heel. Where is this redundancy? As a simple example in English, wherever the letter q appears, the u that follows is redundant. (Or almost—it would be entirely redundant were it not for rare borrowed items like qin and Qatar.) After q, a u is expected. There is no surprise. It contributes no information. After the letter t, an h has a certain amount