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The Code War This article was published in 2003 and has not been updated or revised. BEYONDBEYOND DISCOVERYDISCOVERY® THE PATH FROM RESEARCH TO HUMAN BENEFIT THE CODE WAR efore the widespread use of computers and the about 25 years ago. The security of many public-key sys- Internet, cryptography—from the Greek kryptos tems (there are several kinds) is explicitly based on a B (hidden) and graphein (writing)—was largely long-standing challenge in the branch of mathematics the domain of the military or diplomats. Indeed, the known as number theory—the study of the properties earliest recorded instance of encryption dates to about and patterns of integers, whole numbers such as -2, -1, 400 BC, when the Spartans used a device called a scytale 0, 1, 2,…100,… Number theory has for centuries been to send coded messages between military commanders. widely regarded as the purest of the pure sciences, but in A strip of parchment or leather was wrapped spirally recent years it has found many applications. around a baton or staff of a certain diameter. The Number theory has played an essential role in the sender wrote the message down the length of the staff, development of public-key cryptography. Without the and then unwrapped the parchment, effectively scram- basic inquiry carried out by early theorists, today’s com- bling the order of the letters. To decode the message, the puter transactions would be easy pickings for would-be recipient had to wrap the parchment around a staff of thieves and swindlers. The number theorists’ challenge the same diameter, whereupon the transposed letters to potential intruders is this: Given a (very, very large) returned to their original order. number obtained by multiplying two other numbers, In today’s information age, we make use of data find the two (very large) numbers that were multiplied scrambling whenever we use a password to check e-mail, to produce it. These numbers can be thought of as the withdraw money from an automated bank teller keys that lock and unlock the encryption code. The task machine, make a cellular phone call, or charge a pur- of finding these keys is so difficult that the snoops must chase over the Internet. We rely on encryption to ensure either be bizarrely lucky (as lucky, say, as one person the validity of our financial transactions, prove our winning 20 state lotteries simultaneously) or they must identity, and safeguard our solve a problem that has privacy. Although some stumped the smartest people people hesitate to conduct in the world for more than business over the Internet, 2000 years. Every time we most of us engage in online conduct an encrypted transactions with confi- transaction, we’re betting dence that encryption pro- that the snoops will lose. tects our activities. This faith is generally well founded. Many of the new methods of encrypting Modern e-commerce is pro- and decrypting information tected by reliable encryption involve public-key cryptogra- techniques. Photo ©PhotoDisc phy, which was invented This article was published in 2003 and has not been updated or revised. Plaintext: THE BUTCHER THE BAKER AND THE CANDLESTICK MAKER Early Ciphers Key: BIG BIGBIGB IGB IGBIG BIG BIG BIGBIGBIGBI GBIGB Encryption’s history has been one of unceasing efforts to devise uncrackable codes—and equally Ciphertext: UPK CCZDPKS BNF JGLMX BVJ unceasing efforts to crack them. Early ciphers were UPK DITETKTBODS SBSKS relatively simple systems, easy for both sender and Knowing that “BIG” was the key, the recipient receiver to use. Julius Caesar, for instance, encoded could easily decipher the message by shifting its letters messages with a “substitution cipher” in which each back the corresponding amounts. letter is replaced by the third letter after it in the For many years Vigenère’s cipher was considered alphabet: A is replaced by D, B by E, etc. At the end unbreakable, but Charles Babbage, an independently of the alphabet, the pattern wraps around to the wealthy Englishman known mostly for his pioneer- beginning: X becomes A, Y becomes B, and Z ing work in computer science, showed in the 1850s becomes C. Unfortunately, simplicity of use is a dou- that it was not so. Babbage hacked the system by ble-edged sword: The ciphertext thus encoded is looking for repeated strings of letters. Of course, highly susceptible to being decoded. Caesar’s cipher the strength of Vigenère’s cipher was supposed to can be cracked simply by moving each letter in the be that it encoded letters differently in different encoded message back three spaces in the alphabet. places. The first “THE” in the message above is More sophisticated substitution ciphers, in which the rendered as “UPK” and the second as “BNF”. alphabet is thoroughly scrambled, are nevertheless Also, the two “AKER”s code differently. But the easy enough for amateurs to break, as fans of the first and third “THE”s both code as “UPK.” The “Cryptoquote” puzzles in today’s magazines and “T” in the first “THE” is coded with a “B,” and so newspapers can attest. In any sufficiently long pas- is the “T” in the third “THE.” This happens sage of English text, the most common letter is usual- because the third “THE” begins 21 letters after the ly “E,” the second most common is “T,” and a three- first “THE”; hence the 3-letter keyword “BIG” has letter word that appears repeatedly is probably cycled around 7 times and is back to the beginning “THE.” By applying this type of “frequency analy- again. sis,” an eavesdropper can easily guess which letters in In any message that is much longer than the key, the ciphertext represent “E,” “T” and so on. some repeats of this sort are bound to occur. How Over the years, people who wanted greater secre- would an eavesdropper exploit this fact? If, say, the cy came up with more elaborate coding schemes. ciphertext “UPK” appeared twice, 21 letters apart, In the 1500s, Blaise de Vigenère, a French diplomat, then he could deduce that 21 was probably a multiple invented a method for encrypting different letters in of the keyword’s length. Or to put it another way, a message with different ciphers. Thus, an “E” in the number of letters in the keyword was a divisor of one position might be coded as “M,” while an “E” 21. (A divisor or factor of a number is a number that in another position might be coded as “K,” thereby goes into it with no remainder. The divisors of 21 are foiling anyone attempting to decode the message 1, 3, 7, and 21.) using frequency analysis. Given enough clues of this sort, an eavesdropper In the Vigenère cipher, the sender and recipient could pin down the exact length of the keyword. had to agree on a keyword (or perhaps a literary pas- Once he knew the length, he could do ordinary fre- sage) whose letters told them how far forward or quency analysis to decode the message. Notice that backward to shift the alphabet for every letter in the the math comes first: The eavesdropper figures out message. If the keyword “BIG” was used for example, the length of the keyword before even attempting to the sender would code the message in sets of three let- figure out what its letters are. ters. The first letter of the first trio would need to be Babbage’s ingenious technique broke new ground shifted forward by one (since “B” is one letter after in cryptography by introducing mathematical tools to “A”), the second letter would need to be shifted for- a subject that previously had seemed to be about ward by eight (“I” is eight letters after “A”), and the words. Even if an encryption system does not use third letter would need to be shifted forward by six mathematics explicitly, its hidden patterns can often (“G” is six letters after “A”). After that, the pattern be teased out that way. Mathematics is, after all, the would repeat itself as in the following example: science of patterns. 2 BEYOND DISCOVERY This article was published in 2003 and has not been updated or revised. to be sent. This might just be feasible for messages to a single spy, but it would never be practical for The Enigma Challenge widespread military or commercial use. The “key distribution problem,” as this difficulty is known, Although perhaps not fully appreciated, would remain an obstacle until the latter part of the mathematical decryption techniques made a huge twentieth century, when mathematics once again contribution to the Allied victory in World War II. came to the rescue. In that war, Germany encrypted most of its military transmissions with a machine called “Enigma.” Part electrical, part mechanical, it was like a combination lock with more than 1023 possible combinations. In Cryptography We Trust (For comparison, this is roughly the number of tablespoons of water in all the world’s oceans.) All encryption systems invented before 1970 had Moreover, the Germans changed the combination one thing in common: They were symmetric. In every day—sometimes several times a day. other words, the keys for encryption and decryption Recipients of the transmissions needed to possess were the same, so a person in possession of the key not only a duplicate Enigma machine, but also to could either send or receive messages. But in the know the correct combination. early 1970s, Whitfield Diffie of Stanford University If the Allies had had to rely solely on frequency realized that for some applications this two-way capa- analysis or trial and error, they would still be hunting bility was superfluous.
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