By Paul Hoffman
An creation to the delights and demanding situations of contemporary arithmetic.
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Additional resources for Archimedes' Revenge: The Joys and Perils of Mathematics
A person who is able to encode messages need not be able to decode them. To encode a message, he need know only the 200-digit product. But to decode the message, he has to know the two prime divisors; knowledge of the product isn't enough. Such a code is called public key cryptography because it can be used in a highly public way. If I want to receive secret messages, I simply publish the 200-digit number (and an explanation of how it's used for encryption). Then anyone who wishes can send me a coded message.
Stripped to its mathematical essentials, the problem so far is to solve seven equations that involve eight unknowns (four groups of bulls specified by color and four groups of cows of corresponding color). It turns out that these equations are not hard to solve. Indeed, they admit infinitely many solutions, the smallest involving a total herd of 50,389,082 cattle, a number that could comfortably graze on Sicily's 6,358,400 acres. Archimedes, however, did not stop there. He made the problem much more difficult by imposing two additional constraints on the number of bulls: 30 ARCHIMEDES' REVENGE 8.
Remarkably, almost two thousand years later, the Confederate generals A. S. Johnson and Pierre Beauregard, resurrected this simple cipher during the Battle of Shiloh. A cipher found in the Old Testament is just as simple. In Jeremiah (25:26 and 51:41), the prophet wrote Sheshach for Babel. The second letter of the Hebrew alphabet (b) was replaced by the second-to-last letter (sh), and 48 The Cryptic Case of a Swarthy Stranger 49 the twelfth letter (I) was replaced by the twelfth-to-last letter (ch).
Archimedes' Revenge: The Joys and Perils of Mathematics by Paul Hoffman