Unformatted text preview: miltonian Cycle problem—see Section 5.1.) — ThreeWay Marriage Problem. In a room are n men, n women, and n clergymen (priests, rabbis, whatever). There is also a list of acceptable marriages, which consists of one man, one woman, and one clergyman willing to officiate. Given this list of possible triples, is it possible to arrange n marriages such that everyone is either marrying one person or officiating at one marriage? — ThreeSatisfiability. There is a list of n logical statements, each with three variables. For example: if (x and y) then z, (x and w) or (not z), if ((not u and not x) or (z and (u or not x))) then (not z and u) or x), and so on. Is there a truth assignment for all the variables that satisfies all the statements? (This is a special case of the Satisfiability problem previously mentioned.) 11.3 Number Theory
This isn’t a book on number theory, so I’m just going to sketch a few ideas that apply to cryptography. If you want a detailed mathematical text on number theory, consult one of these books: [1430, 72, 1171, 12, 959, 681, 742, 420]. My two favorite books on the mathematics of finite fields are [971, 1042]. See also [88, 1157, 1158, 1060]. Modular Arithmetic
You all learned modular arithmetic in school; it was called “clock arithmetic.” Remember these word problems? If Mildred says she’ll be home by 10:00, and she’s 13 hours late, what time does she get home and for how many years does her father ground her? That’s arithmetic modulo 12. Twentythree modulo 12 equals 11. (10 + 13) mod 12 = 23 mod 12 = 11 mod 12 Another way of writing this is to say that 23 and 11 are equivalent, modulo 12: 23 a 11 (mod 12) Basically, a a b (mod n) if a = b + kn for some integer k. If a is nonnegative and b is between 0 and n, you can think of b as the remainder of a when divided by n. Sometimes, b is called the residue of a, modulo n. Sometimes a is called congruent to b, modulo n (the triple equals sign, a, denotes congruence). These are just different ways of saying the same thing. The set of integers from 0...
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 Fall '10
 ALIULGER
 Cryptography, Bruce Schneier, Applied Cryptography, EarthWeb, Search Search Tips

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