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applied cryptography - protocols, algorithms, and source code in c

5k operations if k is the length of the number x in

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Unformatted text preview: miltonian Cycle problem—see Section 5.1.) — Three-Way 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? — Three-Satisfiability. 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. Twenty-three 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 non-negative 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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