Unformatted text preview: a key, k, and in polynomial time runs the encryption algorithm on inputs X and k and checks whether the result is equal to C. This is important theoretically, because it puts an upper bound on the complexity of cryptanalysis for these algorithms. In practice, of course, it is a deterministic polynomial-time algorithm that the cryptanalyst seeks. Furthermore, this argument is not applicable to all classes of ciphers; in particular, it is not applicable to one-time pads—for any C, there are many X, k pairs that yield C when run through the encryption algorithm, but most of these X s are nonsense, not legitimate plaintexts. Figure 11.1 Complexity classes. The class NP includes the class P, because any problem solvable in polynomial time on a deterministic Turing machine is also solvable in polynomial time on a nondeterministic Turing machine; the guessing stage can simply be omitted. If all NP problems are solvable in polynomial time on a deterministic machine, then P = NP. Although it seems obvious that some NP problems are much harder than others (a brute-force attack against an encryption algorithm versus encrypting a random block of plaintext), it has never been proven that P ` NP (or that P = NP). However, most people working in complexity theory believe that they are unequal. Stranger still, specific problems in NP can be proven to be as difficult as any problem in the class. Steven Cook  proved that the Satisfiability problem (given a propositional Boolean formula, is there a way to assign truth values to the variables that makes the formula true?) is NP-complete . This means that, if Satisfiability is solvable in polynomial time, then P = NP. Conversely, if any problem in NP can be proven not to have a deterministic polynomial-time algorithm, the proof will show that Satisfiability does not have a deterministic polynomial-time algorithm either. No problem is harder than Satisfiability in NP. Since Cook’s seminal paper was published, a huge number of problems have been shown to be equivalent to Satisfiability; hundreds are listed in , and some examples follow. By equivalent, I m...
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This note was uploaded on 10/18/2010 for the course MATH CS 301 taught by Professor Aliulger during the Fall '10 term at Koç University.
- Fall '10