Unformatted text preview: . (Excellent introductions to this topic are [600, 211, 1226]; see also [1096, 27, 739].) The theory looks at the minimum time and space required to solve the hardest instance of a problem on a theoretical computer known as a Turing machine . A Turing machine is a finite-state machine with an infinite read-write memory tape. It turns out that a Turing machine is a realistic model of computation. Problems that can be solved with polynomial-time algorithms are called tractable, because they can usually be solved in a reasonable amount of time for reasonable-sized inputs. (The exact definition of “reasonable” depends on the circumstance.) Problems that cannot be solved in polynomial time are called intractable, because calculating their solution quickly becomes infeasible. Intractable problems are sometimes just called hard. Problems that can only be solved with algorithms that are superpolynomial are computationally intractable, even for relatively small values of n. It gets worse. Alan Turing proved that some problems are undecidable . It is impossible to devise any algorithm to solve them, regardless of the algorithm’s time complexity. Problems can be divided into complexity classes, which depend on the complexity of their solutions. Figure 11.1 shows the more important complexity classes and their presumed relationships. (Unfortunately, not much about this material has been proved mathematically.) On the bottom, the class P consists of all problems that can be solved in polynomial time. The class NP consists of all problems that can be solved in polynomial time only on a nondeterministic Turing machine: a variant of a normal Turing machine that can make guesses. The machine guesses the solution to the problem—either by making “lucky guesses” or by trying all guesses in parallel—and checks its guess in polynomial time. NP ’s relevance to cryptography is this: Many symmetric algorithms and all public-key algorithms can be cracked in nondeterministic polynomial time. Given a ciphertext C, the cryptanalyst simply guesses a plaintext, X, and...
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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