Unformatted text preview: ly, algorithms are classified according to their time or space complexities. An algorithm is constant if its complexity is independent of n: O(1). An algorithm is linear, if its time complexity is O(n). Algorithms can also be quadratic, cubic, and so on. All these algorithms are polynomial; their complexity is O(nm), when m is a constant. The class of algorithms that have a polynomial time complexity are called polynomial-time algorithms. Algorithms whose complexities are O(t f(n)), where t is a constant greater than 1 and f (n) is some polynomial function of n, are called exponential. The subset of exponential algorithms whose complexities are O(c f(n)), where c is a constant and f (n) is more than constant but less than linear, is called superpolynomial. Ideally, a cryptographer would like to be able to say that the best algorithm to break this encryption algorithm is of exponential-time complexity. In practice, the strongest statements that can be made, given the current state of the art of computational complexity theory, are of the form “all known cracking algorithms for this cryptosystem are of superpolynomial-time complexity.” That is, the cracking algorithms that we know are of superpolynomial-time complexity, but it is not yet possible to prove that no polynomial-time cracking algorithm could ever be discovered. Advances in computational complexity may some day make it possible to design algorithms for which the existence of polynomial-time cracking algorithms can be ruled out with mathematical certainty. As n grows, the time complexity of an algorithm can make an enormous difference in whether the algorithm is practical. Table 11.2 shows the running times for different algorithm classes in which n equals one million. The table ignores constants, but also shows why ignoring constants is reasonable. Table 11.2 Running Times of Different Classes of Algorithms # of Operations for n = 106 1 106 1012 1018 10301,030 Time at 106 O/S 1 µsec. 1 sec. 11.6 days 32, 000 yrs. 10301,006 times the age of the universe Class Constant Linear Quadratic Cubic Exponential Complexity O(1) O(n) O(n2) O(n3) O(2n) A...
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- Fall '10
- Cryptography, Bruce Schneier, Applied Cryptography, EarthWeb, Search Search Tips