Algebraic Aspects of Cryptography (Algorithms and by Neal Koblitz

By Neal Koblitz

From the experiences: "This is a textbook in cryptography with emphasis on algebraic equipment. it truly is supported by means of many workouts (with solutions) making it acceptable for a path in arithmetic or computing device technology. [...] total, this is often an exceptional expository textual content, and may be very precious to either the coed and researcher." Mathematical reports

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Extra info for Algebraic Aspects of Cryptography (Algorithms and Computation in Mathematics, Vol. 3)

Example text

Thus, P1 reduces to P2 . 6. A decision problem P in NP is said to be "NP-complete" if every other problem Q in NP can be reduced to P in polynomial time. To put it another way, if one had a polynomial time algorithm for an NP­ complete problem P, then one would also have polynomial time algorithms for all other NP problems Q. This would mean that P equals NP, and the P/NP conjecture would be false. For this reason, no one is likely to come up with a polynomial time algorithm for any NP-complete problem.

Definition 4. 1 attempts to capture a class of problems that in practice can be solved rapidly. It is not a priori clear that P is the right class to take for this purpose. For instance, an algorithm with running time n 100 , where n is the input length, is slower than one with running time e0·000 1 n until n is greater than about ten million, even though the first algorithm is polynomial time and the second one is exponential time. 2 of Chapter 1 . However, the experience has been that if a problem of practical interest i s i n P, then there is an algorithm for it whose running time is bounded by a small power of the input length.

For example, in the Traveling Salesrep search problem we want a path of minimal length that passes through all the cities. ) There may be many different minimal tours. 4. An instance of a decision problem version of Integer Factorization is as follows: INPUT: Positive integers N and k. QUESTION: Does N have a factor M satisfying 2 :::; M :::; k? The problem of actually finding a nontrivial factor M of N is called the Integer Factorization search problem. 5. An instance of the Traveling Salesrep decision problem has the form INPUT: An integer m, a map from the set of pairs (i, j), 1 :::; i < j :::; m, to the natural numbers, and an integer k.

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