By Neal Koblitz
From the reports: "This is a textbook in cryptography with emphasis on algebraic tools. it's supported by means of many routines (with solutions) making it acceptable for a direction in arithmetic or machine technology. [...] total, this can be an exceptional expository textual content, and may be very necessary to either the coed and researcher." Mathematical stories
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Additional info for Algebraic Aspects of Cryptography (Algorithms and Computation in Mathematics)
To answer this question we have to think about how adding and multi plying affect the length of numbers. It is easy to see that the sum of two numbers has length either equal to the length of the larger number or else equal to 1 plus the length of the larger number. If we add n numbers each of length at most k - that is, each less than 2k then the sum will be less than n2k. Hence, the length of the sum will be at most k + length(n). To deal with multiplication, we use the fact that a number m of length k satisfies: 2k-I ::; m < 2k.
There is no known algorithm that can determine primality of the m-th Fermat number in time that is bounded by a fixed power of log2 m. §3. Time Estimates 31 One class o f algorithms that are very far from polynomial time i s the class of exponential time algorithms. These have a time estimate of the form O ( eck) , where c is a constant. Here k is the total binary length of the integers to which the algorithm is being applied. For example, the "trial division" algorithm for factoring an integer n can easily be shown to take time O(n l /Z+< ) (where E: > 0 can be arbitrarily small).
It is almost certain that NP is a much bigger class of problems than P, but this has not been proved. The claim that P#NP is the most famous conjecture in computer science. 3. Let P 1 and P2 be two decision problems. We say that P 1 reduces to P2 (more precisely, reduces to P2 in polynomial time) if there exists an algorithm that is polynomial time as a function of the input length of P 1 and that, given any instance P1 of P1 , constructs an instance P2 of P2 such that the answer for P1 is the same as the answer for P2 .