By Neal Koblitz
From the stories: "This is a textbook in cryptography with emphasis on algebraic equipment. it truly is supported by means of many workouts (with solutions) making it applicable for a path in arithmetic or laptop technological know-how. [...] total, this can be an exceptional expository textual content, and may be very important to either the scholar and researcher." Mathematical experiences
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9. The Traveling Salesrep decision problem is almost certainly not in P (we'll have more to say about that later). However, it is in NP. That is, suppose that an extraterrestrial being finds the most economical tour for the traveling salesrep, §4. P, NP, and NP-Completeness 39 and it turns out to have length less than or equal to k. She tells you that the answer is "yes", and then shows you the route, at which point you can rapidly verify that her "yes" answer is in fact correct. In the same way, one easily sees that the 3-Coloring decision problem is in NP.
1. If · · · In n lim - = 0 . 2. Let f(n) be the number of base-b digits in n, that is, the length of the number n when it is written to the base b. Here variable. Then f(n) = 1 + 1ogb n = 1 + [ ] b is constant, and n is our [�:� J , where [ ] denotes the greatest integer function. Since b (and hence ln b) is a constant, we conclude that f(n) = O(ln n). Roughly speaking, "the number-of digits function behaves like a logarithm". Example 1 . 3 (see Remark 2 above). This example shows the importance of being clear about what the variable is when one uses big-0, little-a, and asymptotic equality.
Roughly speaking, "the number-of digits function behaves like a logarithm". Example 1 . 3 (see Remark 2 above). This example shows the importance of being clear about what the variable is when one uses big-0, little-a, and asymptotic equality. We consider the sum of the first n positive integers raised to the k-th power: L�; 1 ik . If we are considering k to be constant and letting n get large, then we have that is, f ::=:: g with g(n) = nk+l j(k + 1 ). ) On the other hand, if we regard n as constant and k as the variable that gets large, then the statement is false.
Algebraic Aspects of Cryptography (Algorithms and Computation in Mathematics) by Neal Koblitz