By Neal Koblitz

From the studies: "This is a textbook in cryptography with emphasis on algebraic equipment. it's supported by means of many workouts (with solutions) making it acceptable for a direction in arithmetic or computing device technology. [...] total, this is often a superb expository textual content, and should be very important to either the scholar and researcher." Mathematical experiences

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A) INPUT: A positive integer N . QUESTION: Is 1r(N) an even number? Recall that 1r(N) denotes the number of primes less than or equal to N. (b) INPUT: A list of cities and distances between any two cities, and an integer k. QUESTION: Do all tours that pass through all of the cities have length greater than k? (c) INPUT: A graph and an integer k. QUESTION: Does the graph have k or more different 3-colorings? 12. 2) is found for a n NP complete problem, then any NP problem has a subexponential time algorithm.

M n . ) If we multiply together all n inequalities i= then we find that < 2nk-n - II m ' < 1, . . , n , 2nk , so that the length of the product is between nk - (n - 1 ) and nk. 23 §2. Length of Numbers Usually we're not interested in the exact length, but only in a bound for the length. In that case we can say simply that multiplying together n numbers of length at most k results in a number of length at most nk. A similar discussion applies to subtraction and division (see Exercise 1 below).

There are two other commonly used symbols that are closely related to big0: fl and e. The notation f = fl(g) means exactly the same thing as g = O(f). The notation f = 8(g) means that both f = O(g) and f = fl(g); in other words, there exist positive constants C 1 , C2, and n0 such that C1g(n) S f(n) S C2g(n) for n 2: no. 20 Chapter 2. Complexity of Computations 9. These symbols are often used in the middle of formulas rather than right after an equal sign. For example, if we say that a function is n°(lnIn n) , we mean that there exists a constant C such that for n 2: n0 the function is ::; n° InInn .