By Neal Koblitz

From the stories: "This is a textbook in cryptography with emphasis on algebraic tools. it really is supported via many workouts (with solutions) making it applicable for a direction in arithmetic or laptop technological know-how. [...] total, this can be a good expository textual content, and may be very important to either the scholar and researcher." Mathematical experiences

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Example 4. 7. Given an algorithm for the Integer Factorization decision problem, here is how we use it to factor 9 1 . First question to algorithm: Does 9 1 have a factor between 2 and 63? Answer: YES. Second question: Does 9 1 have a factor between 2 and 3 1 ? Answer: YES. Third question: Does 9 1 have a factor between 2 and 1 5 ? Answer: YES. Fourth question: Does 9 1 have a factor between 2 and 7? Answer: YES. §4. P, NP, and NP-Completeness 37 Fifth question: Does 9 1 have a factor between 2 and 3?

10. Arrange the following numbers in increasing order, if n is equal to the number of mosquitos in New Jersey: (a) the time required to solve a Chinese Remainder Theorem problem with ap­ proximately ln n congruences whose moduli satisfy n < m, < 2n . (b) the time required to find the value at n of a quintic polynomial whose coeffi­ cients are 20-bit integers; (c) the time required to convert n (which is initially written in binary) to hexade­ cimal (base 1 6); (d) the time required to find the least nonnegative residue of m!

8(n In n) == . Exercises for § 2 1 . Suppose that a k-bit integer a is divided by an l-bit integer b (where l :::; k) to get a quotient q and a remainder r: a== qb + r , O:Sr

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