By Neal Koblitz

From the reports: "This is a textbook in cryptography with emphasis on algebraic tools. it really is supported by way of many routines (with solutions) making it applicable for a path in arithmetic or computing device technological know-how. [...] total, this is often a great expository textual content, and may be very precious to either the coed and researcher." Mathematical stories

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**Example text**

Thus, the total number of bit operations required to do all of the necessary divisions is O(k / l) O(kl) = 0(k 2 ). This turns out to be the same answer as in Example 3. 1 . That is, our estimate for the conversion time does not depend upon the base to which we're converting (no matter how large it may be). This is because the greater time required to find each digit is offset by the fact that there are fewer digits to be found. · Example 3. 3. Estimate the time required to compute n ! Solution.

Suppose that P1 is the problem INPUT: Two vectors in 3-dimensional space. QUESTION: Are they proportional? Suppose that P2 is the problem INPUT: Two pairs of (non-proportional) vectors in 3-dimensional space. QUESTION: Do both pairs of vectors span the same plane? Show that P2 reduces to P 1 by constructing a reduction of instances of one problem to instances of the other. 8. Let Pt be the problem INPUT: A polynomial p(X) with integer coefficients. QUESTION: Is there any interval of the real number line on which p(X) decreases?

I, and so on, until N has been written as a product of prime powers. Once we have the prime factorization of N, we can immediately determine whether or not N has a factor in the interval [ 2 k ] . Namely, the answer to this question is "yes" if and only if the smallest prime divisor of N is in that interval. Conversely, suppose that we have an algorithm to do the decision problem. In that case we can use the method of "20 questions" (also called binary search) in order to zero in on the exact value of a factor, thereby solving the Integer Factorization search problem.