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Chapter 1: Q21E (page 49)
How many integers modulo have inverses?
The number of integers coprime to are 1210
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Show that if and if Divides
Letdenote the set. For each of the following families of hash functions, say whether or not it is universal, and determine how many random bits are needed to choose a function from the family.
(a) , whereis a fixed prime and
Notice that each of these functions has signaturethat is, it maps a pair of integers into a single integer in.
(b) is as before, except that nowis some fixed power of.
(c) is the set of all functions.
Consider the problem of computing x y for given integers x and y: we want the whole answer, not modulo a third integer. We know two algorithms for doing this: the iterative algorithm which performs y − 1 multiplications by x; and the recursive algorithm based on the binary expansion of y. Compare the time requirements of these two algorithms, assuming that the time to multiply an n-bit number by an m-bit number is O(mn).
Unlike a decreasing geometric series, the sum of the diverges; that is,
It turns out that, for large n , the sum of the first n terms of this series can be well approximated as
where is natural logarithm (log base ) and y is a particular constant . Show that
(Hint: To show an upper bound, decrease each denominator to the next power of two. For a lower bound, increase each denominator to the next power of 2 .)
Is divisible by ?
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