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Chapter 1: Q20E (page 49)
Find the inverse of:.
Inverse of the given numbers is obtained.
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Prove that the grade-school multiplication algorithm (page 24), when applied to binary numbers, always gives the right answer.
The grade-school algorithm for multiplying two n-bit binary numbers x and y consist of addingtogethern copies of r, each appropriately left-shifted. Each copy, when shifted, is at most 2n bits long.
In this problem, we will examine a scheme for adding n binary numbers, each m bits long, using a circuit or a parallel architecture. The main parameter of interest in this question is therefore the depth of the circuit or the longest path from the input to the output of the circuit. This determines the total time taken for computing the function.
To add two m-bit binary numbers naively, we must wait for the carry bit from position i-1before we can figure out the ith bit of the answer. This leads to a circuit of depth. However, carry-lookahead circuits (seewikipedia.comif you want to know more about this) can add indepth.
Suppose you want to compute the nth Fibonacci number , modulo an integer . Can you find an efficient way to do this?
How many integers modulo have inverses?
Calculate using any method you choose. (Hint: 127 is prime.)
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