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hwsol1.5 - CSci 5403 Spring 2010 Homework 1.5 Sample...

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CSci 5403, Spring 2010 Homework 1.5 Sample Solution Ruoyu Sun 1. Question 5 (a) Prove that given a polynomial-time decision algorithm for fact we can design a polynomial-time factoring algorithm. Suppose Pfact is a polynomial-time decision algorithm for FACT. Given an integer N . We describe a polynomial-time algorithm to re- cover all of its prime factors. 1. Use binary search to find p 1 , which is the largest proper prime factor of N .(A proper factor of N is a factor of N that is not equal to N ). Such p 1 does not exist iff N is a prime. Note that h N, k i ∈ FACT N has a proper prime factor p k p 1 k . Call Pfact to determine whether h N, d N/ 2 ei ∈ FACT , i.e. whether d N/ 2 e ≤ p 1 < N . If d N/ 2 e ≤ p 1 < N , call Pfact ( h N, d 3 N/ 4 ei ) to determine whether d N/ 2 e ≤ p 1 < d 3 N/ 4 e or d 3 N/ 4 e ≤ p 1 < N ; else (1 p 1 < d N/ 2 e or p 1 does not exist), call Pfact ( h N, d N/ 4 ei ) to determine whether d N/ 4 e ≤ p 1 < d N/ 2 e . Continue doing so until we can determine the largest proper prime factor p 1
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