lect10 - 6.841 Advanced Complexity Theory March 9 2009...

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6.841 Advanced Complexity Theory March 9, 2009 Lecture 10 Lecturer: Madhu Sudan Scribe: Asilata Bapat Meeting to talk about final projects on Wednesday, 11 March 2009, from 5pm to 7pm. Location: TBA. Includes food. 1 Overview of today’s lecture Randomized computation. Complexity classes: RP, coRP, BPP, ZPP. Basic properties of these complexity classes. So far, we know that P is a computationally feasible class. We could try and expand this notion, and then study where the expanded notions lie in relation with P, NP, etc. 2 Examples of problems which have randomized algorithms 1. Problem: Find an n -bit prime. Input: N N , N > 3 such that 2 n - 1 < N 2 n . Output: A prime p , such that N p < 2 N . A polynomial-time algorithm for this problem is as follows. This algorithm is randomized. No deter- ministic algorithm is known. 1: loop { n times } 2: Pick k randomly and uniformly between N (inclusive) and 2 N (exclusive). 3: if k is prime then 4: return k 5: else 6: continue loop. 7: return a random value between N (inclusive) and 2 N (exclusive). A sketch of the proof of correctness of this algorithm is as follows. Sketch of Proof First we observe that we can always find such a prime. This is the following lemma, which we state without proof. Lemma 2.1 (Bertrand’s Postulate) If n is a natural number greater than 3, then there exists a prime number p such that n p < 2 n . Apart from Lemma 2.1 the algorithm depends on the Prime Number Theorem, which we state without proof. Theorem 2.2 (Prime Number Theorem) For any real number x , let π ( x ) be the number of primes less than or equal to x . Then, lim x →∞ π ( x ) x/ ln x = 1 . 10-1
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In this context, the Prime Number Theorem implies that the number of primes between N and 2 N is about 2 N n +1 - N n , which is approximately N n . So the probability of k being prime is approximately 1 n . Since the algorithm is repeated n times, the probability of it not returning a prime is approximately n - 1 n n = 1 - 1 n n 1 e . We will see later that this error probability is small enough for our purposes. 2. Problem: Square-root modulo primes. Input: An n -bit long prime p , an integer a such that 0 a p . Output: An integer α such that α 2 = a (mod p ). Berlekamp, and later Adleman, Manders and Miller, gave randomized polynomial-time algorithms to solve this problem. A deterministic polynomial-time algorithm is not known. A randomized polynomial-time algorithm to solve this problem is as follows. First, β is chosen randomly and uniformly from [ p - 1]. If we can solve the equation γ 2 = β 2 α (mod p ) for γ , then α = β/γ . For this, θ is picked randomly and uniformly from [ p - 1], and the following equation can be solved, ( x - θ ) 2 = β 2 α (mod p ). To do this, we use (without proof) the fact that gcd( x 2 - 2 + θ 2 - β 2 α, x p - 1 2 - 1) is linear in x with probability 1/2.
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