lect10

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 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 xθ + θ 2- β 2 α,x p- 1 2- 1) is linear in x with probability 1/2....
View Full Document

This note was uploaded on 04/02/2010 for the course CS 6.841 taught by Professor Madhusudan during the Spring '09 term at MIT.

Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online