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Unformatted text preview: 6.841 Advanced Complexity Theory April 1, 2009 Lecture 15 Lecturer: Madhu Sudan Scribe: Rotem Oshman 1 Today’s topics • Private coins ≡ public coins (that is, IP[ k ] ≡ AM) – GoldwasserSipser approximate counting protocol • Towards protocols for the Permanent, in the goal of showing that #P ⊆ IP. 2 Review of last lecture 2.1 Graph NonIsomorphism The graph nonisomorphism problem is to decide the language GNI = { ( G ,G 1 )  G negationslash∼ G 1 } . Last lecture we saw a privatecoin 2round protocol for GNI: • The Verifier chooses a permutation π ∈ R S n , where n is the number of nodes in the graph, and a bit b ∈ R { , 1 } . It sends H = π ( G b ) to the Prover. • The Prover returns with a bit b prime . • The Verifier accepts if b = b prime . If G ∼ G 1 , then the bit b is independent of H , which means that the Prover is essentially guessing b prime and has a probability of 1 / 2 of being correct. If G negationslash∼ G 1 , then H identifies b uniquely and the Prover will always be correct. 2.2 Kilian’s protocol: IP[ k ] ⊆ AM[poly] Last time we saw a publiccoin protocol through which the Prover could convince the Verifier that there are “many” coin tosses that would have made the Verifier accept in the original privatecoin protocol. Kilian’s protocol is public coin and has completeness of 1, but the number of rounds in the protocol depends on the number of random coins used in the original privatecoin protocol. For example, the publiccoin protocol we would get for GNI would have O ( n log n ) rounds instead of 2. 151 3 The GoldwasserSipser Protocol Let S ⊆ { , 1 } n be a set, such that membership in S is verifiable in AM. We are interested in solving the promise problem given by Π YES = { S :  S  ≥ f ( n ) } Π NO = braceleftbigg S :  S  < f ( n ) 10 n 2 bracerightbigg 3.1 Initial attempt Suppose that f ( n ) is “very large”: f ( n ) ≈ 2 n (actually a little less than that). In this case, for YES instances we have  S  { , 1 } n  ≈ 1, and for NO instances we have  S  { , 1 } n  < ∼ 1 n 2 . In other words, when we choose a random string, for YES instances our chance of hitting a member of...
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 Spring '09
 MadhuSudan
 ObjectOriented Programming, hash function, Cryptographic hash function, verifier, prover

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