# Lec10 - Zero-Knowledge Proofs Sheng Zhong 1 Graph...

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1 Zero-Knowledge Proofs Sheng Zhong

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2 Graph Isomorphism Suppose G1 and G2 are two graphs known to both you and me. Furthermore, I know they are isomorphic. But you don’t know and can’t figure it out. How can I prove to you that they are isomorphic? We need to interact with each other to achieve this goal.
3 Interactive Proof More generally, suppose L is a language and x is in L. (Both of us know x.) How can I prove this fact to you? An interactive proof system (P, V) is two (interactive) PPT algorithms P and V such that P and V have a common input x. (P/V may have its own private input with length poly(| x|)) Each algorithm runs interleavingly and sends message to the other. Finally, V outputs “accept” or “reject”.

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4 Completeness and Soundness The interactive proof system is complete for language L if for all x in L, the output of (P,V) is accept with high probability. The interactive proof system is sound for language L if for all x not in L, for all P*, the output of (P*,V) is reject with high probability.
5 Example: Proof for Graph Isomorphism Consider the example of graph isomorphic map. We can have: P sends the isomorphic map directly to V; V accepts if and only if it is indeed an isomorphic map. This is complete because V accepts with probability 1 when G1 is isomorphic to G2. This is sound because V rejects with probability 1 when G1 is not isomorphic to G2.

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6 What Languages Have Interactive Proofs? For every language L in NP, we can construct an interactive proof system for L. In fact, Shamir showed that IP=PSPACE. Where IP is the set of languages having interactive proofs. PSPACE is the set of languages that can be recognized with polynomial-size space.
Knowledge in Proof The example proof for graph isomorphism is simple, complete and sound. However, it gives V too much knowledge. The isomorphic map is completely revealed to V. In cryptography, we often want to convince somebody without giving him knowledge. Such a proof is called a zero-knowledge (ZK)

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## This note was uploaded on 09/27/2010 for the course CSE 664 taught by Professor Shengzhong during the Spring '10 term at SUNY Buffalo.

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Lec10 - Zero-Knowledge Proofs Sheng Zhong 1 Graph...

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