Complexity-Theoretic Aspects of Interactive Proof Systems

In 1985, Goldwasser, Micali and Racko formulated interactive proof systems as a tool for developing cryptographic protocols. Indeed, many exciting cryptographic results followed from studying interactive proof systems and the related concept of zero-knowledge. Interactive proof systems also have an important part in complexity theory merging the well established concepts of probabilistic and nondeterministic computation. This thesis will study the complexity of various models of interactive proof systems. A perfect zero-knowledge interactive protocol convinces a veri er that a string is in a language without revealing any additional knowledge in an information theoretic sense. This thesis will show that for any language that has a perfect zero-knowledge proof system, its complement has a short interactive protocol. This result implies that there are not any perfect zero-knowledge protocols for NP-complete languages unless the polynomial-time hierarchy collapses. Thus knowledge complexity can show a language is easy to prove. Interesting models of interactive proof systems arise by restricting the power of the veri er. This thesis examines the proof systems with a veri er required to run in logarithmic space as well as polynomial time. Relationships with circuit complexity and log-space Turing machines are developed. We can increase the power of interactive proof systems by allowing many provers that can not communicate among themselves during the protocol. This thesis shows the equivalence between this multi-prover model and probabilistic Turing machines with an untrustworthy oracle. We additionally give an oracle under which co-NP does not have multi-prover interactive protocols. This result implies an oracle where co-NP does not have standard interactive protocols. Another natural model occurs when the veri er has only linear time. Towards this direction, this thesis examines probabilistic machines and linear time. We show an oracle under which linear time probabilistic Turing machines can accept all BPP languages, an unusual collapse of a complexity time hierarchy. We exhibit many other related relativized results. Finally we show probabilistic linear time does not contain all languages accepted by interactive proof systems.

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