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Complexity-Theoretic Aspects of Interactive Proof Systems

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ComplexityTheoretic Asp ects of Interactive Pro of Systems by Lance Jeremy Fortnow BA Mathematics and Computer Science Cornell University Submitted to the Department of Mathematics in partial fulllment of the requirements for the degree of Do ctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June c Massachusetts Institute of Technology All rights reserved Signature of Author ::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::: Department of Mathematics May Certied by ::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::: Michael Sipser Asso ciate Professor Mathematics Thesis Sup ervisor Accepted by ::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::: Daniel Kleitman Chairman Applied Mathematics Committee Accepted by ::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::: Sigurdur Helgason Chairman Departmental Graduate Committee Abstract In Goldwasser Micali and Racko formulated interactive pro of systems as a to ol for de veloping cryptographic proto cols Indeed many exciting cryptographic results followed from studying interactive pro of systems and the related concept of zeroknowledge Interactive pro of systems also have an imp ortant part in complexity theory merging the well established concepts of probabilistic and nondeterministic computation This thesis will study the com plexity of various mo dels of interactive pro of systems A p erfect zeroknowledge interactive proto col convinces a verier 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 p erfect zeroknowledge pro of system its complement has a short interactive proto col This result implies that there are not any p erfect zeroknowledge proto cols for NPcomplete languages unless the p olynomialtime hierarchy collapses Thus knowledge complexity can show a language is easy to prove Interesting mo dels of interactive pro of systems arise by restricting the p ower of the verier This thesis examines the pro of systems with a verier required to run in logarithmic space as well as p olynomial time Relationships with circuit complexity and logspace Turing machines are develop ed We can increase the p ower of interactive pro of systems by allowing many provers that can not communicate among themselves during the proto col This thesis shows the equivalence b etween this multiprover mo del and probabilistic Turing machines with an untrustworthy oracle We additionally give an oracle under which coNP do es not have multiprover interac tive proto cols This result implies an oracle where coNP do es not have standard interactive proto cols Another natural mo del o ccurs when the verier has only linear time Towards this direc tion 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 Fi nally we show probabilistic linear time do es not contain all languages accepted by interactive pro of systems Keywords Computational Complexity Interactive Pro of Systems ZeroKnowledge Probabilistic Computation Oracles Acknowledgments Foremost I would like to thank my advisor Michael Sipser Mike has collab orated on much of my research and virtually every problem I have ever lo oked I have discussed with Mike Mike has greatly inuenced my graduate career and I will b e forever grateful for these four years I have sp ent working with him I sp ent the rst year of my graduate scho oling at the University of California at Berkeley I started working with Mike Sipser at Berkeley and some of the research in this thesis o ccurred during that p erio d I greatly thank the Oce of Naval Research for their generous fellowship that supp orted me for my rst three years in graduate scho ol as well as indirectly covering my travel exp enses through out my graduate career Thanks also to the American So ciety for Engineering Edu cation for eciently administering the fellowship Thanks to all of the graduate students I have worked and so cialized with during these four years In particular Eric Schwab e has b een a great friend ro ommate and pro ofreader since I arrived at MIT Thanks to Marcy App ell for her companionship over the last year and a half and the many new exp eriences she has given to me like skiing and coee Finally thanks to Juris Hartmanis whose courses at Cornell aroused my interests in theoretical computer science and complexity theory If Juris had not b een at Cornell I would probably b e writing a thesis on Hopf algebras and missing out of the excitement of complexity theory Contents Intro duction Victor and the Great Pulu A Short History Denitions Deterministic Time and Space Complexity Nondeterministic Turing Machines Completeness Probabilistic Turing Machines Oracles and The Polynomial Time Hierarchy Circuits and Nonuniform Computation Interactive Pro of Systems Relativization Basic Results An Example Graph Nonisomorphism Perfect ZeroKnowledge A Cryptographic Side of Interactive Pro of Systems Notation and Denitions Related Results Showing Sets are Large and Small Lower Bound Proto col Upp er Bound Proto col Comparison Proto col Main Theorem Structure of Pro of An Example Graph Isomorphism The Proto col Pro of of the Proto cols Correctness Extensions and Corollaries Further Research LogarithmicSpace Veriers Reducing the Power of the Verier Log Space Veriers and BPNL A Circuit Mo del for BPNL BPNL Contains LOGCFL Further Directions of Research Multiple Provers Corrob orating Susp ects Denitions Probabilistic Oracle Machines Are there MultiProver Proto cols for coNP Languages Bounded Round Proto cols Further Research Probabilistic Computation and Linear Time LinearTime Veriers Our Results and Related Results Deterministic Nondeterministic and Probabilistic Linear Time Pro of of the Main Theorem Structure of the Oracle A Simple Case Inuencing and Simulating Strings Order of Enco ding Creating the Dep endency Graph Pro cessing the Dep endency Graph Generalizing the Pro of for All BPP Machines Other Results Conclusions and Further Research Chapter Intro duction Victor and the Great Pulu Victor a venture capitalist had everything a man could desire money women and p ower But he felt something missing He decided he lacked knowledge So Victor packed up his bags and headed to the Himalayas in search of ultimate truths The natives p ointed Victor to a tall mountain and mentioned rumors of a great man full of wisdom Victor who smartly brought some climbing equipment tackled the mountain until he reached a small cave near the summit Victor found the great Pulu grand guru of all that is known Victor inquired to some ultimate truths and Pulu resp onded I wil l teach you but you must not trust my words Victor agreed and found he learned much even though he had to verify all the sayings of the great Pulu Victor though lacked complete happiness and he asked if he could learn knowledge b eyond what he could learn in this manner The grand guru replied You may ask and I wil l answer Victor p ondered this idea for a minute and said Since you know all that is known why can you not predict my questions A silence reigned over the mountain for a short while until the Guru nally sp oke You must use other implementssymbols of your past life Victor thought for a while and reached into his backpack and brought out some spare change he had unwittingly carried with him Even the great Pulu can not predict the ip of a coin He started ipping the coins to ask the guru and wondered what can I learn now In this thesis we will study the very question of what Victor can learn On his own Victor can only decide simple problems With the help of Pulu even when he can not trust the answers Victor found he could learn much more than b efore With a coin Victor could learn even more still We will formalize these interactions by lo oking at Victor as a computer with a restricted amount of p ower
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