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The Mechanism Design Approach to Interactive Proofs A Dissertation presented by Shikha Singh to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Computer Science Stony Brook University August 2018 Stony Brook University The Graduate School Shikha Singh We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation Michael Bender - Dissertation Advisor Professor, Computer Science Jing Chen - Dissertation Advisor Assistant Professor, Computer Science Steven Skiena - Chairperson of Defense Professor, Computer Science Jie Gao Associate Professor, Computer Science Martin Farach-Colton Professor, Computer Science, Rutgers University This dissertation is accepted by the Graduate School Charles Taber Dean of the Graduate School ii Abstract of the Dissertation The Mechanism Design Approach to Interactive Proofs by Shikha Singh Doctor of Philosophy in Computer Science Stony Brook University 2018 Most computation today is not done locally by a client but rather outsourced to third- party service providers in exchange for money. Trading computation for money brings up two problems|(a) how can the client guarantee correctness of the outsourced computation efficiently, and (b) how to design an effective payment scheme. The two problems are closely related|ideally, we want the payment scheme to incentivize the service providers to perform the computation correctly. Interactive proofs (IP) are a fundamental theoretical framework used to study verifiable computation outsourcing. In an IP, the weak client (or verifier) interacts with powerful service providers (or provers) to determine the truthfulness of their claim. IPs give strong guarantees; however they do not admit a non-trivial payment scheme. This dissertation aims to answer the following question: if computation is being traded for money in financially-driven marketplaces, how do we use payments to leverage correctness? Rational proofs are payment-based interactive proofs for computation outsourcing which incentivize the service provider to do the computation correctly. In a rational proof, the prover is rational in the economic sense and acts to maximize its payment. In this dissertation, we introduce and analyze the model of rational proofs with multiple service providers. Multiple rational agents pose new game-theoretic challenges: how do iii we design and analyze their joint incentive structure? Provers may be cooperative (work together as a team to maximize their total payment) or non-cooperative (act to maximize their own individual payment). Using principles from game theory and mechanism design, we show how the incentives of multiple rational provers can used against each other to design simple, efficient and extremely powerful proof systems. Besides the main work on rational proofs, this work also addresses an important algorith- mic problem faced by memory-constrained devices that store most of their data on external storage (accessing which is expensive). In particular, we study the following problem: how do we efficiently test membership and perform updates on a set that is stored remotely? We present improved variants of a Bloom filter, a data structure that is widely used to speed up membership queries to a remote set. iv Dedicated to my parents, Nirupma and Subhash Singh, for teaching me to work hard and to dream big. v Table of Contents 1 Introduction 1 1.1 Motivation and Dissertation Overview . .1 1.2 Rational Interactive Proofs . .1 1.2.1 Results and Contributions . .4 1.3 Minimizing Remote Accesses: Adaptive Bloom Filters . .6 1.4 Dissertation Outline . .7 2 Rational Proofs: Background and Related Work 8 2.1 Interactive Proofs . .8 2.2 Review of Game Theoretic Concepts . 10 2.3 Previous Work on Game-theoretic Interactive Proofs . 12 2.3.1 Refereed Games . 12 2.3.2 Single-Prover Rational Proofs . 13 2.4 Review of Uniform Circuits and Complexity Classes . 16 3 Rational Proofs with Cooperative Provers 19 3.1 Introduction . 19 3.1.1 Overview of Results and Contributions . 21 3.2 Multi-Prover Rational Interactive Proofs (MRIP) . 23 3.2.1 MRIP with Utility Gap . 26 3.3 MRIP Protocols for NEXP ............................ 27 3.4 Constant and Noticeable Utility Gap . 32 3.5 Characterizing the Full Power of MRIP . 35 3.6 Optimal Number of Provers and Rounds . 45 4 Scaled-Down Cooperative Rational Proofs 50 4.1 Introduction . 50 4.1.1 Overview of Results and Contributions . 50 4.2 Preliminaries . 52 4.3 Verification in Logarithmic Time . 53 4.4 Verification in Logarithmic Space . 58 vi 5 Relationship Between Classical and Rational Proofs 61 5.1 Introduction . 61 5.2 Converting a Rational Proof to a Classical Proof . 62 6 Rational Proofs with Non-Cooperative Provers 67 6.1 Introduction . 67 6.2 Model and Preliminaries . 69 6.3 Strong Sequential Equilibrium and its Properties . 71 6.4 Recursive-Maximum SSE . 78 6.5 Utility Gap in ncRIP Protocols . 79 7 Tight Characterizations of ncRIP Classes 81 7.1 Introduction . 81 7.2 Lower Bounds on ncRIP Classes . 83 7.3 Upper Bounds on ncRIP Classes . 90 7.4 Optimal Number of Provers and Rounds . 100 8 Scaled-Down Non-Cooperative Rational Proofs 103 8.1 Introduction . 103 8.1.1 Overview of Results and Contributions . 104 8.2 Model and Preliminaries . 105 8.3 Warm up: O(log n)-time ncRIP Protocols for P and NP ............ 106 8.4 A Lower Bound on O(log n)-time ncRIP Protocols . 110 8.5 A Upper Bound on O(log n)-time ncRIP Protocols . 113 9 Adaptive Bloom Filters to Minimize Remote Accesses 115 9.1 Introduction . 115 9.2 Related Work . 118 9.3 Preliminaries . 119 9.4 A Lower Bound on Local AMQs . 122 9.4.1 Notation and Adversary Model . 123 9.4.2 Analysis . 124 9.5 Broom Filter: An Adaptive Bloom filter . 130 9.5.1 Extending Fingerprints Using Adaptivity Bits . 131 9.5.2 Broom Filter Design for the Small-Remainder Case . 140 9.5.3 Broom Filter Design for the Large-Remainder Case . 142 9.6 How to Turn any AMQ into an Adaptive AMQ . 144 vii List of Figures 1 Complexity classes with scaled-down rational proof protocols. .5 2 The computation power of MRIP vs. classical interactive proof systems. It is widely believed that PSPACE 6= EXP, EXP 6= NEXP, and NEXP 6= coNEXP.. 22 3 An MRIP protocol for L using MRIP protocol for L.............. 25 4 A simple MRIP protocol for NEXP........................ 28 5 A simple and efficient MRIP protocol for Oracle-3SAT............. 29 6 An MRIP protocol for PjjNEXP[γ(n)]......................... 33 7 An MRIP protocol for EXP............................ 36 poly−NEXP 8 An MRIP protocol for EXPjj ....................... 39 9 Simulating any MRIP protocol with 2 provers and 3 rounds. 47 10 Simulating an RIP protocol using 2 provers and reduced communication. 55 NP[γ(n)] 11 An RIP protocol for Ljj ........................... 59 12 Summary of the tight characterizations of ncRIP classes. 82 13 The computation power hierarchy of classical, rational cooperative and ratio- NEXP NEXP nal non-cooperative interactive proof systems, assuming Pjj 6= P ... 82 14 A O(1)-utility gap ncRIP protocol for NEXP................... 84 15 An O(γ(n))-utility gap ncRIP protocol for PNEXP[γ(n)].............. 86 16 Simulating any MRIP using an ncRIP protocol with exponential utility gap. 89 17 A constant-utility gap, O(log n)-time ncRIP protocol for P........... 107 18 A constant-utility gap O(log n)-time ncRIP protocol for 3-SAT......... 109 19 An O(γ(n))-utility gap ncRIP protocol for PNEXP[γ(n)].............. 111 20 Algorithms for Lookup and Adapt in an adaptive Bloom filter. 145 viii Acknowledgements I have been fortunate to have had amazing teachers and mentors in my life, and I am grateful to each one of them even if they are not explicitly mentioned here. I would like to start by acknowledging my fantastic advisors, Jing Chen and Michael Ben- der, for their unwavering support and guidance during the last five years. From Jing I learnt the value of patience and rigor when attacking challenging research problems. From Michael I learnt to appreciate the elegance in simple proofs and the importance of communicating research ideas well. It has been an immensely rewarding experience to observe Michael and Jing at work and it is to them that I owe my research aesthetics, my writing and teaching style and my love for collaboration. I am also grateful to Steven Skiena, Martin Farach-Colton and Rob Johnson for their mentorship and time; to Steve, for trusting me with his classes and helping me figure out what I wanted to do with my life; and to Martin and Rob, for making my ideas feel valued in meetings and for making research fun with their endless repertoire of puns and sarcasm. I was fortunate to have the opportunity to visit several research labs during my PhD. Working in these new environments with new people has helped me grow as a researcher. I would like to thank Manoj Gupta, Mayank Goswami and Nguyen Kim Thang for hosting me in their labs, introducing me to new research problems and advising me during my visit. I want to thank the Algorithms Reading Group, especially Estie Arkin, Joe Mitchell, Rezaul Chowdhury, Jie Gao, Rob Patro, Michael Bender, Jing Chen, and Steve Skiena, for sharing fun problems and creating a vibrant algorithms culture at Stony Brook. A special mention to our admin team, Kathy Germana, Betty Knitweiss and Cindy SantaCruz-Scalzo; thanks for all the paperwork, and making sure that I get paid and that I graduate. I want to thank everyone I have interacted with at workshops and conferences, as these discussions have enriched my grad school experience. Among them, I would especially like to mention Suresh Venkatasubramanian, Jeremy Fineman, Ben Moseley, Cindy Phillips, Andrew McGregor, Tsvi Kopelowitz, Nodari Sitchinava, and Don Sheehy.
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