On the Power of Many One-Bit Provers
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Interactive Proof Systems and Alternating Time-Space Complexity
Theoretical Computer Science 113 (1993) 55-73 55 Elsevier Interactive proof systems and alternating time-space complexity Lance Fortnow” and Carsten Lund** Department of Computer Science, Unicersity of Chicago. 1100 E. 58th Street, Chicago, IL 40637, USA Abstract Fortnow, L. and C. Lund, Interactive proof systems and alternating time-space complexity, Theoretical Computer Science 113 (1993) 55-73. We show a rough equivalence between alternating time-space complexity and a public-coin interactive proof system with the verifier having a polynomial-related time-space complexity. Special cases include the following: . All of NC has interactive proofs, with a log-space polynomial-time public-coin verifier vastly improving the best previous lower bound of LOGCFL for this model (Fortnow and Sipser, 1988). All languages in P have interactive proofs with a polynomial-time public-coin verifier using o(log’ n) space. l All exponential-time languages have interactive proof systems with public-coin polynomial-space exponential-time verifiers. To achieve better bounds, we show how to reduce a k-tape alternating Turing machine to a l-tape alternating Turing machine with only a constant factor increase in time and space. 1. Introduction In 1981, Chandra et al. [4] introduced alternating Turing machines, an extension of nondeterministic computation where the Turing machine can make both existential and universal moves. In 1985, Goldwasser et al. [lo] and Babai [l] introduced interactive proof systems, an extension of nondeterministic computation consisting of two players, an infinitely powerful prover and a probabilistic polynomial-time verifier. The prover will try to convince the verifier of the validity of some statement. -
Interactive Proofs
Interactive proofs April 12, 2014 [72] L´aszl´oBabai. Trading group theory for randomness. In Proc. 17th STOC, pages 421{429. ACM Press, 1985. doi:10.1145/22145.22192. [89] L´aszl´oBabai and Shlomo Moran. Arthur-Merlin games: A randomized proof system and a hierarchy of complexity classes. J. Comput. System Sci., 36(2):254{276, 1988. doi:10.1016/0022-0000(88)90028-1. [99] L´aszl´oBabai, Lance Fortnow, and Carsten Lund. Nondeterministic ex- ponential time has two-prover interactive protocols. In Proc. 31st FOCS, pages 16{25. IEEE Comp. Soc. Press, 1990. doi:10.1109/FSCS.1990.89520. See item 1991.108. [108] L´aszl´oBabai, Lance Fortnow, and Carsten Lund. Nondeterministic expo- nential time has two-prover interactive protocols. Comput. Complexity, 1 (1):3{40, 1991. doi:10.1007/BF01200056. Full version of 1990.99. [136] Sanjeev Arora, L´aszl´oBabai, Jacques Stern, and Z. (Elizabeth) Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. In Proc. 34th FOCS, pages 724{733, Palo Alto CA, 1993. IEEE Comp. Soc. Press. doi:10.1109/SFCS.1993.366815. Conference version of item 1997:160. [160] Sanjeev Arora, L´aszl´oBabai, Jacques Stern, and Z. (Elizabeth) Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. J. Comput. System Sci., 54(2):317{331, 1997. doi:10.1006/jcss.1997.1472. Full version of 1993.136. [111] L´aszl´oBabai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. In Proc. -
The Limits of Post-Selection Generalization
The Limits of Post-Selection Generalization Kobbi Nissim∗ Adam Smithy Thomas Steinke Georgetown University Boston University IBM Research – Almaden [email protected] [email protected] [email protected] Uri Stemmerz Jonathan Ullmanx Ben-Gurion University Northeastern University [email protected] [email protected] Abstract While statistics and machine learning offers numerous methods for ensuring gener- alization, these methods often fail in the presence of post selection—the common practice in which the choice of analysis depends on previous interactions with the same dataset. A recent line of work has introduced powerful, general purpose algorithms that ensure a property called post hoc generalization (Cummings et al., COLT’16), which says that no person when given the output of the algorithm should be able to find any statistic for which the data differs significantly from the population it came from. In this work we show several limitations on the power of algorithms satisfying post hoc generalization. First, we show a tight lower bound on the error of any algorithm that satisfies post hoc generalization and answers adaptively chosen statistical queries, showing a strong barrier to progress in post selection data analysis. Second, we show that post hoc generalization is not closed under composition, despite many examples of such algorithms exhibiting strong composition properties. 1 Introduction Consider a dataset X consisting of n independent samples from some unknown population P. How can we ensure that the conclusions drawn from X generalize to the population P? Despite decades of research in statistics and machine learning on methods for ensuring generalization, there is an increased recognition that many scientific findings do not generalize, with some even declaring this to be a “statistical crisis in science” [14]. -
The Computational Complexity of Nash Equilibria in Concisely Represented Games∗
The Computational Complexity of Nash Equilibria in Concisely Represented Games¤ Grant R. Schoenebeck y Salil P. Vadhanz August 26, 2009 Abstract Games may be represented in many di®erent ways, and di®erent representations of games a®ect the complexity of problems associated with games, such as ¯nding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payo®s, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payo®s are computed by a given boolean circuit, and graph games, where each agent's payo® is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, ¯nding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payo®s to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes. 1 Introduction In recent years, there has been a surge of interest at the interface between computer science and game theory. On one hand, game theory and its notions of equilibria provide a rich framework for modeling the behavior of sel¯sh agents in the kinds of distributed or networked environments that often arise in computer science and o®er mechanisms to achieve e±cient and desirable global outcomes in spite of the sel¯sh behavior. -
Signature Redacted
On Foundations of Public-Key Encryption and Secret Sharing by Akshay Dhananjai Degwekar B.Tech., Indian Institute of Technology Madras (2014) S.M., Massachusetts Institute of Technology (2016) Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September 2019 @Massachusetts Institute of Technology 2019. All rights reserved. Signature redacted Author ............................................ Department of Electrical Engineering and Computer Science June 28, 2019 Signature redacted Certified by....................................... VWi dVaikuntanathan Associate Professor of Electrical Engineering and Computer Science Thesis Supervisor Signature redacted A ccepted by . ......... ...................... MASSACLislie 6jp lodziejski OF EHs o fTE Professor of Electrical Engineering and Computer Science Students Committee on Graduate OCT Chair, Department LIBRARIES c, On Foundations of Public-Key Encryption and Secret Sharing by Akshay Dhananjai Degwekar Submitted to the Department of Electrical Engineering and Computer Science on June 28, 2019, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Since the inception of Cryptography, Information theory and Coding theory have influenced cryptography in myriad ways including numerous information-theoretic notions of security in secret sharing, multiparty computation and statistical zero knowledge; and by providing a large toolbox used extensively in cryptography. This thesis addresses two questions in this realm: Leakage Resilience of Secret Sharing Schemes. We show that classical secret sharing schemes like Shamir secret sharing and additive secret sharing over prime order fields are leakage resilient. Leakage resilience of secret sharing schemes is closely related to locally repairable codes and our results can be viewed as impossibility results for local recovery over prime order fields. -
Input-Oblivious Proof Systems and a Uniform Complexity Perspective on P/Poly
Electronic Colloquium on Computational Complexity, Report No. 23 (2011) Input-Oblivious Proof Systems and a Uniform Complexity Perspective on P/poly Oded Goldreich∗ and Or Meir† Department of Computer Science Weizmann Institute of Science Rehovot, Israel. February 16, 2011 Abstract We initiate a study of input-oblivious proof systems, and present a few preliminary results regarding such systems. Our results offer a perspective on the intersection of the non-uniform complexity class P/poly with uniform complexity classes such as NP and IP. In particular, we provide a uniform complexity formulation of the conjecture N P 6⊂ P/poly and a uniform com- plexity characterization of the class IP∩P/poly. These (and similar) results offer a perspective on the attempt to prove circuit lower bounds for complexity classes such as NP, PSPACE, EXP, and NEXP. Keywords: NP, IP, PCP, ZK, P/poly, MA, BPP, RP, E, NE, EXP, NEXP. Contents 1 Introduction 1 1.1 The case of NP ....................................... 1 1.2 Connection to circuit lower bounds ............................ 2 1.3 Organization and a piece of notation ........................... 3 2 Input-Oblivious NP-Proof Systems (ONP) 3 3 Input-Oblivious Interactive Proof Systems (OIP) 5 4 Input-Oblivious Versions of PCP and ZK 6 4.1 Input-Oblivious PCP .................................... 6 4.2 Input-Oblivious ZK ..................................... 7 Bibliography 10 ∗Partially supported by the Israel Science Foundation (grant No. 1041/08). †Research supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. ISSN 1433-8092 1 Introduction Various types of proof systems play a central role in the theory of computation. -
A Study of the NEXP Vs. P/Poly Problem and Its Variants by Barıs
A Study of the NEXP vs. P/poly Problem and Its Variants by Barı¸sAydınlıoglu˘ A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Computer Sciences) at the UNIVERSITY OF WISCONSIN–MADISON 2017 Date of final oral examination: August 15, 2017 This dissertation is approved by the following members of the Final Oral Committee: Eric Bach, Professor, Computer Sciences Jin-Yi Cai, Professor, Computer Sciences Shuchi Chawla, Associate Professor, Computer Sciences Loris D’Antoni, Asssistant Professor, Computer Sciences Joseph S. Miller, Professor, Mathematics © Copyright by Barı¸sAydınlıoglu˘ 2017 All Rights Reserved i To Azadeh ii acknowledgments I am grateful to my advisor Eric Bach, for taking me on as his student, for being a constant source of inspiration and guidance, for his patience, time, and for our collaboration in [9]. I have a story to tell about that last one, the paper [9]. It was a late Monday night, 9:46 PM to be exact, when I e-mailed Eric this: Subject: question Eric, I am attaching two lemmas. They seem simple enough. Do they seem plausible to you? Do you see a proof/counterexample? Five minutes past midnight, Eric responded, Subject: one down, one to go. I think the first result is just linear algebra. and proceeded to give a proof from The Book. I was ecstatic, though only for fifteen minutes because then he sent a counterexample refuting the other lemma. But a third lemma, inspired by his counterexample, tied everything together. All within three hours. On a Monday midnight. I only wish that I had asked to work with him sooner. -
Zk-Snarks: a Gentle Introduction
zk-SNARKs: A Gentle Introduction Anca Nitulescu Abstract Zero-Knowledge Succinct Non-interactive Arguments of Knowledge (zk-SNARKs) are non-interactive systems with short proofs (i.e., independent of the size of the witness) that enable verifying NP computa- tions with substantially lower complexity than that required for classical NP verification. This is a short, gentle introduction to zk-SNARKs. It recalls some important advancements in the history of proof systems in cryptography following the evolution of the soundness notion, from first interactive proof systems to arguments of knowledge. The main focus of this introduction is on zk-SNARKs from first constructions to recent efficient schemes. For the latter, it provides a modular presentation of the frameworks for state-of-the-art SNARKs. In brief, the main steps common to the design of two wide classes of SNARKs are: • finding a "good" NP characterisation, or arithmetisation • building an information-theoretic proof system • compiling the above proof system into an efficient one using cryptographic tools Arithmetisation is translating a computation or circuit into an equivalent arithmetic relation. This new relation allows to build an information-theoretic proof system that is either inefficient or relies on idealized components called oracles. An additional cryptographic compilation step will turn such a proof system into an efficient one at the cost of considering only computationally bounded adversaries. Depending on the nature of the oracles employed by the initial proof system, two classes of SNARKs can be considered. This introduction aims at explaining in detail the specificity of these general frameworks. QAP-based Compilation step in a trusted setup CRS Information-Theoretic Proof Computation/ Arithmetisation • Oracle Proofs / PCPs Circuit • Interactive Oracle Proofs ROM PIOP-based Compilation step with Polynomial Commitments and Fiat-Shamir Transformation 1 Contents 1 Introduction 3 1.1 Proof Systems in Cryptography . -
Pseudodistributions That Beat All Pseudorandom Generators
Electronic Colloquium on Computational Complexity, Report No. 19 (2021) Pseudodistributions That Beat All Pseudorandom Generators Edward Pyne∗ Salil Vadhany Harvard University Harvard University [email protected] [email protected] February 17, 2021 Abstract A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a pseudorandom pseudodistribution generator (PRPG), which amounts to a pseudorandom gen- erator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of PRPGs for ordered branching programs whose seed length has a better dependence on the error parameter " than the classic PRG construction of Nisan (STOC 1990 and Combinatorica 1992). In this work, we give an explicit construction of PRPGs that achieve parameters that are impossible to achieve by a PRG. In particular, we construct a PRPG for ordered permuta- tion branching programs of unbounded width with a single accept state that has seed length O~(log3=2 n) for error parameter " = 1= poly(n), where n is the input length. In contrast, re- cent work of Hoza et al. (ITCS 2021) shows that any PRG for this model requires seed length Ω(log2 n) to achieve error " = 1= poly(n). As a corollary, we obtain explicit PRPGs with seed length O~(log3=2 n) and error " = 1= poly(n) for ordered permutation branching programs of width w = poly(n) with an arbi- trary number of accept states. Previously, seed length o(log2 n) was only known when both the width and the reciprocal of the error are subpolynomial, i.e. -
Hardness of Non-Interactive Differential Privacy from One-Way
Hardness of Non-Interactive Differential Privacy from One-Way Functions Lucas Kowalczyk* Tal Malkin† Jonathan Ullman‡ Daniel Wichs§ May 30, 2018 Abstract A central challenge in differential privacy is to design computationally efficient non-interactive algorithms that can answer large numbers of statistical queries on a sensitive dataset. That is, we would like to design a differentially private algorithm that takes a dataset D Xn consisting of 2 some small number of elements n from some large data universe X, and efficiently outputs a summary that allows a user to efficiently obtain an answer to any query in some large family Q. Ignoring computational constraints, this problem can be solved even when X and Q are exponentially large and n is just a small polynomial; however, all algorithms with remotely similar guarantees run in exponential time. There have been several results showing that, under the strong assumption of indistinguishability obfuscation (iO), no efficient differentially private algorithm exists when X and Q can be exponentially large. However, there are no strong separations between information-theoretic and computationally efficient differentially private algorithms under any standard complexity assumption. In this work we show that, if one-way functions exist, there is no general purpose differen- tially private algorithm that works when X and Q are exponentially large, and n is an arbitrary polynomial. In fact, we show that this result holds even if X is just subexponentially large (assuming only polynomially-hard one-way functions). This result solves an open problem posed by Vadhan in his recent survey [Vad16]. *Columbia University Department of Computer Science. -
A Note on NP ∩ Conp/Poly Copyright C 2000, Vinodchandran N
BRICS Basic Research in Computer Science BRICS RS-00-19 V. N. Variyam: A Note on A Note on NP \ coNP=poly NP \ coNP = Vinodchandran N. Variyam poly BRICS Report Series RS-00-19 ISSN 0909-0878 August 2000 Copyright c 2000, Vinodchandran N. Variyam. BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy. See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK–8000 Aarhus C Denmark Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] BRICS publications are in general accessible through the World Wide Web and anonymous FTP through these URLs: http://www.brics.dk ftp://ftp.brics.dk This document in subdirectory RS/00/19/ A Note on NP ∩ coNP/poly N. V. Vinodchandran BRICS, Department of Computer Science, University of Aarhus, Denmark. [email protected] August, 2000 Abstract In this note we show that AMexp 6⊆ NP ∩ coNP=poly, where AMexp denotes the exponential version of the class AM.Themain part of the proof is a collapse of EXP to AM under the assumption that EXP ⊆ NP ∩ coNP=poly 1 Introduction The issue of how powerful circuit based computation is, in comparison with Turing machine based computation has considerable importance in complex- ity theory. There are a large number of important open problems in this area. -
Zero Knowledge and Soundness Are Symmetric∗
Zero Knowledge and Soundness are Symmetric∗ Shien Jin Ong Salil Vadhan Division of Engineering and Applied Sciences Harvard University Cambridge, Massachusetts, USA. E-mail: {shienjin,salil}@eecs.harvard.edu November 14, 2006 Abstract We give a complexity-theoretic characterization of the class of problems in NP having zero- knowledge argument systems that is symmetric in its treatment of the zero knowledge and the soundness conditions. From this, we deduce that the class of problems in NP ∩ coNP having zero-knowledge arguments is closed under complement. Furthermore, we show that a problem in NP has a statistical zero-knowledge argument system if and only if its complement has a computational zero-knowledge proof system. What is novel about these results is that they are unconditional, i.e. do not rely on unproven complexity assumptions such as the existence of one-way functions. Our characterization of zero-knowledge arguments also enables us to prove a variety of other unconditional results about the class of problems in NP having zero-knowledge arguments, such as equivalences between honest-verifier and malicious-verifier zero knowledge, private coins and public coins, inefficient provers and efficient provers, and non-black-box simulation and black-box simulation. Previously, such results were only known unconditionally for zero-knowledge proof systems, or under the assumption that one-way functions exist for zero-knowledge argument systems. Keywords: zero-knowledge argument systems, statistical zero knowledge, complexity classes, clo- sure under complement, distributional one-way functions. ∗Both the authors were supported by NSF grant CNS-0430336 and ONR grant N00014-04-1-0478.