Quadratic Span Programs and Succinct Nizks Without Pcps
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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. -
2020 SIGACT REPORT SIGACT EC – Eric Allender, Shuchi Chawla, Nicole Immorlica, Samir Khuller (Chair), Bobby Kleinberg September 14Th, 2020
2020 SIGACT REPORT SIGACT EC – Eric Allender, Shuchi Chawla, Nicole Immorlica, Samir Khuller (chair), Bobby Kleinberg September 14th, 2020 SIGACT Mission Statement: The primary mission of ACM SIGACT (Association for Computing Machinery Special Interest Group on Algorithms and Computation Theory) is to foster and promote the discovery and dissemination of high quality research in the domain of theoretical computer science. The field of theoretical computer science is the rigorous study of all computational phenomena - natural, artificial or man-made. This includes the diverse areas of algorithms, data structures, complexity theory, distributed computation, parallel computation, VLSI, machine learning, computational biology, computational geometry, information theory, cryptography, quantum computation, computational number theory and algebra, program semantics and verification, automata theory, and the study of randomness. Work in this field is often distinguished by its emphasis on mathematical technique and rigor. 1. Awards ▪ 2020 Gödel Prize: This was awarded to Robin A. Moser and Gábor Tardos for their paper “A constructive proof of the general Lovász Local Lemma”, Journal of the ACM, Vol 57 (2), 2010. The Lovász Local Lemma (LLL) is a fundamental tool of the probabilistic method. It enables one to show the existence of certain objects even though they occur with exponentially small probability. The original proof was not algorithmic, and subsequent algorithmic versions had significant losses in parameters. This paper provides a simple, powerful algorithmic paradigm that converts almost all known applications of the LLL into randomized algorithms matching the bounds of the existence proof. The paper further gives a derandomized algorithm, a parallel algorithm, and an extension to the “lopsided” LLL. -
Program Sunday Evening: Welcome Recep- Tion from 7Pm to 9Pm at the Staff Lounge of the Department of Computer Science, Ny Munkegade, Building 540, 2Nd floor
Computational ELECTRONIC REGISTRATION Complexity The registration for CCC’03 is web based. Please register at http://www.brics.dk/Complexity2003/. Registration Fees (In Danish Kroner) Eighteenth Annual IEEE Conference Advance† Late Members‡∗ 1800 DKK 2200 DKK ∗ Sponsored by Nonmembers 2200 DKK 2800 DKK Students+ 500 DKK 600 DKK The IEEE Computer Society ∗The registration fee includes a copy of the proceedings, Technical Committee on receptions Sunday and Monday, the banquet Wednesday, and lunches Monday, Tuesday and Wednesday. Mathematical Foundations +The registration fee includes a copy of the proceedings, of Computing receptions Sunday and Monday, and lunches Monday, Tuesday and Wednesday. The banquet Wednesday is not included. †The advance registration deadline is June 15. ‡ACM, EATCS, IEEE, or SIGACT members. Extra proceedings/banquet tickets Extra proceedings are 350 DKK. Extra banquet tick- ets are 300 DKK. Both can be purchased when reg- istering and will also be available for sale on site. Alternative registration If electronic registration is not possible, please con- tact the organizers at one of the following: E-mail: [email protected] Mail: Complexity 2003 c/o Peter Bro Miltersen In cooperation with Department of Computer Science University of Aarhus ACM-SIGACT and EATCS Ny Munkegade, Building 540 DK 8000 Aarhus C, Denmark Fax: (+45) 8942 3255 July 7–10, 2003 Arhus,˚ Denmark Conference homepage Conference Information Information about this year’s conference is available Location All sessions of the conference and the on the Web at Kolmogorov workshop will be held in Auditorium http://www.brics.dk/Complexity2003/ F of the Department of Mathematical Sciences at Information about the Computational Complexity Aarhus University, Ny Munkegade, building 530, 1st conference is available at floor. -
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. -
SIGACT News Complexity Theory Column 93
SIGACT News Complexity Theory Column 93 Lane A. Hemaspaandra Dept. of Computer Science University of Rochester Rochester, NY 14627, USA Introduction to Complexity Theory Column 93 This issue This issue features Swastik Kopparty and Shubhangi Saraf's column, \Local Testing and Decoding of High-Rate Error-Correcting Codes." Warmest thanks to the authors for their fascinating, exciting column! Future issues And please stay tuned to future issues' Complexity Theory Columns, for articles by Neeraj Kayal and Chandan Saha (tentative topic: arithmetic circuit lower bounds); Stephen Fenner and Thomas Thierauf (tentative topic: related to the STOC 2016 quasi-NC bipartite perfect matching algorithm of Fenner, Gurjar, and Thierauf); Srinivasan Arunachalam and Ronald de Wolf (tentative topic: quantum machine learning); and Ryan O'Donnell and John Wright (tentative topic: learning and testing quantum states). 2016{2017 As this issue reaches you, the 2016{2017 academic year will be starting. For those of you who are faculty members, I wish you a theorem-filled year, including the treat of working on research with bright graduate and undergraduate students. For those you who are students (bright ones, of course|you're reading this column!), if you are not already working with a faculty member on research and would like to, don't be shy! Even if you're a young undergraduate, it makes a lot of sense to pop by your favorite theory professor's office, and let him or her know what has most interested you so far within theory, if something has (perhaps a nifty article you read by Kopparty and Saraf?), and that you find theory thrilling, if you do (admit it, you do!), and that you'd love to be doing theory research as soon as possible and would deeply appreciate any advice as to what you can do to best make that happen. -
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 . -
Toward Probabilistic Checking Against Non-Signaling Strategies with Constant Locality
Electronic Colloquium on Computational Complexity, Report No. 144 (2020) Toward Probabilistic Checking against Non-Signaling Strategies with Constant Locality Mohammad Mahdi Jahanara Sajin Koroth Igor Shinkar [email protected] sajin [email protected] [email protected] Simon Fraser University Simon Fraser University Simon Fraser University September 7, 2020 Abstract Non-signaling strategies are a generalization of quantum strategies that have been studied in physics over the past three decades. Recently, they have found applications in theoretical computer science, including to proving inapproximability results for linear programming and to constructing protocols for delegating computation. A central tool for these applications is probabilistically checkable proof (PCPs) systems that are sound against non-signaling strategies. In this paper we show, assuming a certain geometrical hypothesis about noise robustness of non-signaling proofs (or, equivalently, about robustness to noise of solutions to the Sherali- Adams linear program), that a slight variant of the parallel repetition of the exponential-length constant-query PCP construction due to Arora et al. (JACM 1998) is sound against non-signaling strategies with constant locality. Our proof relies on the analysis of the linearity test and agreement test (also known as the direct product test) in the non-signaling setting. Keywords: direct product testing; linearity testing; non-signaling strategies; parallel repetition; proba- bilistically checkable proofs 1 ISSN 1433-8092 Contents 1 Introduction 3 1.1 Informal statement of the result . .5 1.2 Roadmap . .6 2 Preliminaries 6 2.1 Probabilistically Checkable Proofs . .6 2.2 Parallel repetition . .7 2.3 Non-signaling functions . .7 2.4 Permutation folded repeated non-signaling functions . -
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. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Hardness of Maximum Constraint Satisfaction Permalink https://escholarship.org/uc/item/5x33g1k7 Author Chan, Siu On Publication Date 2013 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Hardness of Maximum Constraint Satisfaction by Siu On Chan A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Elchanan Mossel, Chair Professor Luca Trevisan Professor Satish Rao Professor Michael Christ Spring 2013 Hardness of Maximum Constraint Satisfaction Creative Commons 3.0 BY: C 2013 by Siu On Chan 1 Abstract Hardness of Maximum Constraint Satisfaction by Siu On Chan Doctor of Philosophy in Computer Science University of California, Berkeley Professor Elchanan Mossel, Chair Maximum constraint satisfaction problem (Max-CSP) is a rich class of combinatorial op- timization problems. In this dissertation, we show optimal (up to a constant factor) NP- hardness for maximum constraint satisfaction problem with k variables per constraint (Max- k-CSP), whenever k is larger than the domain size. This follows from our main result con- cerning CSPs given by a predicate: a CSP is approximation resistant if its predicate contains a subgroup that is balanced pairwise independent. Our main result is related to previous works conditioned on the Unique-Games Conjecture and integrality gaps in sum-of-squares semidefinite programming hierarchies. Our main ingredient is a new gap-amplification technique inspired by XOR-lemmas. Using this technique, we also improve the NP-hardness of approximating Independent-Set on bounded-degree graphs, Almost-Coloring, Two-Prover-One-Round-Game, and various other problems.