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Two Query PCP with Sub-Constant Error

Dana Moshkovitz, Ran Raz
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
We show that the N P-Complete language 3SAT has a PCP verifier that makes two queries to a proof of almost-linear size and achieves sub-constant probability of error o(1).  ...  There were also PCP Theorems with sub-constant error and almost-linear size, but a constant number of queries that is larger than 2 [26]. As a corollary, we obtain a host of new results.  ...  ., [3] ): Are there PCP Theorems with two queries and sub-constant error ? How about two-query projection tests and sub-constant error ? Are there such PCPs with almost-linear size ?  ... 
doi:10.1109/focs.2008.60 dblp:conf/focs/MoshkovitzR08 fatcat:agpggrno5ncuxnvr4vimozvvfy

Composition of Low-Error 2-Query PCPs Using Decodable PCPs

Irit Dinur, Prahladh Harsha
2009 2009 50th Annual IEEE Symposium on Foundations of Computer Science  
The main result of this paper is a generic composition theorem for low error two-query probabilistically checkable proofs (PCPs).  ...  Prior to this work, composition of PCPs was wellunderstood only in the constant error regime.  ...  Indeed, this is how one constructs PCPs with sub-constant error and a constant number of queries for the entire range of Ω(1) ≤ |Σ| ≤ exp((log n) 1−ε ) [RS97, AS03, DFK + 99].  ... 
doi:10.1109/focs.2009.8 dblp:conf/focs/DinurH09 fatcat:4ymy3vyienguzmn4wgwobw3j4e

Composition of Low-Error 2-Query PCPs Using Decodable PCPs

Irit Dinur, Prahladh Harsha
2013 SIAM journal on computing (Print)  
The main result of this paper is a generic composition theorem for low error two-query probabilistically checkable proofs (PCPs).  ...  Prior to this work, composition of PCPs was wellunderstood only in the constant error regime.  ...  Indeed, this is how one constructs PCPs with sub-constant error and a constant number of queries for the entire range of Ω(1) ≤ |Σ| ≤ exp((log n) 1−ε ) [RS97, AS03, DFK + 99].  ... 
doi:10.1137/100788161 fatcat:3k2qhgnkvrahri42ynnykctyle

Composition of Low-Error 2-Query PCPs Using Decodable PCPs [chapter]

Irit Dinur, Prahladh Harsha
2010 Lecture Notes in Computer Science  
The main result of this paper is a generic composition theorem for low error two-query probabilistically checkable proofs (PCPs).  ...  Prior to this work, composition of PCPs was wellunderstood only in the constant error regime.  ...  Indeed, this is how one constructs PCPs with sub-constant error and a constant number of queries for the entire range of Ω(1) ≤ |Σ| ≤ exp((log n) 1−ε ) [RS97, AS03, DFK + 99].  ... 
doi:10.1007/978-3-642-16367-8_21 fatcat:57tg7catbndm7dyeaelrfgee54

Efficient Arguments without Short PCPs

Yuval Ishai, Eyal Kushilevitz, Rafail Ostrovsky
2007 Proceedings - IEEE Conference on Computational Complexity  
Current constructions of efficient argument systems combine a short (polynomial size) PCP with a cryptographic hashing technique.  ...  (JACM 1998) yields a simple argument system for NP in which the communication from the prover to the verifier only includes a constant number of short encryptions.  ...  We are grateful to Eli Ben-Sasson and Prahladh Harsha for enlightening discussions on the efficiency of short PCPs, and to Rafael Pass for pointing out the relevance of [33] .  ... 
doi:10.1109/ccc.2007.10 dblp:conf/coco/IshaiKO07 fatcat:cxpvnlwrw5clnmll6yhgmhuyse

Low-degree test with polynomially small error

Dana Moshkovitz
2016 Computational Complexity  
We show that the Sliding Scale Conjecture in PCP, namely the conjecture that there are PCP verifiers whose error is exponentially small in their randomness, would follow from a derandomization of our parallel  ...  This is known as low degree testing, and is the core of the algebraic approach to construction of PCP.  ...  With small constant number of queries, we have PCP theorems with soundness error 2 −(log n) β for some constant β > 0 and 3 queries [38, 4] .  ... 
doi:10.1007/s00037-016-0149-4 fatcat:sgzrdmg7hzfmdfh323r54n7mhq

Rigid Matrices From Rectangular PCPs [article]

Amey Bhangale, Prahladh Harsha, Orr Paradise, Avishay Tal
2020 arXiv   pre-print
Our construction of rectangular PCPs starts with an analysis of how randomness yields queries in the Reed–Muller-based outer PCP of Ben-Sasson, Goldreich, Harsha, Sudan and Vadhan [SICOMP, 2006; CCC, 2005  ...  We introduce a variant of PCPs, that we refer to as rectangular PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets  ...  For every constants s ∈ (0, 1/2) and τ ∈ (0, 1), there exists a constant-query, smooth and τ -almost rectangular PCP for L over the Boolean alphabet with perfect completeness, soundness error s, proof  ... 
arXiv:2005.03123v4 fatcat:7icnjyt7fzfovixm5puja3fxui

Short Locally Testable Codes and Proofs: A Survey in Two Parts [chapter]

Oded Goldreich
2010 Lecture Notes in Computer Science  
We survey known results regarding locally testable codes and locally testable proofs (known as PCPs), with emphasis on the length of these constructs.  ...  The survey consists of two independent (i.e., self-contained) parts that cover the same material at different levels of rigor and detail.  ...  Applying this reduction log log n times yields a PCP of query complexity poly(log n) and length O(n), which in turn yields a 3-query "PCP with soundness error 1 − 1/poly(log n)".  ... 
doi:10.1007/978-3-642-16367-8_6 fatcat:x4strlpapzhcxdulsd3qjzbfje

Universal Arguments and their Applications

Boaz Barak, Oded Goldreich
2009 SIAM journal on computing (Print)  
indistinguishable proof systems, error-correcting codes, tree hashing.  ...  Furthermore, these protocols have a constant number of rounds and are of the public-coin type.  ...  Loosely speaking, such a commitment scheme is an e cient two-phase two-party protocol through which one party, called the sender, can commit itself to a value so the following two con icting requirements  ... 
doi:10.1137/070709244 fatcat:hfvkr5g455hhdiawk3v7rksaaq

On Efficient Zero-Knowledge PCPs [chapter]

Yuval Ishai, Mohammad Mahmoody, Amit Sahai
2012 Lecture Notes in Computer Science  
Simplifying Bounded-Query ZK-PCPs. The bounded-query zeroknowledge PCP of Kilian et al. starts from a weakly-sound boundedquery ZK-PCP of Dwork et al.  ...  The existence of ZK-PCPs for NP with efficient oracles and arbitrary polynomial-time malicious verifiers was left open.  ...  [40] gave an elegant way of using locking schemes to convert a ZK-PCP with 1−1/ poly(k) soundness error into a standard ZK-PCP of constant or even negligible error.  ... 
doi:10.1007/978-3-642-28914-9_9 fatcat:by5mj3efbnb6xdqvdkvcgtp5qa

Probabilistic Checking Against Non-Signaling Strategies from Linearity Testing

Alessandro Chiesa, Peter Manohar, Igor Shinkar, Michael Wagner
2018 Innovations in Theoretical Computer Science  
In this paper we prove that the exponential-length constant-query PCP construction due to Arora et al. (JACM 1998) is sound against non-signaling strategies.  ...  Our result offers a new length-vs-query tradeoff when compared to the non-signaling PCP of Kalai, Raz, and Rothblum (STOC 2013 and 2014) and, moreover, may serve as an intermediate step to a proof of a  ...  sub-constant soundness error of 2 −O(log S) .  ... 
doi:10.4230/lipics.itcs.2019.25 dblp:conf/innovations/ChiesaMS19 fatcat:7io7utcuhbb4pd6awocpevulqy

Subquadratic SNARGs in the Random Oracle Model [article]

Alessandro Chiesa, Eylon Yogev
2021 IACR Cryptology ePrint Archive  
We present a SNARG in the ROM with a sub-quadratic argument size: Õ(log(t/ ) • log t). Our construction relies on a strong soundness notion for PCPs and a weak binding notion for commitments.  ...  A SNARG in the ROM is (t, )-secure if every t-query malicious prover can convince the verifier of a false statement with probability at most .  ...  For example, a PCP with this soundness error can be obtained by repeating O(log(t/ )) times the verifier of any constant-query constant-soundness base PCP.  ... 
dblp:journals/iacr/ChiesaY21a fatcat:5urql44a7bbchfmfjd4x2lmlle

Robust PCPs of Proximity, Shorter PCPs, and Applications to Coding

Eli Ben‐Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, Salil Vadhan
2006 SIAM journal on computing (Print)  
In both cases, false assertions are rejected with constant probability (which may be set to be arbitrarily close to 1).  ...  For every ε > 0, we present PCPs of length exp(log ε n) · n that can be verified by making a constant number of Boolean queries.  ...  Acknowledgments We are grateful to Avi Wigderson for collaborating with us at early stages of this research and to Irit Dinur for inspiring discussions at late stages of this research.  ... 
doi:10.1137/s0097539705446810 fatcat:lzcftg6gvngmhbdw7gnkmltwlu

Robust pcps of proximity, shorter pcps and applications to coding

Eli Ben-Sasson, Oded Goldreich, Prahladh Harsha, Madhu Sudan, Salil Vadhan
2004 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing - STOC '04  
In both cases, false assertions are rejected with constant probability (which may be set to be arbitrarily close to 1).  ...  For every ε > 0, we present PCPs of length exp(log ε n) · n that can be verified by making a constant number of Boolean queries.  ...  Acknowledgments We are grateful to Avi Wigderson for collaborating with us at early stages of this research and to Irit Dinur for inspiring discussions at late stages of this research.  ... 
doi:10.1145/1007352.1007361 dblp:conf/stoc/Ben-SassonGHSV04 fatcat:l7iua5ues5gwvc54mpdzjvcjqa

Derandomized Parallel Repetition via Structured PCPs

Irit Dinur, Or Meir
2011 Computational Complexity  
In this work we construct such PCP verifiers, i.e., PCPs that make only two queries and have sub-constant soundness error.  ...  Constructing PCPs with sub-constant soundness error and, at the same time, a minimal number of queries into the proof (namely two) is especially important due to applications for inapproximability.  ...  In this work we give a new construction of a PCP with sub-constant soundness error and two queries. This setting is particularly important for inapproximability, as will be discussed shortly below.  ... 
doi:10.1007/s00037-011-0013-5 fatcat:d66hpe5weje6hbpic5twqtf5re
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