Local proofs approaching the witness length [extended abstract]

Noga Ron-Zewi, Ron D. Rothblum

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

10 Scopus citations

Abstract

Interactive oracle proofs (IOPs) are a hybrid between interactive proofs and PCPs. In an IOP the prover is allowed to interact with a verifier (like in an interactive proof) by sending relatively long messages to the verifier, who in turn is only allowed to query a few of the bits that were sent (like in a PCP). Efficient IOPs are at the core of leading practical implementations of highly efficient proof-systems. In this work we construct, for a large class of N P relations, IOPs in which the communication complexity approaches the witness length. More precisely, for any N P relation for which membership can be decided in polynomial-time and bounded polynomial space (e.g., SAT, Hamiltonicity, Clique, Vertex-Cover, etc.) and for any constant gamma > 0, we construct an IOP with communication complexity (1+ gamma) cdot n, where n is the original witness length. The number of rounds, as well as the number of queries made by the IOP verifier, are constant. This result improves over prior works on short IOPs/PCPs in two ways. First, the communication complexity in these short IOPs is proportional to the complexity of verifying the NP witness, which can be polynomially larger than the witness size. Second, even ignoring the difference between witness length and non-deterministic verification time, prior works incur (at the very least) a large constant multiplicative overhead to the communication complexity. In particular, as a special case, we also obtain an IOP for CircuitSAT with communication complexity (1+ gamma) cdot t, for circuits of size t and any constant gamma > 0. This improves upon the prior state-of-the-art work of Ben Sasson et al. (ICALP, 2017) who construct an IOP for CircuitSAT with communication length c cdot t for a large (unspecified) constant c geq 1. Our proof leverages the local testability and (relaxed) local correctability of high-rate tensor codes, as well as their support of a sumcheck-like procedure. In particular, we bypass the barrier imposed by the low rate of multiplication codes (e.g., Reed-Solomon, Reed-Muller or AG codes)- A key building block of all known short PCP/IOP constructions.

Original languageEnglish
Title of host publicationProceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020
PublisherIEEE Computer Society
Pages846-857
Number of pages12
ISBN (Electronic)9781728196213
DOIs
StatePublished - 1 Nov 2020
Externally publishedYes
Event61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States
Duration: 16 Nov 202019 Nov 2020

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2020-November
ISSN (Print)0272-5428

Conference

Conference61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020
Country/TerritoryUnited States
CityVirtual, Durham
Period16/11/2019/11/20

Keywords

  • computational complexity

ASJC Scopus subject areas

  • Computer Science (all)

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