Query Complexity Lower Bounds for Local List-Decoding and Hard-Core Predicates (Even for Small Rate and Huge Lists)

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Abstract

A binary code Enc: {0, 1}k → {0, 1}n is (12 - ε, L)-list decodable if for every w ∈ {0, 1}n, there exists a set List(w) of size at most L, containing all messages m ∈ {0, 1}k such that the relative Hamming distance between Enc(m) and w is at most 12 - ε. A q-query local list-decoder for Enc is a randomized procedure Dec that when given oracle access to a string w, makes at most q oracle calls, and for every message m ∈ List(w), with high probability, there exists j ∈ [L] such that for every i ∈ [k], with high probability, Decw(i, j) = mi. We prove lower bounds on q, that apply even if L is huge (say L = 2k0.9) and the rate of Enc is small (meaning that n ≥ 2k): For ε = 1/kν for some constant 0 < ν < 1, we prove a lower bound of q = Ω(log(1 ε2/δ)), where δ is the error probability of the local list-decoder. This bound is tight as there is a matching upper bound by Goldreich and Levin (STOC 1989) of q = O(log(1 ε2/δ)) for the Hadamard code (which has n = 2k). This bound extends an earlier work of Grinberg, Shaltiel and Viola (FOCS 2018) which only works if n ≤ 2 and the number of coins tossed by Dec is small (and therefore does not apply to the Hadamard code, or other codes with low rate). For smaller ε, we prove a lower bound of roughly q = Ω(√1ε). To the best of our knowledge, this is the first lower bound on the number of queries of local list-decoders that gives q ≥ k for small ε. Local list-decoders with small ε form the key component in the celebrated theorem of Goldreich and Levin that extracts a hard-core predicate from a one-way function. We show that black-box proofs cannot improve the Goldreich-Levin theorem and produce a hard-core predicate that is hard to predict with probability 12 + 1(1) when provided with a one-way function f : {0, 1}` → {0, 1}`, where f is such that circuits of size poly(`) cannot invert f with probability ρ = 1/2 √` (or even ρ = 1/2Ω(`)). This limitation applies to any proof by black-box reduction (even if the reduction is allowed to use nonuniformity and has oracle access to f).

Original languageEnglish
Title of host publication12th Innovations in Theoretical Computer Science Conference (ITCS 2021)
EditorsJames R. Lee
Place of PublicationDagstuhl, Germany
PublisherSchloss Dagstuhl -- Leibniz-Zentrum für Informatik
Pages33:1-33:18
Volume185
ISBN (Electronic)9783959771771
ISBN (Print)978-3-95977-177-1
DOIs
StatePublished - 2021
Event12th Innovations in Theoretical Computer Science Conference, ITCS 2021 - Virtual, Online
Duration: 6 Jan 20218 Jan 2021

Publication series

NameLeibniz International Proceedings in Informatics (LIPIcs)
PublisherSchloss Dagstuhl--Leibniz-Zentrum für Informatik

Conference

Conference12th Innovations in Theoretical Computer Science Conference, ITCS 2021
CityVirtual, Online
Period6/01/218/01/21

Bibliographical note

Publisher Copyright:
© Noga Ron-Zewi, Ronen Shaltiel, and Nithin Varma.

Keywords

  • Black-box reduction
  • Hadamard code
  • Hard-core predicates
  • Local list-decoding

ASJC Scopus subject areas

  • Software

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