文章基本信息
- 标题：Query Complexity Lower Bounds for Local List-Decoding and Hard-Core Predicates (Even for Small Rate and Huge Lists)
- 本地全文：下载
- 作者：Noga Ron-Zewi ; Ronen Shaltiel ; Nithin Varma 等
- 期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
- 电子版ISSN：1868-8969
- 出版年度：2021
- 卷号：185
- 页码：33:1-33:18
- DOI：10.4230/LIPIcs.ITCS.2021.33
- 出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
- 摘要：A binary code Enc:{0,1}^k â' {0,1}â¿ is (1/2-ε,L)-list decodable if for every w â^^ {0,1}â¿, there exists a set List(w) of size at most L, containing all messages m â^^ {0,1}^k such that the relative Hamming distance between Enc(m) and w is at most 1/2-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 â^^ List(w), with high probability, there exists j â^^ [L] such that for every i â^^ [k], with high probability, Dec^w(i,j) = m_i. We prove lower bounds on q, that apply even if L is huge (say L = 2^{k^{0.9}}) and the rate of Enc is small (meaning that n ⥠2^{k}): - For ε = 1/k^{ν} for some constant 0 < ν < 1, we prove a lower bound of q = Ω(log(1/δ)/ε²), 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/δ)/ε²) for the Hadamard code (which has n = 2^k). This bound extends an earlier work of Grinberg, Shaltiel and Viola (FOCS 2018) which only works if n ⤠2^{k^ν} 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(1/(â^Sε)). 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 1/2 1/ð"^Ï(1) when provided with a one-way function f:{0,1}^ð" â' {0,1}^ð", where f is such that circuits of size poly(ð") cannot invert f with probability Ï = 1/2^â^Sð" (or even Ï = 1/2^Ω(ð")). 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).
- 关键词：Local list-decoding; Hard-core predicates; Black-box reduction; Hadamard code
