## Exponential lower bound for 2-query locally decodable codes via a quantum argument (2003)

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Venue: | Journal of Computer and System Sciences |

Citations: | 118 - 15 self |

### BibTeX

@INPROCEEDINGS{Kerenidis03exponentiallower,

author = {Iordanis Kerenidis and Ronald De Wolf},

title = {Exponential lower bound for 2-query locally decodable codes via a quantum argument},

booktitle = {Journal of Computer and System Sciences},

year = {2003},

pages = {106--115}

}

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### Abstract

Abstract A locally decodable code encodes n-bit strings x in m-bit codewords C(x) in such a way that one can recover any bit xi from a corrupted codeword by querying only a few bits of that word. We use a quantum argument to prove that LDCs with 2 classical queries require exponential length: m = 2 \Omega (n). Previously this was known only for linear codes (Goldreich et al. 02). The

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Citation Context ... a few positions of that word. Such schemes are called locally decodable codes (LDCs). They have found various applications in complexity theory and cryptography, such as self-correcting computations =-=[5, 24, 17, 16, 18]-=-, Probabilistically Checkable Proofs [2], worst-case to average-case reductions [3, 34], private information retrieval [11], and extractors [25]. Informally, LDCs are described as follows: A (q; ffi; ... |

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Citation Context ..., then there exist protocols with significantly less communication. Chor et al. [11] exhibited a 2-server PIR scheme with communication complexity O(n 1=3 ) and one with O(n 1=k ) for k ? 2. Ambainis =-=[1]-=- improved the latter to O(n 1=(2k\Gamma 1) ). Beimel et al. [8] improved the communication complexity to O(n 2 log log k=k log k ). Their results improve the previous best bounds for all k * 3 but not... |

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Citation Context ...that m * 2 ffi"n=8 if C is a linear code. Obata [29] subsequently strengthened the dependence on " to m * 2 \Omega (ffin=(1\Gamma 2")) , which is essentially optimal. Very recently, Ben-Sasson et al. =-=[9]-=- studied a relaxed notion of LDCs where the decoder is allowed to output "don't know" for a constant fraction of the indices. They construct relaxed LDCs with a constant number of queries and size m =... |

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Citation Context ...h a constant number of queries. There is still a large gap between the best known upper and lower bounds. In particular, it is open whether m = poly(n) is achievable with constant q. Goldreich et al. =-=[20]-=- examined the case q = 2, and showed that m * 2 ffi"n=8 if C is a linear code. Obata [29] subsequently strengthened the dependence on " to m * 2 \Omega (ffin=(1\Gamma 2")) , which is essentially optim... |

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Citation Context ...hy, such as self-correcting computations [5, 24, 17, 16, 18], Probabilistically Checkable Proofs [2], worst-case to average-case reductions [3, 34], private information retrieval [11], and extractors =-=[25]-=-. Informally, LDCs are described as follows: A (q; ffi; ")-locally decodable code encodes n-bit strings x into m-bit codewords C(x), such that, for each i, the bit xi can be recovered with probability... |

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Citation Context ...or all k * 3 but not for k = 2. No general lower bounds better than \Omega (log n) are known for PIRs with k * 2 servers. For the case of 2 servers, the best known lower bound is 4 log n, due to Mann =-=[26]-=-. A PIR scheme is linear if for every query that the user makes, the answer bits are linear combinations of the bits of x. Goldreich et al. [20] proved that linear 2-server PIRs with t-bit queries and... |

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Citation Context ... n, q, δ, and ε. For q = polylog(n), Babai et al. [2] showed how to achieve m = n 1+o(1) , for some fixed δ,ε. For constant q, the best known upper bounds are of the form m = 2 O(n1/(q−1) ) (see e.g. =-=[3]-=-). The study of lower bounds on m was initiated by Katz and Trevisan [13]. They proved that for q = 1, LDCs do not exist if n is larger than some constant depending on δ and ε. For q ≥ 2, they proved ... |

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Citation Context ... Chor et al. [7] exhibited a 2-server PIR with communication complexity O(n 1/3 ) and with O(n 1/k ) for k > 2. Ambainis [1] improved the latter to O(n 1/(2k−1) ), and some more recent references are =-=[3, 4]-=-. No general lower bounds better than Ω(log n) are known for PIRs with k ≥ 2 servers. Goldreich et al. [11] proved that linear 2-server PIRs with t-bit queries, and a-bit answers where the user looks ... |

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Citation Context ... in fact stronger than the previous classical lower bounds of Buhrman et al. [10]. Sen and Venkatesh did the same for data structures for the predecessor problem [32, quant-ph version]. Klauck et al. =-=[23]-=- proved lower bounds for the k-round quantum communication complexity of the tree-jumping problem that are somewhat stronger than the previous best classical lower bounds. In cryptography, Gisin, Renn... |

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Citation Context ...nd lower bounds. In particular, it is open whether m = poly(n) is achievable with constant q. Goldreich et al. [20] examined the case q = 2, and showed that m * 2 ffi"n=8 if C is a linear code. Obata =-=[29]-=- subsequently strengthened the dependence on " to m * 2 \Omega (ffin=(1\Gamma 2")) , which is essentially optimal. Very recently, Ben-Sasson et al. [9] studied a relaxed notion of LDCs where the decod... |

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Citation Context ...depending on ffi and ". For q * 2, they proved a bound of m = \Omega (n 1+1=(q\Gamma 1) ) if the q queries are made non-adaptively; this bound was generalized to the adaptive case by Deshpande et al. =-=[14]-=-. This establishes superlinear (but at most quadratic) lower bounds on the length of LDCs with a constant number of queries. There is still a large gap between the best known upper and lower bounds. I... |

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3 |
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2 |
Nearly tight bounds for private information retrieval systems.Technical Report 2002-L001N
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Citation Context ...tz and Trevisan give a 4:4 log n lower bound for the general 2-server PIR. This is the first, very modest improvement on the bound of Mann [26]. Subsequently to our work, Beigel, Fortnow, and Gasarch =-=[7]-=- found a classical proof that a 2-server PIR with perfect recovery (" = 1=2) and 1-bit answers needs query length * n \Gammas2. However, their proof does not seem to extend to the case " ! 1=2, or to ... |

2 | Exact bounds on any 1-round private information retrieval with 1-bit answers. Unpublished manuscript - Beigel, Fortnow, et al. - 2002 |

1 |
A Optimal 1-Query Quantum Algorithms for 2-Bit Functions In this appendix we show that every 2-bit Boolean function f can be computed with success probability 9=10 using only one quantum query, and that this is optimal for functions like AND and OR. If f
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- 2002
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