computational complexity

, Volume 15, Issue 3, pp 263–296

Lower bounds for linear locally decodable codes and private information retrieval

Authors

    • Computer Science DepartmentWeizmann Institute of Science
  • Howard Karloff
    • AT&T Labs–Research
  • Leonard J. Schulman
    • Caltech, MC256-80
  • Luca Trevisan
    • Computer Science DivisionUniversity of California at Berkeley
Article

DOI: 10.1007/s00037-006-0216-3

Cite this article as:
Goldreich, O., Karloff, H., Schulman, L.J. et al. comput. complex. (2006) 15: 263. doi:10.1007/s00037-006-0216-3

Abstract.

We prove that if a linear error-correcting code C:{0, 1}n→{0, 1}m is such that a bit of the message can be probabilistically reconstructed by looking at two entries of a corrupted codeword, then m = 2Ω (n). We also present several extensions of this result.

We show a reduction from the complexity of one-round, information-theoretic Private Information Retrieval Systems (with two servers) to Locally Decodable Codes, and conclude that if all the servers’ answers are linear combinations of the database content, then t  =  Ω (n/2a), where t is the length of the user’s query and a is the length of the servers’ answers. Actually, 2a can be replaced by O(ak), where k is the number of bit locations in the answer that are actually inspected in the reconstruction.

Keywords.

Error-correcting codeslower boundslocally decodable codesprivate information retrieval

Subject classification.

68P30

Copyright information

© Birkhäuser Verlag, Basel 2006