Skip to main content
SpringerLink
Log in
Book cover

International Workshop on Approximation Algorithms for Combinatorial Optimization

International Workshop on Randomization and Approximation Techniques in Computer Science

APPROX 2007, RANDOM 2007: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques pp 311–325Cite as

On Locally Decodable Codes, Self-correctable Codes, and t-Private PIR

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4627))

Abstract

A k-query locally decodable code (LDC) allows to probabilistically decode any bit of an encoded message by probing only k bits of its corrupted encoding. A stronger and desirable property is that of self-correction, allowing to efficiently recover not only bits of the message but also arbitrary bits of its encoding. In contrast to the initial constructions of LDCs, the recent and most efficient constructions are not known to be self-correctable. The existence of self-correctable codes of comparable efficiency remains open.

A closely related problem with a very different motivation is that of private information retrieval (PIR). A k-server PIR protocol allows a user to retrieve the i-th bit of a database, which is replicated among k servers, without revealing information about i to any individual server. A natural generalization is t-private PIR, which keeps i hidden from any t colluding servers. In contrast to the initial PIR protocols, it is not known how to generalize the recent and most efficient protocols to yield t-private protocols of comparable efficiency.

In this work we study both of the above questions, showing that they are in fact related. We start by presenting a general transformation of any 1-private PIR protocol (equivalently, LDC) into a t-private protocol with a similar amount of communication per server. Combined with the recent result of Yekhanin (STOC 2007), this yields a significant improvement over previous t-private PIR protocols. A major weakness of our transformation is that the number of servers in the resulting t-private protocols grows exponentially with t. We show that if the underlying LDC satisfies the stronger self-correction property, then there is a similar transformation in which the number of servers grows only linearly with t, which is the best one can hope for. Finally, we study the question of closing the current gap between the complexity of the best known LDC and that of self-correctable codes, and relate this question to a conjecture of Hamada concerning the algebraic rank of combinatorial designs.

Research supported by grant 1310/06 from the Israel Science Foundation, grant 2004361 from the U.S.-Israel Binational Science Foundation, and the Technion VPR fund. Part of this research was done while visiting IPAM.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Ambainis, A.: Upper bound on the communication complexity of private information retrieval. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 401–407. Springer, Heidelberg (1997)

    Chapter  Google Scholar 

  2. Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. of the ACM 45(1), 70–122 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof Verification and the Hardness of Approximation Problems. J. of the ACM 45(3), 501–555 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  4. Assmus, E.F., Key, J.D.: Designs and codes: An update. Designs, Codes and Cryptography 9(1), 7–27 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Babai, L., Fortnow, L., Levin, L.A., Szegedy, M.: Checking Computations in Polylogarithmic Time. In: Proc. STOC 1991, pp. 21–31 (1991)

    Google Scholar 

  6. Babai, L., Fortnow, L., Nisan, N., Wigderson, A.: BPP Has Subexponential Time Simulations Unless EXPTIME has Publishable Proofs. Computational Complexity 3, 307–318 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Beaver, D., Feigenbaum, J.: Hiding instances in multioracle queries. In: Proc. of STACS 1990, pp. 37–48 (1990)

    Google Scholar 

  8. Beimel, A., Ishai, Y.: Information-theoretic private information retrieval: A unified construction. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 912–926. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  9. Beimel, A., Ishai, Y., Kushilevitz, E.: General constructions for information-theoretic private information retrieval. J. of Computer and Systems Sciences 71(2), 213–247 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Beimel, A., Ishai, Y., Kushilevitz, E., Raymond, J.F.: Breaking the \(O(n\frac{1}{2k-1})\) barrier for information-theoretic private information retrieval. In: FOCS. Proc. of the 43rd Annual IEEE Symposium on Foundations of Computer Science 2002, pp. 261–270. IEEE Computer Society Press, Los Alamitos (2002)

    Google Scholar 

  11. Beth, T., Jungnickel, D., Lenz, H.: Design Theory, 2nd edn., vol. 1. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  12. Blum, M., Kannan, S.: Designing programs that check their work. J. of the ACM 42(1), 269–291 (1995)

    Article  MATH  Google Scholar 

  13. Chor, B., Gilboa, N.: Computationally private information retrieval. In: Proc. of STOC 1997, pp. 304–313 (1997)

    Google Scholar 

  14. Chor, B., Goldreich, O., Kushilevitz, E., Sudan, M.: Private information retrieval. In: FOCS 1995. Proc. of the 36th Annual IEEE Symposium on Foundations of Computer Science, pp. 41–50. IEEE Computer Society Press, Los Alamitos (1995)

    Google Scholar 

  15. Di-Crescenzo, G., Ishai, Y., Ostrovsky, R.: Universal service-providers for private information retrieval. J. of Cryptology 14(1), 37–74 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Feige, U., Goldwasser, S., Lovasz, L., Safra, S., Szegedy, M.: Interactive Proofs and the Hardness of Approximating Cliques. J. of the ACM 43(2), 268–292 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gasarch, W.: A survey on private information retrieval. Bulletin of the European Association for Theoretical Computer Science 82, 72–107 (2004)

    MathSciNet  MATH  Google Scholar 

  18. Hamada, N.: On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its application to error-correcting codes. Hiroshima Math J. 3, 153–226 (1973)

    MathSciNet  MATH  Google Scholar 

  19. Hamada, N.: The geometric structure and the p-rank of an affine triple system derived from a nonassociative moufang loop with the maximum associative center. J. Comb. Theory, Ser. A 30(3), 285–297 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hirt, M., Maurer, U.M.: Player simulation and general adversary structures in perfect multiparty computation. J. of Cryptology 13(1), 31–60 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ishai, Y., Kushilevitz, E.: Improved upper bounds on information-theoretic private information retrieval. In: Proc. STOC 1999, pp. 79–88 (1999)

    Google Scholar 

  22. Ishai, Y., Kushilevitz, E.: On the Hardness of Information-Theoretic Multiparty Computation. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 439–455. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  23. Katz, J., Trevisan, L.: On the efficiency of local decoding procedures for error-correcting codes. In: STOC 2000, pp. 80–86 (2000)

    Google Scholar 

  24. Kerenidis, I., de Wolf, R.: Exponential lower bound for 2-query locally decodable codes. J. of Computer and Systems Sciences, 395–420 (2004)

    Google Scholar 

  25. Kushilevitz, E., Ostrovsky, R.: Replication is not needed: Single database, computationally-private information retrieval. In: FOCS 1997, pp. 364–373 (1997)

    Google Scholar 

  26. Lipton, R.: Efficient checking of computations. In: STACS 1990, pp. 207–215 (1990)

    Google Scholar 

  27. Lu, C.-J., Reingold, O., Vadhan, S.P., Wigderson, A.: Extractors: optimal up to constant factors. In: Proc.STOC 2003, pp. 602–611 (2003)

    Google Scholar 

  28. Raghavendra, P.: A note on Yekhanin’s locally decodable codes. In: Electronic Colloquium on Computational Complexity (ECCC) (2007)

    Google Scholar 

  29. Shamir, A.: How to share a secret. Communications of the ACM 22, 612–613 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sudan, M., Trevisan, L., Vadhan, S.P.: Pseudorandom Generators without the XOR Lemma. J. of Computer and Systems Sciences 62(2), 236–266 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. Tonchev, V.D.: Linear perfert codes and a characterization of the classical designs. Designs, Codes and Cryptography 17, 121–128 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  32. Trevisan, L.: Some applications of coding theory in computational complexity. Quaderni di Matematica 13, 347–424 (2004)

    MathSciNet  MATH  Google Scholar 

  33. Wehner, S., de Wolf, R.: Improved lower bounds for locally decodable codes and private information retrieval. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1424–1436. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  34. Woodruff, D.: New lower bounds for general locally decodable codes. In: ECCC, Report No. 6 (2007)

    Google Scholar 

  35. Woodruff, D., Yekhanin, S.: A geometric approach to information-theoretic private information retrieval. In: proc. of CCC 2005, pp. 275–284 (2005)

    Google Scholar 

  36. Yekhanin, S.: Towards 3-Query Locally Decodable Codes of Subexponential Length. In: Proc. STOC 2007 (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Science, Technion, Haifa, Israel

    Omer Barkol, Yuval Ishai & Enav Weinreb

Authors
  1. Omer Barkol
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuval Ishai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Enav Weinreb
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computer Science Dept., Princeton University, 35 Olden Street, NJ 08540-5233, Princeton, USA

    Moses Charikar

  2. Institut für Informatik, Christian-Albrechts-Universität zu Kiel,, Olshausenstr. 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Omer Reingold

  4. Centre Universitaire d’Informatique,, Battelle bâtiment A, route de Drize 7, 1227, Geneva, Carouge, Switzerland

    José D. P. Rolim

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Barkol, O., Ishai, Y., Weinreb, E. (2007). On Locally Decodable Codes, Self-correctable Codes, and t-Private PIR. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2007 2007. Lecture Notes in Computer Science, vol 4627. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74208-1_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-74208-1_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-74207-4

  • Online ISBN: 978-3-540-74208-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation