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How to Correct Errors in Multi-server PIR

  • Kaoru KurosawaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11922)

Abstract

Suppose that there exist a user and \(\ell \) servers \(S_1,\ldots ,S_{\ell }\). Each server \(S_j\) holds a copy of a database \(\mathbf {x}=(x_1, \ldots , x_n) \in \{0,1\}^n\), and the user holds a secret index \(i_0 \in \{1, \ldots , n\}\). A b error correcting \(\ell \) server PIR (Private Information Retrieval) scheme allows a user to retrieve \(x_{i_0}\) correctly even if and b or less servers return false answers while each server learns no information on \(i_0\) in the information theoretic sense. Although there exists such a scheme with the total communication cost \( O(n^{1/(2k-1)} \times k\ell \log {\ell } ) \) where \(k=\ell -2b\), the decoding algorithm is very inefficient.

In this paper, we show an efficient decoding algorithm for this b error correcting \(\ell \) server PIR scheme. It runs in time \(O(\ell ^3)\).

Keywords

Private information retrieval Information theoretic Error correcting 

References

  1. 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).  https://doi.org/10.1007/3-540-63165-8_196CrossRefGoogle Scholar
  2. 2.
    Banawan, K., Ulukus, S.: Private information retrieval from Byzantine and colluding databases, Allerton, pp. 1091–1098 (2017)Google Scholar
  3. 3.
    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).  https://doi.org/10.1007/3-540-48224-5_74CrossRefGoogle Scholar
  4. 4.
    Beimel, A., Ishai, Y., Kushilevitz, E., Raymond, J.F.: Breaking the O(n1/(2k ]1)) barrier for information-theoretic private information retrieval. In: FOCS f02, pp. 261–270 (2002)Google Scholar
  5. 5.
    Beimel, A., Stahl, Y.: Robust information-theoretic private information retrieval. J. Cryptol. 20(3), 295–321 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chee, Y.M., Feng, T., Ling, S., Wang, H., Zhang, L.F.: Query-efficient locally decodable codes of subexponential length. Comput. Complex. 22(1), 159–189 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Chor, B., Gilboa, N.: Comput. Private Information Retrieval, STOC (1997)Google Scholar
  8. 8.
    Chor, B., Goldreich, O., Kushilevitz, E., Sudan, M.: Private information retrieval. J. ACM 45(6), 965–981 (1998)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Cachin, C., Micali, S., Stadler, M.: Computationally private information retrieval with polylogarithmic communication. In: Stern, J. (ed.) EUROCRYPT 1999. LNCS, vol. 1592, pp. 402–414. Springer, Heidelberg (1999).  https://doi.org/10.1007/3-540-48910-X_28CrossRefGoogle Scholar
  10. 10.
    Dvir, Z., Gopi, S.: 2-server PIR with subpolynomial communication. J. ACM 63(4), 39 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Devet, C., Goldberg, I., Heninger, N.: Optimally Robust Private Information Retrieval. In: USENIX Security Symposium, pp. 269–283 (2012)Google Scholar
  12. 12.
    Efremenko, K.: 3-query locally decodable codes of subexponential length. SIAM J. Comput. 41(6), 1694–1703 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gasarch, W.: A survey on private information retrieval. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.8246
  14. 14.
    Goldberg, I.: Improving the robustness of private information retrieval. IEEE Symp. Secur. Priv. 131–148, 131–148 (2007)Google Scholar
  15. 15.
    Gentry, C., Ramzan, Z.: Single-database private information retrieval with constant communication rate. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 803–815. Springer, Heidelberg (2005).  https://doi.org/10.1007/11523468_65CrossRefGoogle Scholar
  16. 16.
    Itoh, T., Suzuki, Y.: Improved constructions for query-efficient locally decodable codes of subexponential length. IEICE Trans. 93(2), 263–270 (2010)CrossRefGoogle Scholar
  17. 17.
    Kushilevitz, E., Ostrovsky, R.: Replication is not needed: single database, computationally private information retrieval. In: FOCS (1997)Google Scholar
  18. 18.
    Lipmaa, H.: An oblivious transfer protocol with log-squared communication. In: Zhou, J., Lopez, J., Deng, R.H., Bao, F. (eds.) ISC 2005. LNCS, vol. 3650, pp. 314–328. Springer, Heidelberg (2005).  https://doi.org/10.1007/11556992_23CrossRefGoogle Scholar
  19. 19.
    Sun, H., Jafar, S.A.: The capacity of private information retrieval. IEEE Trans. Information Theory 63(7), 4075–4088 (2017)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sun, H., Jafar, S.A.: The capacity of robust private information retrieval with colluding databases. IEEE Trans. Information Theory 64(4), 2361–2370 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Woodruff, D., Yekhanin, S.: A geometric approach to information-theoretic private information retrieval. SIAM J. Comput. 37(4), 1046–1056 (2007)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Yekhanin, S.: Towards 3-query locally decodable codes of subexponential length. J. ACM 55, 1 (2008)MathSciNetCrossRefGoogle Scholar
  23. 23.
  24. 24.
    Lecture 10 Reed Solomon Codes Decoding: Berlekamp-Welch. http://people.ece.umn.edu/~arya/EE5583/lecture10.pdf

Copyright information

© International Association for Cryptologic Research 2019

Authors and Affiliations

  1. 1.Ibaraki UniversityHitachiJapan

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