Abstract
We consider the problem of constructing efficient locally decodable codes in the presence of a computationally bounded adversary. Assuming the existence of one-way functions, we construct efficient locally decodable codes with positive information rate and low (almost optimal) query complexity which can correctly decode any given bit of the message from constant channel error rate ρ. This compares favorably to our state of knowledge locally-decodable codes without cryptographic assumptions. For all our constructions, the probability for any polynomial-time adversary, that the decoding algorithm incorrectly decodes any bit of the message is negligible in the security parameter.
Part of this work was done when all the authors were at IPAM. The first author is supported in part by NSF Cybertrust grant No. 0430254, Xerox Innovation group Award and IBM Faculty Award. The second and third authors were supported in part from grants from the NSF ITR and Cybertrust programs (including grants 0627781, 0456717, and 0205594), a subgrant from SRI as part of the Army Cyber-TA program, an equipment grant from Intel, and an Alfred P. Sloan Foundation Research Fellowship.
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References
Babai, L., Fortnow, L., Levin, L.A., Szegedy, M.: Checking computations in polylogarithmic time. In: STOC, pp. 21–31 (1991)
Beimel, A., Ishai, Y.: Information-theoretic private information retrieval: A unified construction. In: ICALP, pp. 912–926 (2001)
Beimel, A., Ishai, Y., Kushilevitz, E., Raymond, J.-F.: Breaking the o(n1/(2k-1)) barrier for information-theoretic private information retrieval. In: FOCS, pp. 261–270 (2002)
Chor, B., Kushilevitz, E., Goldreich, O., Sudan, M.: Private information retrieval. J. ACM 45(6), 965–981 (1998)
Deshpande, A., Jain, R., Kavitha, T., Radhakrishnan, J., Lokam, S.V.: Better lower bounds for locally decodable codes. In: IEEE Conference on Computational Complexity, pp. 184–193. IEEE Computer Society Press, Los Alamitos (2002)
Gemmell, P., Lipton, R.J., Rubinfeld, R., Sudan, M., Wigderson, A.: Self-testing/correcting for polynomials and for approximate functions. In: STOC, pp. 32–42 (1991)
Goldreich, O., Goldwasser, S., Micali, S.: How to construct random functions. J. ACM 33(4), 792–807 (1986)
Goldreich, O., Karloff, H.J., Schulman, L.J., Trevisan, L.: Lower bounds for linear locally decodable codes and private information retrieval. In: IEEE Conference on Computational Complexity, pp. 175–183. IEEE Computer Society Press, Los Alamitos (2002)
Gopalana, P., Lipton, R.J., Ding, Y.Z.: Error correction against computationally bounded adversaries. In Manuscript (2004)
Justesen, J.: A class of constructive asymptotically good algebraic codes. IEEE Transactions on Information Theory 18, 652–656 (1972)
Katz, J., Trevisan, L.: On the efficiency of local decoding procedures for error-correcting codes. In: STOC, pp. 80–86 (2000)
Kerenidis, I., de Wolf, R.: Exponential lower bound for 2-query locally decodable codes via a quantum argument. In: STOC, pp. 106–115 (2003)
Kushilevitz, E., Ostrovsky, R.: Replication is not needed: Single database, computationally-private information retrieval. In: FOCS, pp. 364–373 (1997)
Langberg, M.: Private codes or succinct random codes that are (almost) perfect. In: FOCS, pp. 325–334 (2004)
Lipton, R.J.: A new approach to information theory. In: STACS, pp. 699–708 (1994)
Micali, S., Peikert, C., Sudan, M., Wilson, D.A.: Optimal Error Correction Against Computationally Bounded Noise. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, Springer, Heidelberg (2006)
Obata, K.: Optimal lower bounds for 2-query locally decodable linear codes. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 39–50. Springer, Heidelberg (2002)
Ostrovsky, R., Pandey, O., Sahai, A.: Private Locally Decodable Codes. available at http://eprint.iacr.org/2007/025/
Shiowattana, D., Lokam, S.V.: An optimal lower bound for 2-query locally decodable linear codes. Inf. Process. Lett. 97(6), 244–250 (2006)
Sipser, M., Spielman, D.A.: Expander codes. In: FOCS, pp. 566–576 (1994)
Smith, A.: Scrambling adversarial errors using few random bits. In: SODA (2007)
Sudan, M.: Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems, PhD Thesis, University of California at Berkley (1992)
Wehner, S., de Wolf, R.: Improved lower bounds for locally decodable codes and private information retrieval. In: ICALP, pp. 1424–1436 (2005)
Woodruff, D.: New lower bounds for general locally decodable codes. In: ECCC TR07-006 (2007)
Yekhanin, S.: Towards 3-query locally decodable codes of subexponential length. In: STOC (2007) Also appears on ECCC as TR06-127 under a different title.
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Ostrovsky, R., Pandey, O., Sahai, A. (2007). Private Locally Decodable Codes. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds) Automata, Languages and Programming. ICALP 2007. Lecture Notes in Computer Science, vol 4596. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73420-8_35
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DOI: https://doi.org/10.1007/978-3-540-73420-8_35
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