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Probabilistically Checkable Proofs of Proximity with Zero-Knowledge

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Part of the Lecture Notes in Computer Science book series (LNSC,volume 8349)


A probabilistically Checkable Proof (PCP) allows a randomized verifier, with oracle access to a purported proof, to probabilistically verify an input statement of the form “x ∈ L” by querying only few bits of the proof. A PCP of proximity (PCPP) has the additional feature of allowing the verifier to query only few bits of the input x, where if the input is accepted then the verifier is guaranteed that (with high probability) the input is close to some x′ ∈ L.

Motivated by their usefulness for sublinear-communication cryptography, we initiate the study of a natural zero-knowledge variant of PCPP (ZKPCPP), where the view of any verifier making a bounded number of queries can be efficiently simulated by making the same number of queries to the input oracle alone. This new notion provides a useful extension of the standard notion of zero-knowledge PCPs. We obtain two types of results.

  • Constructions. We obtain the first constructions of query-efficient ZKPCPPs via a general transformation which combines standard query-efficient PCPPs with protocols for secure multiparty computation. As a byproduct, our construction provides a conceptually simpler alternative to a previous construction of honest-verifier zero-knowledge PCPs due to Dwork et al. (Crypto ’92).

  • Applications. We motivate the notion of ZKPCPPs by applying it towards sublinear-communication implementations of commit-and-prove functionalities. Concretely, we present the first sublinear-communication commit-and-prove protocols which make a black-box use of a collision-resistant hash function, and the first such multiparty protocols which offer information-theoretic security in the presence of an honest majority.


  • Query Complexity
  • Trusted Third Party
  • Secret Sharing Scheme
  • Random String
  • Commitment Scheme

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Research supported by the European Union’s Tenth Framework Programme (FP10/ 2010-2016) under grant agreement no. 259426 ERC-CaC. The first author was additionally supported by ISF grant 1361/10 and BSF grants 2008411 and 2012366.


  1. Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Electronic Colloquium on Computational Complexity (ECCC) 5(8) (1998)

    Google Scholar 

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

    CrossRef  MATH  MathSciNet  Google Scholar 

  3. Barak, B.: How to go beyond the black-box simulation barrier. In: FOCS, pp. 106–115 (2001)

    Google Scholar 

  4. Barak, B., Goldreich, O.: Universal arguments and their applications. SIAM J. Comput. 38(5), 1661–1694 (2008)

    CrossRef  MATH  MathSciNet  Google Scholar 

  5. Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: STOC, pp. 1–10 (1988)

    Google Scholar 

  6. Ben-Or, M., Rabin, T.: Verifiable secret sharing and multiparty protocols with honest majority. In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing (STOC), Seattle, Washigton, USA, May 14-17, pp. 73–85. ACM (1989)

    Google Scholar 

  7. Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.P.: Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM J. Comput. 36(4), 889–974 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Ben-Sasson, E., Sudan, M.: Short PCPs with polylog query complexity. SIAM J. Comput. 38(2), 551–607 (2008)

    CrossRef  MathSciNet  Google Scholar 

  9. Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: STOC, pp. 11–19 (1988)

    Google Scholar 

  10. Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable secret sharing and achieving simultaneity in the presence of faults (extended abstract). In: FOCS, pp. 383–395 (1985)

    Google Scholar 

  11. Dinur, I.: The PCP theorem by gap amplification. In: STOC, pp. 241–250 (2006)

    Google Scholar 

  12. Dinur, I., Reingold, O.: Assignment testers: Towards a combinatorial proof of the PCP-theorem. In: FOCS, pp. 155–164 (2004)

    Google Scholar 

  13. Dwork, C., Feige, U., Kilian, J., Naor, M., Safra, M.: Low communication 2-prover zero-knowledge proofs for NP. In: Brickell, E.F. (ed.) Advances in Cryptology - CRYPTO 1992. LNCS, vol. 740, pp. 215–227. Springer, Heidelberg (1993)

    Google Scholar 

  14. Feldman, P.: A practical scheme for non-interactive verifiable secret sharing. In: FOCS, pp. 427–437. IEEE Computer Society (1987)

    Google Scholar 

  15. Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18(1), 186–208 (1989)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. Ishai, Y., Kushilevitz, E., Ostrovsky, R., Sahai, A.: Zero-knowledge from secure multiparty computation. In: STOC, pp. 21–30 (2007)

    Google Scholar 

  17. Ishai, Y., Mahmoody, M., Sahai, A.: On efficient zero-knowledge PCPs. In: Cramer, R. (ed.) TCC 2012. LNCS, vol. 7194, pp. 151–168. Springer, Heidelberg (2012)

    CrossRef  Google Scholar 

  18. Ishai, Y., Sahai, A., Viderman, M., Weiss, M.: Zero knowledge LTCs and their applications. In: APPROX-RANDOM, pp. 607–622 (2013)

    Google Scholar 

  19. Kilian, J.: Uses of randomness in algorithms and protocols. MIT Press (1990)

    Google Scholar 

  20. Kilian, J., Naor, M.: On the complexity of statistical reasoning. In: Proceedings of the Third Israel Symposium on the Theory of Computing and Systems, pp. 209–217. IEEE (1995)

    Google Scholar 

  21. Kilian, J., Petrank, E., Tardos, G.: Probabilistically checkable proofs with zero knowledge. In: STOC, pp. 496–505 (1997)

    Google Scholar 

  22. Meir, O.: Combinatorial construction of locally testable codes. SIAM J. Comput. 39(2), 491–544 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

  23. Mie, T.: Short PCPPs verifiable in polylogarithmic time with O(1) queries. Ann. Math. Artif. Intell. 56(3-4), 313–338 (2009)

    CrossRef  MATH  MathSciNet  Google Scholar 

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© 2014 International Association for Cryptologic Research

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Ishai, Y., Weiss, M. (2014). Probabilistically Checkable Proofs of Proximity with Zero-Knowledge. In: Lindell, Y. (eds) Theory of Cryptography. TCC 2014. Lecture Notes in Computer Science, vol 8349. Springer, Berlin, Heidelberg.

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