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Zero-Knowledge Protocols for Search Problems

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Security and Cryptography for Networks (SCN 2018)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11035))

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Abstract

We consider natural ways to extend the notion of Zero-Knowledge (ZK) Proofs beyond decision problems. Specifically, we consider search problems, and define zero-knowledge proofs in this context as interactive protocols in which the prover can establish the correctness of a solution to a given instance without the verifier learning anything beyond the intended solution, even if it deviates from the protocol.

The goal of this work is to initiate a study of Search Zero-Knowledge (search-ZK), the class of search problems for which such systems exist. This class trivially contains search problems where the validity of a solution can be efficiently verified (using a single message proof containing only the solution). A slightly less obvious, but still straightforward, way to obtain zero-knowledge proofs for search problems is to let the prover send a solution and prove in zero-knowledge that the instance-solution pair is valid. However, there may be other ways to obtain such zero-knowledge proofs, and they may be more advantageous.

In fact, we prove that there are search problems for which the aforementioned approach fails, but still search zero-knowledge protocols exist. On the other hand, we show sufficient conditions for search problems under which some form of zero-knowledge can be obtained using the straightforward way.

The full version of this paper can be found at https://eprint.iacr.org/2018/437.pdf. Supported by the Israel Science Foundation (Grant No. 468/14), Binational Science Foundation (Grants No. 2016726, 2014276), and by the European Union Horizon 2020 Research and Innovation Program via ERC Project REACT (Grant 756482) and via Project PROMETHEUS (Grant 780701).

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References

  1. Beimel, A., Carmi, P., Nissim, K., Weinreb, E.: Private approximation of search problems. In: Kleinberg, J.M. (ed.) Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, 21–23 May 2006, pp. 119–128. ACM (2006). https://doi.acm.org/10.1145/1132516.1132533

  2. Ben-Or, M., et al.: Everything provable is provable in zero-knowledge. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 37–56. Springer, New York (1990). https://doi.org/10.1007/0-387-34799-2_4. http://dl.acm.org/citation.cfm?id=646753.704888

    Chapter  Google Scholar 

  3. Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, 2–4 May 1988, pp. 1–10 (1988). https://doi.acm.org/10.1145/62212.62213

  4. Berger, B., Brakerski, Z.: Zero-knowledge protocols for search problems. Cryptology ePrint Archive, Report 2018/437 (2018). https://eprint.iacr.org/2018/437

  5. Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, 2–4 May 1988, pp. 11–19 (1988). http://doi.acm.org/10.1145/62212.62214

  6. Damgård, I., Goldreich, O., Okamoto, T., Wigderson, A.: Honest verifier vs dishonest verifier in public coin zero-knowledge proofs. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 325–338. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-44750-4_26

    Chapter  Google Scholar 

  7. Feigenbaum, J., Ishai, Y., Malkin, T., Nissim, K., Strauss, M.J., Wright, R.N.: Secure multiparty computation of approximations. ACM Trans. Algorithms 2(3), 435–472 (2006). https://doi.org/10.1145/1159892.1159900

    Article  MathSciNet  MATH  Google Scholar 

  8. Gat, E., Goldwasser, S.: Probabilistic search algorithms with unique answers and their cryptographic applications. Electron. Colloq. Comput. Complex. (ECCC) 18, 136 (2011). http://eccc.hpi-web.de/report/2011/136

    Google Scholar 

  9. Goldreich, O., Vadhan, S.: Comparing entropies in statistical zero knowledge with applications to the structure of SZK. In: Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity (Formerly: Structure in Complexity Theory Conference) (Cat.No.99CB36317), pp. 54–73 (1999). https://doi.org/10.1109/CCC.1999.766262

  10. Goldreich, O.: Computational Complexity - A Conceptual Perspective. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  11. Goldreich, O., Goldwasser, S., Ron, D.: On the possibilities and limitations of pseudodeterministic algorithms. In: Proceedings of the 4th Conference on Innovations in Theoretical Computer Science, ITCS 2013, pp. 127–138. ACM, New York (2013). https://doi.acm.org/10.1145/2422436.2422453

  12. Goldreich, O., Micali, S., Wigderson, A.: How to play any mental game or a completeness theorem for protocols with honest majority. In: STOC, pp. 218–229 (1987)

    Google Scholar 

  13. Goldreich, O., Micali, S., Wigderson, A.: Proofs that yield nothing but their validity for all languages in NP have zero-knowledge proof systems. J. ACM 38(3), 691–729 (1991). https://doi.org/10.1145/116825.116852

    Article  MathSciNet  MATH  Google Scholar 

  14. Goldreich, O., Oren, Y.: Definitions and properties of zero-knowledge proof systems. J. Cryptol. 7(1), 1–32 (1994). https://doi.org/10.1007/BF00195207

    Article  MathSciNet  MATH  Google Scholar 

  15. Goldreich, O., Sahai, A., Vadhan, S.P.: Honest-verifier statistical zero-knowledge equals general statistical zero-knowledge. In: Vitter, J.S. (ed.) Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, Dallas, Texas, USA, 23–26 May 1998, pp. 399–408. ACM (1998). https://doi.acm.org/10.1145/276698.276852

  16. Goldreich, O., Sahai, A., Vadhan, S.: Can statistical zero knowledge be made non-interactive? Or on the relationship of SZK and NISZK. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 467–484. Springer, Heidelberg (1999). https://doi.org/10.1007/3-540-48405-1_30. http://dl.acm.org/citation.cfm?id=646764.703982

    Chapter  Google Scholar 

  17. Goldwasser, S., Grossman, O.: Perfect bipartite matching in pseudo-deterministic RNC. Electron. Colloq. Comput. Complex. (ECCC) 22, 208 (2015). http://eccc.hpi-web.de/report/2015/208

    Google Scholar 

  18. Goldwasser, S., Grossman, O.: Bipartite perfect matching in pseudo-deterministic NC. In: Chatzigiannakis, I., Indyk, P., Kuhn, F., Muscholl, A. (eds.) 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, Warsaw, Poland, 10–14 July 2017. LIPIcs, vol. 80, pp. 87:1–87:13. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017). https://doi.org/10.4230/LIPIcs.ICALP.2017.87

  19. Goldwasser, S., Grossman, O., Holden, D.: Pseudo-deterministic proofs. In: Karlin, A.R. (ed.) 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, Cambridge, MA, USA, 11–14 January 2018. LIPIcs, vol. 94, pp. 17:1–17:18. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2018). https://doi.org/10.4230/LIPIcs.ITCS.2018.17

  20. Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18(1), 186–208 (1989). https://doi.org/10.1137/0218012

    Article  MathSciNet  MATH  Google Scholar 

  21. Grossman, O.: Finding primitive roots pseudo-deterministically. Electron. Colloq. Comput. Complex. (ECCC) 22, 207 (2015). http://eccc.hpi-web.de/report/2015/207

    Google Scholar 

  22. Halevi, S., Krauthgamer, R., Kushilevitz, E., Nissim, K.: Private approximation of NP-hard functions. In: Vitter, J.S., Spirakis, P.G., Yannakakis, M. (eds.) Proceedings on 33rd Annual ACM Symposium on Theory of Computing, Heraklion, Crete, Greece, 6–8 July 2001, pp. 550–559. ACM (2001). https://doi.acm.org/10.1145/380752.380850

  23. Okamoto, T.: On relationships between statistical zero-knowledge proofs. In: Miller, G.L. (ed.) Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, Philadelphia, Pennsylvania, USA, 22–24 May 1996, pp. 649–658. ACM (1996). https://doi.acm.org/10.1145/237814.238016

  24. Sahai, A., Vadhan, S.P.: A complete problem for statistical zero knowledge. J. ACM 50(2), 196–249 (2003). http://doi.acm.org/10.1145/636865.636868

    Article  MathSciNet  Google Scholar 

  25. Shamir, A.: IP = PSPACE. J. ACM 39(4), 869–877 (1992). http://doi.acm.org/10.1145/146585.146609

    Article  MathSciNet  Google Scholar 

  26. Yao, A.C.C.: Protocols for secure computations (extended abstract). In: FOCS, pp. 160–164 (1982)

    Google Scholar 

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Acknowledgements

We thank Ofer Grossman and Oded Goldreich for helpful discussions.

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Authors and Affiliations

  1. Weizmann Institute of Science, Rehovot, Israel

    Ben Berger & Zvika Brakerski

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  1. Ben Berger
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  2. Zvika Brakerski
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Correspondence to Ben Berger .

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Editors and Affiliations

  1. University of Catania, Catania, Italy

    Dario Catalano

  2. University of Salerno, Fisciano, Italy

    Roberto De Prisco

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Berger, B., Brakerski, Z. (2018). Zero-Knowledge Protocols for Search Problems. In: Catalano, D., De Prisco, R. (eds) Security and Cryptography for Networks. SCN 2018. Lecture Notes in Computer Science(), vol 11035. Springer, Cham. https://doi.org/10.1007/978-3-319-98113-0_16

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  • DOI: https://doi.org/10.1007/978-3-319-98113-0_16

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