Advertisement

Promise Zero Knowledge and Its Applications to Round Optimal MPC

  • Saikrishna Badrinarayanan
  • Vipul Goyal
  • Abhishek Jain
  • Yael Tauman Kalai
  • Dakshita Khurana
  • Amit Sahai
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10992)

Abstract

We devise a new partitioned simulation technique for MPC where the simulator uses different strategies for simulating the view of aborting adversaries and non-aborting adversaries. The protagonist of this technique is a new notion of promise zero knowledge (ZK) where the ZK property only holds against non-aborting verifiers. We show how to realize promise ZK in three rounds in the simultaneous-message model assuming polynomially hard DDH (or QR or N\(^{th}\)-Residuosity).

We demonstrate the following applications of our new technique:
  • We construct the first round-optimal (i.e., four round) MPC protocol for general functions based on polynomially hard DDH (or QR or N\(^{th}\)-Residuosity).

  • We further show how to overcome the four-round barrier for MPC by constructing a three-round protocol for “list coin-tossing” – a slight relaxation of coin-tossing that suffices for most conceivable applications – based on polynomially hard DDH (or QR or N\(^{th}\)-Residuosity). This result generalizes to randomized input-less functionalities.

Previously, four round MPC protocols required sub-exponential-time hardness assumptions and no multi-party three-round protocols were known for any relaxed security notions with polynomial-time simulation against malicious adversaries.

In order to base security on polynomial-time standard assumptions, we also rely upon a leveled rewinding security technique that can be viewed as a polynomial-time alternative to leveled complexity leveraging for achieving “non-malleability” across different primitives.

Notes

Acknowledgements

We thank Silas Richelson, Shai Halevi, Carmit Hazay, Antigoni Polychroniadou, Muthuramakrishnan Venkitasubramaniam and the anonymous reviewers of STOC 2018 for useful comments in an earlier draft of this paper.

References

  1. 1.
    Ananth, P., Choudhuri, A.R., Jain, A.: A new approach to round-optimal secure multiparty computation. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10401, pp. 468–499. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63688-7_16CrossRefGoogle Scholar
  2. 2.
    Barak, B.: How to go beyond the black-box simulation barrier. In: 2001 Proceedings of the 42nd IEEE Symposium on Foundations of Computer Science, pp. 106–115. IEEE (2001)Google Scholar
  3. 3.
    Barak, B., Sahai, A.: How to play almost any mental game over the net - concurrent composition via super-polynomial simulation. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), 23–25 October 2005, Pittsburgh, PA, USA, pp. 543–552 (2005).  https://doi.org/10.1109/SFCS.2005.43
  4. 4.
    Beaver, D., Micali, S., Rogaway, P.: The round complexity of secure protocols (extended abstract). In: STOC, pp. 503–513 (1990)Google Scholar
  5. 5.
    Benhamouda, F., Lin, H.: k-round multiparty computation from k-round oblivious transfer via garbled interactive circuits. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 500–532. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78375-8_17CrossRefGoogle Scholar
  6. 6.
    Bitansky, N., Kalai, Y.T., Paneth, O.: Multi-collision resistance: a paradigm for keyless hash functions. IACR Cryptology ePrint Archive 2017, 488 (2017). http://eprint.iacr.org/2017/488
  7. 7.
    Brakerski, Z., Halevi, S., Polychroniadou, A.: Four round secure computation without setup. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 645–677. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70500-2_22CrossRefGoogle Scholar
  8. 8.
    Ciampi, M., Ostrovsky, R., Siniscalchi, L., Visconti, I.: Delayed-input non-malleable zero knowledge and multi-party coin tossing in four rounds. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017. LNCS, vol. 10677, pp. 711–742. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70500-2_24CrossRefGoogle Scholar
  9. 9.
    Ciampi, M., Ostrovsky, R., Siniscalchi, L., Visconti, I.: Concurrent non-malleable commitments (and more) in 3 rounds. In: Robshaw, M., Katz, J. (eds.) CRYPTO 2016. LNCS, vol. 9816, pp. 270–299. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53015-3_10CrossRefMATHGoogle Scholar
  10. 10.
    Cleve, R.: Limits on the security of coin flips when half the processors are faulty (extended abstract). In: Hartmanis, J. (ed.) STOC, pp. 364–369. ACM (1986)Google Scholar
  11. 11.
    Dolev, D., Dwork, C., Naor, M.: Non-malleable cryptography (extended abstract). In: STOC, pp. 542–552 (1991)Google Scholar
  12. 12.
    Dwork, C., Naor, M.: Zaps and their applications. In: FOCS, pp. 283–293 (2000)Google Scholar
  13. 13.
    Feige, U., Lapidot, D., Shamir, A.: Multiple non-interactive zero knowledge proofs based on a single random string (extended abstract). In: FOCS, pp. 308–317 (1990)Google Scholar
  14. 14.
    Garg, S., Mukherjee, P., Pandey, O., Polychroniadou, A.: The exact round complexity of secure computation. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016. LNCS, vol. 9666, pp. 448–476. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-49896-5_16CrossRefGoogle Scholar
  15. 15.
    Garg, S., Srinivasan, A.: Two-round multiparty secure computation from minimal assumptions. In: Nielsen, J.B., Rijmen, V. (eds.) EUROCRYPT 2018. LNCS, vol. 10821, pp. 468–499. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-78375-8_16CrossRefGoogle Scholar
  16. 16.
    Goldreich, O.: The Foundations of Cryptography - Volume 2, Basic Applications. Cambridge University Press, New York (2004)MATHGoogle Scholar
  17. 17.
    Goldreich, O., Kahan, A.: How to construct constant-round zero-knowledge proof systems for NP. J. Cryptol. 9(3), 167–190 (1996)MathSciNetCrossRefGoogle Scholar
  18. 18.
    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
  19. 19.
    Goldwasser, S., Micali, S., Rackoff, C.: The knowledge complexity of interactive proof systems. SIAM J. Comput. 18, 186–208 (1989)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Goyal, V.: Constant round non-malleable protocols using one way functions. In: STOC, pp. 695–704 (2011)Google Scholar
  21. 21.
    Goyal, V., Pandey, O., Richelson, S.: Textbook non-malleable commitments. In: STOC, pp. 1128–1141 (2016)Google Scholar
  22. 22.
    Goyal, V., Richelson, S., Rosen, A., Vald, M.: An algebraic approach to non-malleability. In: FOCS, pp. 41–50 (2014)Google Scholar
  23. 23.
    Halevi, S., Hazay, C., Polychroniadou, A., Venkitasubramaniam, M.: Round-optimal secure multi-party computation. IACR Cryptology ePrint Archive. 2017, 1056 (2017). http://eprint.iacr.org/2017/1056. Accepted to CRYPTO 2018
  24. 24.
    Jain, A., Kalai, Y.T., Khurana, D., Rothblum, R.: Distinguisher-dependent simulation in two rounds and its applications. In: Katz, J., Shacham, H. (eds.) CRYPTO 2017. LNCS, vol. 10402, pp. 158–189. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-63715-0_6CrossRefMATHGoogle Scholar
  25. 25.
    Katz, J., Ostrovsky, R.: Round-optimal secure two-party computation. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 335–354. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-28628-8_21CrossRefGoogle Scholar
  26. 26.
    Katz, J., Ostrovsky, R., Smith, A.: Round efficiency of multi-party computation with a dishonest majority. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 578–595. Springer, Heidelberg (2003).  https://doi.org/10.1007/3-540-39200-9_36CrossRefGoogle Scholar
  27. 27.
    Khurana, D.: Round optimal concurrent non-malleability from polynomial hardness. In: Kalai, Y., Reyzin, L. (eds.) TCC 2017, Part II. LNCS, vol. 10678, pp. 139–171. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-70503-3_5CrossRefGoogle Scholar
  28. 28.
    Lin, H., Pass, R., Venkitasubramaniam, M.: Concurrent non-malleable commitments from any one-way function. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 571–588. Springer, Heidelberg (2008).  https://doi.org/10.1007/978-3-540-78524-8_31CrossRefGoogle Scholar
  29. 29.
    Pass, R.: Bounded-concurrent secure multi-party computation with a dishonest majority. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, 13–16 June 2004, pp. 232–241 (2004)Google Scholar
  30. 30.
    Pass, R., Rosen, A.: Concurrent non-malleable commitments. In: FOCS, pp. 563–572 (2005)Google Scholar
  31. 31.
    Prabhakaran, M., Rosen, A., Sahai, A.: Concurrent zero knowledge with logarithmic round-complexity. In: FOCS, pp. 366–375 (2002)Google Scholar
  32. 32.
    Rosen, A.: A note on constant-round zero-knowledge proofs for NP. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 191–202. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-24638-1_11CrossRefGoogle Scholar
  33. 33.
    Sahai, A.: Non-malleable non-interactive zero knowledge and adaptive chosen-ciphertext security. In: FOCS, pp. 543–553 (1999)Google Scholar
  34. 34.
    Yao, A.C.: Protocols for secure computations (extended abstract). In: FOCS (1982)Google Scholar

Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  • Saikrishna Badrinarayanan
    • 1
  • Vipul Goyal
    • 2
  • Abhishek Jain
    • 3
  • Yael Tauman Kalai
    • 4
  • Dakshita Khurana
    • 1
  • Amit Sahai
    • 1
  1. 1.UCLALos AngelesUSA
  2. 2.CMUPittsburghUSA
  3. 3.JHUBaltimoreUSA
  4. 4.Microsoft ResearchMITCambridgeUSA

Personalised recommendations