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On the Power of an Honest Majority in Three-Party Computation Without Broadcast

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Part of the Lecture Notes in Computer Science book series (LNSC,volume 12551)

Abstract

Fully secure multiparty computation (MPC) allows a set of parties to compute some function of their inputs, while guaranteeing correctness, privacy, fairness, and output delivery. Understanding the necessary and sufficient assumptions that allow for fully secure MPC is an important goal. Cleve (STOC’86) showed that full security cannot be obtained in general without an honest majority. Conversely, by Rabin and Ben-Or (FOCS’89), assuming a broadcast channel and an honest majority enables a fully secure computation of any function.

Our goal is to characterize the set of functionalities that can be computed with full security, assuming an honest majority, but no broadcast. This question was fully answered by Cohen et al. (TCC’16) – for the restricted class of symmetric functionalities (where all parties receive the same output). Instructively, their results crucially rely on agreement and do not carry over to general asymmetric functionalities. In this work, we focus on the case of three-party asymmetric functionalities, providing a variety of necessary and sufficient conditions to enable fully secure computation.

An interesting use-case of our results is server-aided computation, where an untrusted server helps two parties to carry out their computation. We show that without a broadcast assumption, the resource of an external non-colluding server provides no additional power. Namely, a functionality can be computed with the help of the server if and only if it can be computed without it. For fair coin tossing, we further show that the optimal bias for three-party (server-aided) r-round protocol remains \(\varTheta \left( 1/r\right) \) (as in the two-party setting).

Keywords

  • Broadcast
  • Point-to-point communication
  • Multiparty computation
  • Coin flipping
  • Impossibility result
  • Honest majority

B. Alon, E. Omri and T. Suad—Research supported by ISF grant 152/17, and by the Ariel Cyber Innovation Center in conjunction with the Israel National Cyber directorate in the Prime Minister’s Office.

R. Cohen—Research supported by NSF grant 1646671.

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Fig. 1.
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Notes

  1. 1.

    The notion of full security is formally captured via the real vs. ideal paradigm, where the protocol is said to be secure if it emulates some ideal setting, in which the capabilities of the adversary are very limited.

  2. 2.

    Convergecast [25] is a three-party functionality where two of the parties start with a non-Boolean input, and the receiver learns exactly one of the input values. The receiver does not learn anything about the other input, and none of the senders learns the receiver’s choice as well as the input of the other sender.

  3. 3.

    The choice of giving \(\mathsf {Adv}_3\) the input \(z_2\) as auxiliary is arbitrary.

References

  1. Agrawal, S., Prabhakaran, M.: On fair exchange, fair coins and fair sampling. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 259–276. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40041-4_15

    CrossRef  Google Scholar 

  2. Alon, B., Omri, E.: Almost-optimally fair multiparty coin-tossing with nearly three-quarters malicious. In: Hirt, M., Smith, A. (eds.) TCC 2016-B, Part I. LNCS, vol. 9985, pp. 307–335. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-662-53641-4_13

    CrossRef  Google Scholar 

  3. Alon, B., Cohen, R., Omri, E., Suad, T.: On the power of an honest majority in three-party computation without broadcast. Cryptology ePrint Archive, Report 2020/1170 (2020). https://eprint.iacr.org/2020/1170

  4. Asharov, G.: Towards characterizing complete fairness in secure two-party computation. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 291–316. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-642-54242-8_13

    CrossRef  Google Scholar 

  5. Asharov, G., Beimel, A., Makriyannis, N., Omri, E.: Complete characterization of fairness in secure two-party computation of boolean functions. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9014, pp. 199–228. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-46494-6_10

    CrossRef  MATH  Google Scholar 

  6. Asokan, N., Shoup, V., Waidner, M.: Optimistic fair exchange of digital signatures. In: Nyberg, K. (ed.) EUROCRYPT 1998. LNCS, vol. 1403, pp. 591–606. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0054156

    CrossRef  Google Scholar 

  7. Beimel, A., Omri, E., Orlov, I.: Protocols for multiparty coin toss with a dishonest majority. J. Cryptol. 28(3), 551–600 (2015)

    MathSciNet  CrossRef  Google Scholar 

  8. Beimel, A., Haitner, I., Makriyannis, N., Omri, E.: Tighter bounds on multi-party coin flipping via augmented weak martingales and differentially private sampling. In: FOCS, pp. 838–849 (2018)

    Google Scholar 

  9. 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 

  10. Blum, M.: Coin flipping by telephone. In: CRYPTO 1981, pp. 11–15 (1981)

    Google Scholar 

  11. Borcherding, M.: Levels of authentication in distributed agreement. In: Babaoğlu, Ö., Marzullo, K. (eds.) WDAG 1996. LNCS, vol. 1151, pp. 40–55. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-61769-8_4

    CrossRef  Google Scholar 

  12. Bracha, G., Toueg, S.: Asynchronous consensus and broadcast protocols. J. ACM 32(4), 824–840 (1985)

    MathSciNet  CrossRef  Google Scholar 

  13. Buchbinder, N., Haitner, I., Levi, N., Tsfadia, E.: Fair coin flipping: tighter analysis and the many-party case. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2580–2600 (2017)

    Google Scholar 

  14. Cachin, C., Camenisch, J.: Optimistic fair secure computation. In: Bellare, M. (ed.) CRYPTO 2000. LNCS, vol. 1880, pp. 93–111. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44598-6_6

    CrossRef  Google Scholar 

  15. Canetti, R.: Security and composition of multiparty cryptographic protocols. J. Cryptol. 13(1), 143–202 (2000)

    MathSciNet  CrossRef  Google Scholar 

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

    Google Scholar 

  17. Cleve, R.: Limits on the security of coin flips when half the processors are faulty (extended abstract). In: STOC, pp. 364–369 (1986)

    Google Scholar 

  18. Cohen, R., Lindell, Y.: Fairness versus guaranteed output delivery in secure multiparty computation. J. Cryptol. 30(4), 1157–1186 (2017)

    MathSciNet  CrossRef  Google Scholar 

  19. Cohen, R., Haitner, I., Omri, E., Rotem, L.: Characterization of secure multiparty computation without broadcast. J. Cryptol. 31(2), 587–609 (2018)

    MathSciNet  CrossRef  Google Scholar 

  20. Cohen, R., Haitner, I., Omri, E., Rotem, L.: From fairness to full security in multiparty computation. In: Catalano, D., De Prisco, R. (eds.) SCN 2018. LNCS, vol. 11035, pp. 216–234. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-98113-0_12

    CrossRef  MATH  Google Scholar 

  21. Dwork, C., Lynch, N.A., Stockmeyer, L.J.: Consensus in the presence of partial synchrony. J. ACM 35(2), 288–323 (1988)

    MathSciNet  CrossRef  Google Scholar 

  22. Fischer, M.J., Lynch, N.A., Merritt, M.: Easy impossibility proofs for distributed consensus problems. Distrib. Comput. 1(1), 26–39 (1986)

    CrossRef  Google Scholar 

  23. Fitzi, M., Gisin, N., Maurer, U., von Rotz, O.: Unconditional byzantine agreement and multi-party computation secure against dishonest minorities from scratch. In: Knudsen, L.R. (ed.) EUROCRYPT 2002. LNCS, vol. 2332, pp. 482–501. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-46035-7_32

    CrossRef  Google Scholar 

  24. Fitzi, M., Gottesman, D., Hirt, M., Holenstein, T., Smith, A.D.: Detectable byzantine agreement secure against faulty majorities. In: PODC, pp. 118–126 (2002)

    Google Scholar 

  25. Fitzi, M., Garay, J.A., Maurer, U.M., Ostrovsky, R.: Minimal complete primitives for secure multi-party computation. J. Cryptol. 18(1), 37–61 (2005)

    MathSciNet  CrossRef  Google Scholar 

  26. Garay, J.A., Kiayias, A., Ostrovsky, R.M., Panagiotakos, G., Zikas, V.: Resource-restricted cryptography: revisiting MPC bounds in the proof-of-work era. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020, Part II. LNCS, vol. 12106, pp. 129–158. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45724-2_5

    CrossRef  Google Scholar 

  27. Goldreich, O.: Foundations of Cryptography - VOLUME 2: Basic Applications. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  28. 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 

  29. Gordon, S.D., Hazay, C., Katz, J., Lindell, Y.: Complete fairness in secure two-party computation. In: STOC, pp. 413–422 (2008)

    Google Scholar 

  30. Haitner, I., Tsfadia, E.: An almost-optimally fair three-party coin-flipping protocol. SICOMP 46(2), 479–542 (2017)

    MathSciNet  CrossRef  Google Scholar 

  31. Haitner, I., Makriyannis, N., Omri, E.: On the complexity of fair coin flipping. In: Beimel, A., Dziembowski, S. (eds.) TCC 2018, Part I. LNCS, vol. 11239, pp. 539–562. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-03807-6_20

    CrossRef  MATH  Google Scholar 

  32. Halevi, S., Ishai, Y., Kushilevitz, E., Makriyannis, N., Rabin, T.: On fully secure MPC with solitary output. In: Hofheinz, D., Rosen, A. (eds.) TCC 2019, Part I. LNCS, vol. 11891, pp. 312–340. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-36030-6_13

    CrossRef  Google Scholar 

  33. Kilian, J.: Founding cryptography on oblivious transfer. In: STOC, pp. 20–31 (1988)

    Google Scholar 

  34. Lamport, L., Shostak, R.E., Pease, M.C.: The byzantine generals problem. ACM Trans. Program. Lang. Syst. (TOPLAS) 4(3), 382–401 (1982)

    CrossRef  Google Scholar 

  35. Makriyannis, N.: On the classification of finite boolean functions up to fairness. In: Abdalla, M., De Prisco, R. (eds.) SCN 2014. LNCS, vol. 8642, pp. 135–154. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-10879-7_9

    CrossRef  Google Scholar 

  36. Moran, T., Naor, M., Segev, G.: An optimally fair coin toss. J. Cryptol. 29(3), 491–513 (2016)

    MathSciNet  CrossRef  Google Scholar 

  37. Pease, M.C., Shostak, R.E., Lamport, L.: Reaching agreement in the presence of faults. J. ACM 27(2), 228–234 (1980)

    MathSciNet  CrossRef  Google Scholar 

  38. Rabin, T., Ben-Or, M.: Verifiable secret sharing and multiparty protocols with honest majority (extended abstract). In: FOCS, pp. 73–85 (1989)

    Google Scholar 

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

    Google Scholar 

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Alon, B., Cohen, R., Omri, E., Suad, T. (2020). On the Power of an Honest Majority in Three-Party Computation Without Broadcast. In: Pass, R., Pietrzak, K. (eds) Theory of Cryptography. TCC 2020. Lecture Notes in Computer Science(), vol 12551. Springer, Cham. https://doi.org/10.1007/978-3-030-64378-2_22

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