Skip to main content
Springer Nature Link
Log in

Asymptotically Free Broadcast in Constant Expected Time via Packed VSS

  • Conference paper
  • First Online:
Theory of Cryptography (TCC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13747))

Included in the following conference series:

  • 467 Accesses

Abstract

Broadcast is an essential primitive for secure computation. We focus in this paper on optimal resilience (i.e., when the number of corrupted parties t is less than a third of the computing parties n), and with no setup or cryptographic assumptions.

While broadcast with worst case t rounds is impossible, it has been shown [Feldman and Micali STOC’88, Katz and Koo CRYPTO’06] how to construct protocols with expected constant number of rounds in the private channel model. However, those constructions have large communication complexity, specifically \({\mathcal {O}}(n^2L+n^6\log n)\) expected number of bits transmitted for broadcasting a message of length L. This leads to a significant communication blowup in secure computation protocols in this setting.

In this paper, we substantially improve the communication complexity of broadcast in constant expected time. Specifically, the expected communication complexity of our protocol is \({\mathcal {O}}(nL+n^4\log n)\). For messages of length \(L=\varOmega (n^3 \log n)\), our broadcast has no asymptotic overhead (up to expectation), as each party has to send or receive \({\mathcal {O}}(n^3 \log n)\) bits. We also consider parallel broadcast, where n parties wish to broadcast L bit messages in parallel. Our protocol has no asymptotic overhead for \(L=\varOmega (n^2\log n)\), which is a common communication pattern in perfectly secure MPC protocols. For instance, it is common that all parties share their inputs simultaneously at the same round, and verifiable secret sharing protocols require the dealer to broadcast a total of \({\mathcal {O}}(n^2\log n)\) bits.

As an independent interest, our broadcast is achieved by a packed verifiable secret sharing, a new notion that we introduce. We show a protocol that verifies \({\mathcal {O}}(n)\) secrets simultaneously with the same cost of verifying just a single secret. This improves by a factor of n the state-of-the-art.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Broadcast extension protocols handle long messages efficiently at the cost of a small number of single-bit broadcasts.

  2. 2.

    Using broadcast extension of [46] we can bring the asymptotic cost to \({\mathcal {O}}(nL)+E({\mathcal {O}}(n^7\log {n}))\) bits. However, the minimum message size to achieve this \(L = \varOmega (n^6\log {n})\). This is prohibitively high even for \(n = 100\).

  3. 3.

    In fact, in each round of the protocol, each party performs \({\mathcal {O}}(n)\) verifiable secret sharings (VSSs), i.e., it has to broadcast \({\mathcal {O}}(n^3 \log n)\) bits. In [2] it has been shown how to reduce it to \({\mathcal {O}}(1)\) VSSs per party, i.e., each party might have to broadcast \({\mathcal {O}}(n^2 \log n)\).

  4. 4.

    We call a bivariate polynomial where the degree in x is 2t and in y is t, i.e., \(S(x,y)=\sum _{i=0}^{2t}\sum _{j=0}^{t}a_{i,j}x^iy^j\) as a (2t, t)-bivariate polynomial.

  5. 5.

    The \(\textsf{happy}\) bits will be used later for Multi-Moderated VSS in Sect. 6.

References

  1. Aws latency monitoring. https://www.cloudping.co/grid. Accessed February 2022

  2. Abraham, I., Asharov, G., Yanai, A.: Efficient perfectly secure computation with optimal resilience. In: Nissim, K., Waters, B. (eds.) TCC 2021. LNCS, vol. 13043, pp. 66–96. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-90453-1_3

    Chapter  Google Scholar 

  3. Abraham, I., Devadas, S., Dolev, D., Nayak, K., Ren, L.: Synchronous byzantine agreement with expected O(1) rounds, expected \(O(n^2)\) communication, and optimal resilience. In: Goldberg, I., Moore, T. (eds.) FC 2019. LNCS, vol. 11598, pp. 320–334. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-32101-7_20

    Chapter  MATH  Google Scholar 

  4. Abraham, I., Nayak, K.: Crusader agreement with \(\le 1/3\) error is impossible for \(n \le 3f\) if the adversary can simulate. Decentralized Thoughts, Blog Post (2021). https://tinyurl.com/decentralizedthougts. Accessed Sept 2021

  5. Applebaum, B., Kachlon, E., Patra, A.: The round complexity of perfect MPC with active security and optimal resiliency. In: FOCS (2020)

    Google Scholar 

  6. Asharov, G., Cohen, R., Shochat, O.: Static vs. adaptive security in perfect MPC: a separation and the adaptive security of BGW. In: Conference on Information-Theoretic Cryptography - ITC 2022. (To Appear) (2022)

    Google Scholar 

  7. Asharov, G., Lindell, Y.: A full proof of the bgw protocol for perfectly secure multiparty computation. J. Cryptol. 30(1), 58–151 (2017). https://doi.org/10.1007/s00145-015-9214-4

    Article  MathSciNet  MATH  Google Scholar 

  8. Asharov, G., Lindell, Y., Rabin, T.: Perfectly-secure multiplication for any t \(<\) n/3. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 240–258. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-22792-9_14

  9. Beerliová-Trubíniová, Z., Hirt, M.: Perfectly-secure MPC with linear communication complexity. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 213–230. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78524-8_13

  10. Ben-Or, M.: Another advantage of free choice: completely asynchronous agreement protocols (extended abstract). In: PODC (1983)

    Google Scholar 

  11. Ben-Or, M., El-Yaniv, R.: Resilient-optimal interactive consistency in constant time. Distrib. Comput. 16, 249–262 (2003). https://doi.org/10.1007/s00446-002-0083-3

    Article  MATH  Google Scholar 

  12. Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: ACM Symposium on Theory of Computing (1988)

    Google Scholar 

  13. Berman, P., Garay, J.A., Perry, K.J.: Bit optimal distributed consensus. In: Baeza-Yates, R., Manber, U. (eds.) Computer Science. Springer, Boston (1992)

    Google Scholar 

  14. Canetti, R.: Asynchronous secure computation. Technion - Computer Science Department - Technical report (1993)

    Google Scholar 

  15. Canetti, R.: Studies in secure multiparty computation and applications. Ph.D. thesis, Citeseer (1996)

    Google Scholar 

  16. Canetti, R.: Security and composition of multiparty cryptographic protocols. J. Cryptology 13(1), 143–202 (2000). https://doi.org/10.1007/s001459910006

    Article  MathSciNet  MATH  Google Scholar 

  17. Canetti, R.: Universally composable security: a new paradigm for cryptographic protocols. In: FOCS (2001)

    Google Scholar 

  18. Canetti, R., Damgaard, I., Dziembowski, S., Ishai, Y., Malkin, T.: On adaptive vs. non-adaptive security of multiparty protocols. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 262–279. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44987-6_17

  19. Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols (extended abstract). In: ACM Symposium on Theory of Computing (1988)

    Google Scholar 

  20. Chen, J.: Optimal error-free multi-valued byzantine agreement. In: DISC (2021)

    Google Scholar 

  21. Chor, B., Goldwasser, S., Micali, S., Awerbuch, B.: Verifiable secret sharing and achieving simultaneity in the presence of faults (extended abstract). In: 26th Annual Symposium on Foundations of Computer Science (1985)

    Google Scholar 

  22. Coan, B.A., Welch, J.L.: Modular construction of nearly optimal byzantine agreement protocols. In: ACM Symposium on Principles of Distributed Computing (1989)

    Google Scholar 

  23. Cohen, R., Coretti, S., Garay, J., Zikas, V.: Probabilistic termination and composability of cryptographic protocols. J. Cryptology 32(3), 690–741 (2018). https://doi.org/10.1007/s00145-018-9279-y

    Article  MathSciNet  MATH  Google Scholar 

  24. Cramer, R., Damgård, I., Maurer, U.: General secure multi-party computation from any linear secret-sharing scheme. In: Preneel, B. (ed.) EUROCRYPT 2000. LNCS, vol. 1807, pp. 316–334. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-45539-6_22

  25. Damgård, I., David, B., Giacomelli, I., Nielsen, J.B.: Compact VSS and efficient homomorphic UC commitments. In: Sarkar, P., Iwata, T. (eds.) ASIACRYPT 2014. LNCS, vol. 8874, pp. 213–232. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-45608-8_12

  26. Das, S., Xiang, Z., Ren, L.: Asynchronous data dissemination and its applications. In: ACM CCS Conference on Computer and Communications Security (2021)

    Google Scholar 

  27. Dolev, D., Reischuk, R.: Bounds on information exchange for byzantine agreement. In: Symposium on Principles of Distributed Computing (1982)

    Google Scholar 

  28. Dolev, D., Reischuk, R.: Bounds on information exchange for byzantine agreement. J. ACM (JACM) 32(1), 191–204 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  29. Feldman, P., Micali, S.: Optimal algorithms for byzantine agreement. In: ACM Symposium on Theory of Computing (1988)

    Google Scholar 

  30. Feldman, P.N.: Optimal Algorithms for Byzantine Agreement. Ph.D. thesis, Massachusetts Institute of Technology (1988)

    Google Scholar 

  31. Feldman, P., Micali, S.: An optimal probabilistic protocol for synchronous byzantine agreement. SIAM J. Comput. 26(4), 873–933 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. Fischer, M.J., Lynch, N.A.: A lower bound for the time to assure interactive consistency. Information Processing Letters (1982)

    Google Scholar 

  33. Fitzi, M., Garay, J.A.: Efficient player-optimal protocols for strong and differential consensus. In: PODC (2003)

    Google Scholar 

  34. Franklin, M.K., Yung, M.: Communication complexity of secure computation (extended abstract). In: ACM Symposium on Theory of Computing (1992)

    Google Scholar 

  35. Ganesh, C., Patra, A.: Optimal extension protocols for byzantine broadcast and agreement. Distrib. Comput. 34, 59–77 (2021). https://doi.org/10.1007/s00446-020-00384-1

    Article  MathSciNet  MATH  Google Scholar 

  36. Gennaro, R., Ishai, Y., Kushilevitz, E., Rabin, T.: The round complexity of verifiable secret sharing and secure multicast. In: STOC (2001)

    Google Scholar 

  37. Gennaro, R., Rabin, M.O., Rabin, T.: Simplified vss and fast-track multiparty computations with applications to threshold cryptography. In: PODC (1998)

    Google Scholar 

  38. Goldreich, O., Petrank, E.: The best of both worlds: guaranteeing termination in fast randomized byzantine agreement protocols. Inf. Process. Lett. 36(1), 45–49 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  39. Goyal, V., Liu, Y., Song, Y.: Communication-efficient unconditional mpc with guaranteed output delivery. In: Boldyreva, A., Micciancio, D. (eds.) CRYPTO 2019. LNCS, vol. 11693, pp. 85–114. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-26951-7_4

  40. Hirt, M., Maurer, U., Przydatek, B.: Efficient secure multi-party computation. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 143–161. Springer, Heidelberg (2000). https://doi.org/10.1007/3-540-44448-3_12

  41. Katz, J., Koo, C.-Y.: On expected constant-round protocols for byzantine agreement. In: Dwork, C. (ed.) CRYPTO 2006. LNCS, vol. 4117, pp. 445–462. Springer, Heidelberg (2006). https://doi.org/10.1007/11818175_27

  42. Katz, J., Koo, C.-Y., Kumaresan, R.: Improving the round complexity of VSS in point-to-point networks. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008. LNCS, vol. 5126, pp. 499–510. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-70583-3_41

  43. Kushilevitz, E., Lindell, Y., Rabin, T.: Information-theoretically secure protocols and security under composition. In: STOC (2006)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  45. Lindell, Y., Lysyanskaya, A., Rabin, T.: Sequential composition of protocols without simultaneous termination. In: PODC (2002)

    Google Scholar 

  46. Nayak, K., Ren, L., Shi, E., Vaidya, N.H., Xiang, Z.: Improved extension protocols for byzantine broadcast and agreement. arXiv preprint. arXiv:2002.11321 (2020)

  47. Patra, A.: Error-free multi-valued broadcast and byzantine agreement with optimal communication complexity. In: Fernàndez Anta, A., Lipari, G., Roy, M. (eds.) OPODIS 2011. LNCS, vol. 7109, pp. 34–49. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25873-2_4

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

    Article  MathSciNet  MATH  Google Scholar 

  49. Rabin, M.O.: Randomized byzantine generals. In: Symposium on Foundations of Computer Science (1983)

    Google Scholar 

  50. Shrestha, N., Bhat, A., Kate, A., Nayak, K.: Synchronous distributed key generation without broadcasts. IACR Cryptology ePrint Archive (2021)

    Google Scholar 

  51. Turpin, R., Coan, B.A.: Extending binary byzantine agreement to multivalued byzantine agreement. Inf. Process. Lett. 18(2), 73–76 (1984)

    Article  Google Scholar 

Download references

Acknowledgements

Gilad Asharov is sponsored by the Israel Science Foundation (grant No. 2439/20), by JPM Faculty Research Award, by the BIU Center for Research in Applied Cryptography and Cyber Security in conjunction with the Israel National Cyber Bureau in the Prime Minister’s Office, and by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 891234. Shravani Patil would like to acknowledge the support of DST National Mission on Interdisciplinary Cyber-Physical Systems (NM-ICPS) 2020-2025. Arpita Patra would like to acknowledge the support of DST National Mission on Interdisciplinary Cyber-Physical Systems (NM-ICPS) 2020-2025, Google India Faculty Award, and SERB MATRICS (Theoretical Sciences) Grant 2020-2023.

Author information

Authors and Affiliations

  1. VMWare Research, Palo Alto, USA

    Ittai Abraham

  2. Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel

    Gilad Asharov

  3. Indian Institute of Science, Bangalore, India

    Shravani Patil & Arpita Patra

Authors
  1. Ittai Abraham
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilad Asharov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Shravani Patil
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Arpita Patra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shravani Patil .

Editor information

Editors and Affiliations

  1. Ruhr University Bochum, Bochum, Germany

    Eike Kiltz

  2. Massachusetts Institute of Technology, Cambridge, MA, USA

    Vinod Vaikuntanathan

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Abraham, I., Asharov, G., Patil, S., Patra, A. (2022). Asymptotically Free Broadcast in Constant Expected Time via Packed VSS. In: Kiltz, E., Vaikuntanathan, V. (eds) Theory of Cryptography. TCC 2022. Lecture Notes in Computer Science, vol 13747. Springer, Cham. https://doi.org/10.1007/978-3-031-22318-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-22318-1_14

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-22317-4

  • Online ISBN: 978-3-031-22318-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics