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Designs, Codes and Cryptography

, Volume 62, Issue 3, pp 259–272 | Cite as

Multi-party computation with conversion of secret sharing

  • Hossein Ghodosi
  • Josef Pieprzyk
  • Ron Steinfeld
Article

Abstract

Classical results in unconditionally secure multi-party computation (MPC) protocols with a passive adversary indicate that every n-variate function can be computed by n participants, such that no set of size t < n/2 participants learns any additional information other than what they could derive from their private inputs and the output of the protocol. We study unconditionally secure MPC protocols in the presence of a passive adversary in the trusted setup (‘semi-ideal’) model, in which the participants are supplied with some auxiliary information (which is random and independent from the participant inputs) ahead of the protocol execution (such information can be purchased as a “commodity” well before a run of the protocol). We present a new MPC protocol in the trusted setup model, which allows the adversary to corrupt an arbitrary number t < n of participants. Our protocol makes use of a novel subprotocol for converting an additive secret sharing over a field to a multiplicative secret sharing, and can be used to securely evaluate any n-variate polynomial G over a field F, with inputs restricted to non-zero elements of F. The communication complexity of our protocol is O( · n 2) field elements, where is the number of non-linear monomials in G. Previous protocols in the trusted setup model require communication proportional to the number of multiplications in an arithmetic circuit for G; thus, our protocol may offer savings over previous protocols for functions with a small number of monomials but a large number of multiplications.

Keywords

Multi-party computation Hybrid secret sharing schemes Unconditional security 

Mathematics Subject Classification (2000)

68Q25 

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References

  1. 1.
    Bar-Ilan J., Beaver D.: Non-cryptographic fault-tolerant computing in a constant number of rounds of interaction. In: 8th Annual ACM Symposium on Principles of Distributed Computing, Edmonton, Alberta, Canada, pp. 201–209. (1989).Google Scholar
  2. 2.
    Beaver D.: Multiparty protocols tolerating half faulty processors. In: Brassard G. (ed.) Advances in Cryptology—Proceedings of CRYPTO’89. Lecture Notes in Computer Science, vol. 435, pp. 560–572. Springer-Verlag, Heidelberg (1990).Google Scholar
  3. 3.
    Beaver D.: Secure multiparty protocols and zero-knowledge proof systems tolerating a faulty minority. J. Cryptol. 4, 75–122 (1991)MATHCrossRefGoogle Scholar
  4. 4.
    Beaver D.: Efficient multiparty protocols using circuit randomization. In: Feigenbaum J. (ed.) Advances in Cryptology—Proceedings of CRYPTO’91. Lecture Notes in Computer Science, vol. 576, pp. 420–432. Springer-Verlag, Heidelberg (1992).Google Scholar
  5. 5.
    Beaver D.: Precomputing oblivious transfer. In: Dwork C. (ed.) Advances in Cryptology—Proceedings of CRYPTO 1995. Lecture Notes in Computer Science vol. 963, pp. 97–109. Springer-Verlag, Heidelberg (1995).Google Scholar
  6. 6.
    Beaver D.: Commodity-based cryptography. In: Proceedings of the 29th ACM Annual Symposium on the Theory of Computing (STOC’97), pp. 446–455. (1997).Google Scholar
  7. 7.
    Beaver D., Goldwasser S.: Multiparty computation with faulty majority. In: The 30th IEEE Symposium on the Foundations of Computer Science (FOCS89), pp. 468–473. (1989).Google Scholar
  8. 8.
    Beaver D., Micali S., Rogaway P.: The round complexity of secure protocols. In: Proceedings of the 22nd ACM Annual Symposium on the Theory of Computing (STOC’90), pp. 503–513. (1990).Google Scholar
  9. 9.
    Beaver D., Wool A.: Quorum-based secure multi-party computation. In: Nyberg K. (ed.) Advances in Cryptology—Proceedings of EUROCRYPT’98. Lecture Notes in Computer Science, vol. 1403, pp. 375–390. Springer-Verlag, Heidelberg (1998).Google Scholar
  10. 10.
    Benaloh J.: Secret sharing homomorphisms: keeping shares of a secret. In: Odlyzko A. (ed.) Advances in Cryptology—Proceedings of CRYPTO’86. Lecture Notes in Computer Science, vol. 263, pp. 251–260. Springer-Verlag, Heidelberg (1987).Google Scholar
  11. 11.
    Ben-Or M., Goldwasser S., Wigderson A.: Completeness theorem for non-cryptographic fault-tolerant distributed computation. In: Proceedings of the 20th ACM Annual Symposium on the Theory of Computing (STOC’88), pp. 1–10. (1988).Google Scholar
  12. 12.
    Chaum D., Crépeau C., Damgård I.: Multiparty unconditionally secure protocols. In: Proceedings of the 20th ACM Annual Symposium on the Theory of Computing (STOC’88), pp. 11–19. (1988).Google Scholar
  13. 13.
    Cramer R., Damgård I., Ishai Y.: Share conversion, pseudorandom secret-sharing, and applications to secure computation. In: Kilian J. (ed.) 2nd Theory of Cryptography Conference TCC 2005. Lecture Notes in Computer Science, vol. 3378, pp. 342–362. Springer-Verlag, Heidelberg (2005).Google Scholar
  14. 14.
    Crépeau C., van de Graaf J., Tapp A.: Committed oblivious transfer and private multi-party computation. In: Dwork C. (ed.) Advances in Cryptology—Proceedings of CRYPTO 1995. Lecture Notes in Computer Science, vol. 963, pp. 110–123. Springer-Verlag, Heidelberg (1995).Google Scholar
  15. 15.
    Damgård I., Ishai Y.: Scalable secure multiparty computation. In: Dwork C. (ed.) Advances in Cryptology—Proceedings of CRYPTO 2006. Lecture Notes in Computer Science, vol. 4117, pp. 501–520. Springer-Verlag, Heidelberg (2006).Google Scholar
  16. 16.
    Damgård I., Nielsen J.: Universally composable efficient multiparty computation from threshold homomorphic encryption. In: Boneh D. (ed.) Advances in Cryptology—Proceedings of CRYPTO 2003. Lecture Notes in Computer Science, vol. 2729, pp. 247–264. Springer-Verlag, Heidelberg (2003).Google Scholar
  17. 17.
    Fitzi M., Garay J., Maurer U., Ostrovsky R.: Minimal complete primitives for secure multi-party computation. In: Kilian J. (ed.) Advances in Cryptology—Proceedings of CRYPTO 2001. Lecture Notes in Computer Science, vol. 2139, pp. 80–100. Springer-Verlag, Heidelberg (2001).Google Scholar
  18. 18.
    Fitzi M., Hirt M., Maurer U.: Trading correctness for privacy in unconditional multi-party computation. In: Krawczyk H. (ed.) Advances in Cryptology—Proceedings of CRYPTO’98. of Lecture Notes in Computer Science, vol. 1462, pp. 121–136. Springer-Verlag, Heidelberg (1998).Google Scholar
  19. 19.
    Fitzi M., Holenstein T., Wullschleger J.: Multi-party Computation with Hybrid Security. In: Cachin C., Camenisch J. (eds.) Advances in Cryptology—Proceedings of EUROCRYPT 2004. Lecture Notes in Computer Science, vol. 3027, pp. 419–438. Springer-Verlag, Heidelberg (2004).Google Scholar
  20. 20.
    Gennaro R., Rabin M., Rabin T.: Simplified VSS and fast-track multiparty computations with applications to threshold cryptography. In: 17th Annual ACM Symposium on Principles of Distributed Computing, pp. 101–111. (1998).Google Scholar
  21. 21.
    Goldreich O.: Foundations of Cryptography, vol. II. Cambridge University Press, Cambridge (2004)CrossRefGoogle Scholar
  22. 22.
    Goldreich O., Micali S., Wigderson A.: How to play any mental game. In: Proceedings of the 19th ACM Annual Symposium on the Theory of Computing (STOC’87), pp. 218–229, 25–27 May 1987.Google Scholar
  23. 23.
    Hirt M., Maurer U.: Complete characterization of adversaries tolerable in secure multi-party computations. In: 16th Annual ACM Symposium on Principles of Distributed Computing (PODC), pp. 25–34. (1997).Google Scholar
  24. 24.
    Hirt M., Maurer U.: Robustness for free in unconditional multi-party computation. In: Kilian J. (ed.) Advances in Cryptology—Proceedings of CRYPTO 2001. Lecture Notes in Computer Science, vol. 2139, pp. 101–118. Springer-Verlag, Heidelberg (2001).Google Scholar
  25. 25.
    Hirt M., Maurer U., Przydatek B.: Efficient secure multi-party computation. In Okamoto T. (ed.) Advances in Cryptology—Proceedings of ASIACRYPT 2000. Lecture Notes in Computer Science, vol. 1976, pp. 143–161. Springer-Verlag, Heidelberg (2000).Google Scholar
  26. 26.
    Hirt M., Nielsen J.: Upper bounds on the communication complexity of optimally resilient cryptographic multiparty computation. In: Roy B. (ed.) Advances in Cryptology—Proceedings of ASIACRYPT 2005. Lecture Notes in Computer Science, vol. 3788, pp. 79–99. Springer-Verlag, Heidelberg (2005).Google Scholar
  27. 27.
    Ishai Y., Prabhakaran M., Sahai A.: Secure arithmetic computation with no honest majority. In: Reingold O. (ed.) 6th Theory of Cryptography Conference TCC 2009. Lecture Notes in Computer Science, vol. 5444, pp. 294–314. Springer-Verlag, Heidelberg (2009).Google Scholar
  28. 28.
    Ishai Y., Prabhakaran M., Sahai A.: Founding cryptography on oblivious transfer—efficiently. In: Wagner D. (ed.) Advances in Cryptology—Proceedings of CRYPTO 2008. Lecture Notes in Computer Science, vol. 5157, pp. 572–591. Springer-Verlag, Heidelberg (2008).Google Scholar
  29. 29.
    Kilian J.: Founding cryptography on oblivious transfer. In: Proceedings of the 20th ACM Annual Symposium on the Theory of Computing (STOC’88), pp. 20–31. (1988).Google Scholar
  30. 30.
    Katz J., Ostrovsky R., Smith A.: Round efficiency of multi-party computation with a dishonest majority. In: Biham, E. (ed.) Advances in Cryptology—Proceedings of EUROCRYPT 2003, pp. 578–595. Springer-Verlag, Heidelberg (2003)CrossRefGoogle Scholar
  31. 31.
    Kushilevitz E.: Privacy and communication complexity. In: the 30th IEEE Symposium on the Foundations of Computer Science (FOCS89), pp. 416–421. (1989).Google Scholar
  32. 32.
    Kushilevitz E., Rosc̀n A.: A randomness-rounds tradeoff in private computation. In: Desmedt Y. (ed.) Advances in Cryptology—Proceedings of CRYPTO’94. Lecture Notes in Computer Science, vol. 839, pp. 397–409. Springer-Verlag, Heidelberg (1994).Google Scholar
  33. 33.
    Rabin T., Ben-Or M.: Verifiable secret sharing and multiparty protocols with honest majority. In: Proceedings of the 21th ACM Annual Symposium on the Theory of Computing (STOC’89), pp. 73–85. (1989).Google Scholar
  34. 34.
    Shamir A.: How to share a secret. Communications of the ACM 22, 612–613 (1979)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Stinson D.: An explication of secret sharing schemes. Designs, Codes and Cryptography 2, 357–390 (1992)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Yao A.: Protocols for secure computations. In: the 23rd IEEE Symposium on the Foundations of Computer Science, pp. 160–164. (1982).Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Hossein Ghodosi
    • 1
  • Josef Pieprzyk
    • 2
  • Ron Steinfeld
    • 2
  1. 1.Department of Information Technology, School of BusinessJames Cook UniversityTownsvilleAustralia
  2. 2.Department of Computing, Center for Advanced Computing—Algorithms and CryptographyMacquarie UniversitySydneyAustralia

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