A Unified Characterization of Completeness and Triviality for Secure Function Evaluation

  • Hemanta K. Maji
  • Manoj Prabhakaran
  • Mike Rosulek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7668)

Abstract

We present unified combinatorial characterizations of completeness for 2-party secure function evaluation (SFE) against passive and active corruptions in the information-theoretic setting, so that all known characterizations appear as special cases.

In doing so we develop new technical concepts. We define several notions of isomorphism of SFE functionalities and define the “kernel” of an SFE functionality. An SFE functionality is then said to be “simple” if and only if it is strongly isomorphic to its kernel. An SFE functionality \({\mathcal F}'\) is a core of an SFE functionality \(\mathcal F\) if it is “redundancy free” and is weakly isomorphic to \(\mathcal F\). Then:

  • An SFE functionality is complete for security against passive corruptions if and only if it is not simple.

  • A deterministic SFE functionality is complete for security against active corruptions if and only if it has a core that is not simple. We conjecture that this characterization extends to randomized SFE as well.

We further give explicit combinatorial characterizations of simple SFE functionalities.

Finally, we apply our new notions of isomorphism to reduce the problem of characterization of trivial functionalities (i.e., those securely realizable without setups) for the case of general SFE to the same problem for the case of simple symmetric SFE.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Bea89]
    Beaver, D.: Perfect privacy for two-party protocols. In: Feigenbaum, J., Merritt, M. (eds.) Proceedings of DIMACS Workshop on Distributed Computing and Cryptography, vol. 2, pp. 65–77. American Mathematical Society (1989)Google Scholar
  2. [BGW88]
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In: Simon, J. (ed.) STOC, pp. 1–10. ACM (1988)Google Scholar
  3. [CCD88]
    Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols. In: Simon, J. (ed.) STOC, pp. 11–19. ACM (1988)Google Scholar
  4. [CK88]
    Crépeau, C., Kilian, J.: Achieving oblivious transfer using weakened security assumptions (extended abstract). In: FOCS, pp. 42–52. IEEE (1988)Google Scholar
  5. [CKL03]
    Canetti, R., Kushilevitz, E., Lindell, Y.: On the Limitations of Universally Composable Two-Party Computation Without Set-Up Assumptions. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 68–86. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. [CMW04]
    Crépeau, C., Morozov, K., Wolf, S.: Efficient Unconditional Oblivious Transfer from Almost Any Noisy Channel. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 47–59. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. [GMW87]
    Goldreich, O., Micali, S., Wigderson, A.: How to play ANY mental game. In: Aho, A.V. (ed.) STOC, pp. 218–229. ACM (1987); See [Gol04, ch. 7] for more detailsGoogle Scholar
  8. [Gol04]
    Goldreich, O.: Foundations of Cryptography: Basic Applications. Cambridge University Press (2004)Google Scholar
  9. [GV87]
    Goldreich, O., Vainish, R.: How to Solve Any Protocol Probleman Efficiency Improvement. In: Pomerance, C. (ed.) CRYPTO 1987. LNCS, vol. 293, pp. 73–86. Springer, Heidelberg (1988)Google Scholar
  10. [HM86]
    Haber, S., Micali, S.: Unpublished Manuscript (1986)Google Scholar
  11. [IPS08]
    Ishai, Y., Prabhakaran, M., Sahai, A.: Founding Cryptography on Oblivious Transfer – Efficiently. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 572–591. Springer, Heidelberg (2008)Google Scholar
  12. [Kil88]
    Kilian, J.: Founding cryptography on oblivious transfer. In: Simon, J. (ed.) STOC, pp. 20–31. ACM (1988)Google Scholar
  13. [Kil91]
    Kilian, J.: A general completeness theorem for two-party games. In: Koutsougeras, C., Vitter, J.S. (eds.) STOC, pp. 553–560. ACM (1991)Google Scholar
  14. [Kil00]
    Kilian, J.: More general completeness theorems for secure two-party computation. In: Frances Yao, F., Luks, E.M. (eds.) STOC, pp. 316–324. ACM (2000)Google Scholar
  15. [KM11]
    Kraschewski, D., Müller-Quade, J.: Completeness Theorems with Constructive Proofs for Finite Deterministic 2-Party Functions. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 364–381. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. [KMR09]
    Künzler, R., Müller-Quade, J., Raub, D.: Secure Computability of Functions in the IT Setting with Dishonest Majority and Applications to Long-Term Security. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 238–255. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. [KTRR03]
    Koulgi, P., Tuncel, E., Regunathan, S.L., Rose, K.: On zero-error coding of correlated sources. IEEE Transactions on Information Theory 49(11), 2856–2873 (2003)MathSciNetCrossRefGoogle Scholar
  18. [Kus89]
    Kushilevitz, E.: Privacy and communication complexity. In: FOCS, pp. 416–421. IEEE (1989)Google Scholar
  19. [MOPR11]
    Maji, H.K., Ouppaphan, P., Prabhakaran, M., Rosulek, M.: Exploring the Limits of Common Coins Using Frontier Analysis of Protocols. In: Ishai, Y. (ed.) TCC 2011. LNCS, vol. 6597, pp. 486–503. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. [MPR09]
    Maji, H.K., Prabhakaran, M., Rosulek, M.: Complexity of Multi-party Computation Problems: The Case of 2-Party Symmetric Secure Function Evaluation. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 256–273. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. [PR08]
    Prabhakaran, M., Rosulek, M.: Cryptographic Complexity of Multi-Party Computation Problems: Classifications and Separations. In: Wagner, D. (ed.) CRYPTO 2008. LNCS, vol. 5157, pp. 262–279. Springer, Heidelberg (2008)Google Scholar
  22. [Wit76]
    Witsenhausen, H.S.: The zero-error side information problem and chromatic numbers (corresp.). IEEE Transactions on Information Theory 22(5), 592–593 (1976)MathSciNetMATHCrossRefGoogle Scholar
  23. [WW06]
    Wolf, S., Wullschleger, J.: Oblivious Transfer Is Symmetric. In: Vaudenay, S. (ed.) EUROCRYPT 2006. LNCS, vol. 4004, pp. 222–232. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. [Yao86]
    Yao, A.C.-C.: How to generate and exchange secrets. In: FOCS, pp. 162–167. IEEE Computer Society (1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Hemanta K. Maji
    • 1
  • Manoj Prabhakaran
    • 2
  • Mike Rosulek
    • 3
  1. 1.University of CaliforniaLos AngelesUSA
  2. 2.University of IllinoisUrbana-ChampaignUSA
  3. 3.University of MontanaUSA

Personalised recommendations