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Towards Characterizing Complete Fairness in Secure Two-Party Computation

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Part of the Lecture Notes in Computer Science book series (LNSC,volume 8349)


The well known impossibility result of Cleve (STOC 1986) implies that in general it is impossible to securely compute a function with complete fairness without an honest majority. Since then, the accepted belief has been that nothing non-trivial can be computed with complete fairness in the two party setting. The surprising work of Gordon, Hazay, Katz and Lindell (STOC 2008) shows that this belief is false, and that there exist some non-trivial (deterministic, finite-domain) boolean functions that can be computed fairly. This raises the fundamental question of characterizing complete fairness in secure two-party computation.

In this work we show that not only that some or few functions can be computed fairly, but rather an enormous amount of functions can be computed with complete fairness. In fact, almost all boolean functions with distinct domain sizes can be computed with complete fairness (for instance, more than 99.999% of the boolean functions with domain sizes 31 ×30). The class of functions that is shown to be possible includes also rather involved and highly non-trivial tasks, such as set-membership, evaluation of a private (Boolean) function and private matchmaking.

In addition, we demonstrate that fairness is not restricted to the class of symmetric boolean functions where both parties get the same output, which is the only known feasibility result. Specifically, we show that fairness is also possible for asymmetric boolean functions where the output of the parties is not necessarily the same. Moreover, we consider the class of functions with non-binary output, and show that fairness is possible for any finite range.

The constructions are based on the protocol of Gordon et. al, and the analysis uses tools from convex geometry.


  • Complete fairness
  • secure two-party computation
  • foundations
  • malicious adversaries

This research was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 239868 and by the israel science foundation (grant No. 189/11).


  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)

    CrossRef  Google Scholar 

  2. Asharov, G.: Towards characterizing complete fairness in secure two-party computation. IACR Cryptology ePrint Archive (to appear)

    Google Scholar 

  3. Asharov, G., Lindell, Y., Rabin, T.: A full characterization of functions that imply fair coin tossing and ramifications to fairness. In: Sahai, A. (ed.) TCC 2013. LNCS, vol. 7785, pp. 243–262. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  4. Beimel, A., Lindell, Y., Omri, E., Orlov, I.: 1/p-secure multiparty computation without honest majority and the best of both worlds. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 277–296. Springer, Heidelberg (2011)

    CrossRef  Google Scholar 

  5. Ben-Or, M., Goldreich, O., Micali, S., Rivest, R.L.: A fair protocol for signing contracts. IEEE Transactions on Information Theory 36(1), 40–46 (1990)

    CrossRef  Google Scholar 

  6. 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 

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

    CrossRef  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  9. 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 

  10. Cleve, R.: Controlled gradual disclosure schemes for random bits and their applications. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 573–588. Springer, Heidelberg (1990)

    Google Scholar 

  11. Even, S., Goldreich, O., Lempel, A.: A randomized protocol for signing contracts. In: CRYPTO, pp. 205–210 (1982)

    Google Scholar 

  12. Even, S., Yacobi, Y.: Relations among public key signature schemes. Technical Report #175, Technion Israel Institute of Technology, Computer Science Department (1980),

  13. Goldreich, O.: The Foundations of Cryptography - Basic Applications, vol. 2. Cambridge University Press (2004)

    Google Scholar 

  14. 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 

  15. Goldwasser, S., Levin, L.: Fair computation of general functions in presence of immoral majority. In: Menezes, A.J., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 77–93. Springer, Heidelberg (1991)

    Google Scholar 

  16. Gordon, S.D.: On fairness in secure computation. PhD thesis, University of Maryland (2010)

    Google Scholar 

  17. Gordon, S.D., Hazay, C., Katz, J., Lindell, Y.: Complete fairness in secure two-party computation. In: STOC, pp. 413–422 (2008); Extended full version available on: . Journal version: [18]

  18. Gordon, S.D., Hazay, C., Katz, J., Lindell, Y.: Complete fairness in secure twoparty computation. J. ACM 58(6), 24 (2011)

    CrossRef  MathSciNet  Google Scholar 

  19. Gordon, S.D., Katz, J.: Partial fairness in secure two-party computation. In: Gilbert, H. (ed.) EUROCRYPT 2010. LNCS, vol. 6110, pp. 157–176. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  20. Grünbaum, B.: Convex Polytopes. In: Kaibel, V., Klee, V., Ziegler, G. (eds.) Graduate Texts in Mathematics, 2nd edn. Springer (May 2003)

    Google Scholar 

  21. Halpern, J.Y., Teague, V.: Rational secret sharing and multiparty computation: extended abstract. In: STOC, pp. 623–632 (2004)

    Google Scholar 

  22. Kahn, J., Komlòs, J., Szemerèdi, E.: On the probability that a random ±1-matrix is singular. Journal of Amer. Math. Soc. 8, 223–240 (1995)

    MATH  Google Scholar 

  23. Komlòs, J.: On the determinant of (0,1) matrices. Studia Sci. Math. Hungar 2, 7–21 (1967)

    MATH  MathSciNet  Google Scholar 

  24. Rabin, M.O.: How to exchange secrets with oblivious transfer. Technical Report TR-81, Aiken Computation Lab, Harvard University (1981)

    Google Scholar 

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

    Google Scholar 

  26. Roman, S.: Advanced Linear Algebra, 3rd edn. Graduate Texts in Mathematics, vol. 135, p. xviii. Springer, New York (2008)

    Google Scholar 

  27. Voigt, T., Ziegler, G.M.: Singular 0/1-matrices, and the hyperplanes spanned by random 0/1-vectors. Combinatorics, Probability and Computing 15(3), 463–471 (2006)

    CrossRef  MATH  MathSciNet  Google Scholar 

  28. Wood, P.J.: On the probability that a discrete complex random matrix is singular. PhD thesis, Rutgers University, New Brunswick, NJ, USA, AAI3379178 (2009)

    Google Scholar 

  29. Yao, A.C.C.: How to generate and exchange secrets (extended abstract). In: FOCS, pp. 162–167 (1986)

    Google Scholar 

  30. Ziegler, G.M.: Lectures on 0/1-polytopes. Polytopes: Combinatorics and Computation, Birkhauser, Basel. DMV Seminar, vol. 29, pp. 1–40 (2000)

    Google Scholar 

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Asharov, G. (2014). Towards Characterizing Complete Fairness in Secure Two-Party Computation. In: Lindell, Y. (eds) Theory of Cryptography. TCC 2014. Lecture Notes in Computer Science, vol 8349. Springer, Berlin, Heidelberg.

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