Random Selection with an Adversarial Majority

  • Ronen Gradwohl
  • Salil Vadhan
  • David Zuckerman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4117)


We consider the problem of random selection, where p players follow a protocol to jointly select a random element of a universe of size n. However, some of the players may be adversarial and collude to force the output to lie in a small subset of the universe. We describe essentially the first protocols that solve this problem in the presence of a dishonest majority in the full-information model (where the adversary is computationally unbounded and all communication is via non-simultaneous broadcast). Our protocols are nearly optimal in several parameters, including the round complexity (as a function of n), the randomness complexity, the communication complexity, and the tradeoffs between the fraction of honest players, the probability that the output lies in a small subset of the universe, and the density of this subset.


  1. 1.
    Alon, N., Naor, M.: Coin-flipping games immune against linear-sized coalitions. SIAM J. Computing 22(2), 403–417 (1993)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Antonakopoulos, S.: Fast leader-election protocols with bounded cheaters’ edge. In: Proc. 38th STOC, pp. 187–196 (2006)Google Scholar
  3. 3.
    Barak, B.: Constant-round coin-tossing with a man in the middle or realizing the shared random string model. In: 43rd FOCS (2002)Google Scholar
  4. 4.
    Blum, M.: Coin flipping by telephone. In: IEEE Spring COMPCOM (1982)Google Scholar
  5. 5.
    Beaver, D., Goldwasser, S.: Multiparty computation with faulty majority. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 589–590. Springer, Heidelberg (1990)Google Scholar
  6. 6.
    Ben-Or, M., Goldwasser, S., Wigderson, A.: Completeness theorems for non-cryptographic fault-tolerant distributed computation. In: 20th STOC, pp. 1–10 (1988)Google Scholar
  7. 7.
    Ben-Or, M., Linial, N.: Collective coin fliping. In: Advances in Computing Research. Randomness and Computation, vol. 5, pp. 91–115. JAI Press, Greenwich, CT (1989)Google Scholar
  8. 8.
    Boppana, R., Narayanan, B.: Perfect-information leader election with optimal resilience. SIAM J. Computing 29(4), 1304–1320 (2000)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bellare, M., Rompel, J.: Randomness-efficient oblivious sampling. In: 35th FOCS (1994)Google Scholar
  10. 10.
    Chaum, D., Crépeau, C., Damgård, I.: Multiparty unconditionally secure protocols. In: 20th STOC, pp. 11–19 (1988)Google Scholar
  11. 11.
    Cooper, J., Linial, N.: Fast perfect-information leader-election protocols with linear immunity. Combinatorica 15, 319–332 (1995)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Damgård, I.B.: Interactive hashing can simplify zero-knowledge protocol design without computational assumptions (extended abstract). In: Stinson, D.R. (ed.) CRYPTO 1993. LNCS, vol. 773, pp. 100–109. Springer, Heidelberg (1994)Google Scholar
  13. 13.
    Damgård, I., Goldreich, O., Wigderson, A.: Hashing functions can simplify zero-knowledge protocol design (too). TR RS-94-39. BRICS (1994)Google Scholar
  14. 14.
    Ding, Y.Z., Harnik, D., Rosen, A., Shaltiel, R.: Constant-round oblivious transfer in the bounded storage model. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 446–472. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  15. 15.
    Feige, U.: Noncryptographic selection protocols. In: 40th FOCS, pp. 142–152 (1999)Google Scholar
  16. 16.
    Goldreich, O.: A sample of samplers - a computational perspective on sampling (survey). Report 97-020, Electronic Colloquium on Computational Complexity (1997)Google Scholar
  17. 17.
    Goldreich, O., Goldwasser, S., Linial, N.: Fault-tolerant computation in the full information model. SIAM J. Computing 27(2) (1998)Google Scholar
  18. 18.
    Goldwasser, S., Levin, L.A.: Fair computation of general functions in presence of immoral majority. In: Menezes, A., Vanstone, S.A. (eds.) CRYPTO 1990. LNCS, vol. 537, pp. 77–93. Springer, Heidelberg (1991)Google Scholar
  19. 19.
    Goldwasser, S., Lindell, Y.: Secure computation without agreement. In: Malkhi, D. (ed.) DISC 2002. LNCS, vol. 2508, pp. 17–32. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  20. 20.
    Goldreich, O., Micali, S., Wigderson, A.: How to play ANY mental game. In: 19th STOC, pp. 218–229 (1987)Google Scholar
  21. 21.
    Goldreich, O., Sahai, A., Vadhan, S.: Honest-verifier statistical zero-knowledge equals general statistical zero-knowledge. In: 30th STOC (1998)Google Scholar
  22. 22.
    Gradwohl, R., Vadhan, S., Zuckerman, D.: Random Selection with an Adversarial Majority. Report TR06-26, Electronic Colloquium on Computational Complexity (February 2006)Google Scholar
  23. 23.
    Katz, J., Ostrovsky, R.: Round-optimal secure two-party computation. In: Franklin, M. (ed.) CRYPTO 2004. LNCS, vol. 3152, pp. 335–354. Springer, Heidelberg (2004)Google Scholar
  24. 24.
    Katz, J., Ostrovsky, R., Smith, A.: Round efficiency of multi-party computation with a dishonest majority. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 578–595. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  25. 25.
    Lindell, Y.: Parallel coin-tossing and constant-round secure two-party computation. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, p. 171. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  26. 26.
    Pease, M., Shostak, R., Lamport, L.: Reaching agreement in the presence of faults. J. ACM 27(2), 228–234 (1980)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Lu, C., Reingold, O., Vadhan, S., Wigderson, A.: Extractors: Optimal up to constant factors. In: 35th STOC (2003)Google Scholar
  28. 28.
    Naor, M., Ostrovsky, R., Venkatesan, R., Yung, M.: Perfect zero-knowledge arguments for NP can be based on general complexity assumptions. J. Cryptology 11 (1998)Google Scholar
  29. 29.
    Nisan, N., Zuckerman, D.: Randomness is linear in space. J. Computer and System Sci. 52(1), 43–52 (1996)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Okamoto, T.: On relationships between statistical zero-knowledge proofs. J. Computer and System Sci. 60(1), 47–108 (2000)MATHCrossRefGoogle Scholar
  31. 31.
    Ostrovsky, R., Rajagopalan, S., Vazirani, U.: Simple and efficient leader election in the full information model. In: Proc. 26th STOC, pp. 234–242 (1994)Google Scholar
  32. 32.
    Ostrovsky, R., Venkatesan, R., Yung, M.: Interactive hashing simplifies zero-knowledge protocol design. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 267–273. Springer, Heidelberg (1994)Google Scholar
  33. 33.
    Raz, R., Reingold, O., Vadhan, S.: Extracting all the randomness and reducing the error in Trevisan’s extractors. J. Computer and System Sc. 65(1), 97–128 (2002)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Raz, R., Reingold, O., Vadhan, S.: Error Reduction for Extractors. In: 40th FOCS (1999)Google Scholar
  35. 35.
    Russell, A., Saks, M., Zuckerman, D.: Lower bounds for leader election and collective coin- flipping in the perfect information model. SIAM J. Computing 31, 1645–1662 (2002)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Radhakrishnan, J., Ta-Shma, A.: Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM J. Discrete Math. 13(1), 2–24 (2000)MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Russell, A., Zuckerman, D.: Perfect-information leader election in log* n + O(1) rounds. J. Computer and System Sci. 63, 612–626 (2001)MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Saks, M.: A robust noncryptographic protocol for collective coin flipping. SIAM J. Discrete Math. 2(2), 240–244 (1989)MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Sanghvi, S., Vadhan, S.: The round complexity of two-party random selection. In: 37th STOC (2005)Google Scholar
  40. 40.
    Vadhan, S.: Constructing locally computable extractors and cryptosystems in the bounded-storage model. J. Cryptology 17(1), 43–77 (2004)MATHCrossRefMathSciNetGoogle Scholar
  41. 41.
    Wigderson, A., Zuckerman, D.: Expanders that beat the eigenvalue bound: explicit construction and applications. Combinatorica 19(17), 125–138 (1999)MATHCrossRefMathSciNetGoogle Scholar
  42. 42.
    Yao, A.: How to generate and exchange secrets. In: Proc. 27th FOCS (1986)Google Scholar
  43. 43.
    Zuckerman, D.: Randomness-optimal oblivious sampling. Random Structures and Algorithms 11(4), 345–367 (1997)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ronen Gradwohl
    • 1
  • Salil Vadhan
    • 2
  • David Zuckerman
    • 3
  1. 1.Department of Computer Science and Applied MathWeizmann Institute of Science 
  2. 2.Division of Engineering & Applied SciencesHarvard University 
  3. 3.Department of Computer ScienceUniversity of Texas at Austin 

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