A Uniform Min-Max Theorem with Applications in Cryptography

  • Salil Vadhan
  • Colin Jia Zheng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8042)

Abstract

We present a new, more constructive proof of von Neumann’s Min-Max Theorem for two-player zero-sum game — specifically, an algorithm that builds a near-optimal mixed strategy for the second player from several best-responses of the second player to mixed strategies of the first player. The algorithm extends previous work of Freund and Schapire (Games and Economic Behavior ’99) with the advantage that the algorithm runs in poly(n) time even when a pure strategy for the first player is a distribution chosen from a set of distributions over {0, 1} n . This extension enables a number of additional applications in cryptography and complexity theory, often yielding uniform security versions of results that were previously only proved for nonuniform security (due to use of the non-constructive Min-Max Theorem).

We describe several applications, including a more modular and improved uniform version of Impagliazzo’s Hardcore Theorem (FOCS ’95), showing impossibility of constructing succinct non-interactive arguments (SNARGs) via black-box reductions under uniform hardness assumptions (using techniques from Gentry and Wichs (STOC ’11) for the nonuniform setting), and efficiently simulating high entropy distributions within any sufficiently nice convex set (extending a result of Trevisan, Tulsiani and Vadhan (CCC ’09)).

References

  1. [BHK]
    Barak, B., Hardt, M., Kale, S.: The uniform hardcore lemma via approximate bregman projections. In: SODA 2009: Proceedings of the Nineteenth Annual ACM -SIAM Symposium on Discrete Algorithms, Philadelphia, PA, USA, pp. 1193–1200. Society for Industrial and Applied Mathematics (2009)Google Scholar
  2. [BSW]
    Barak, B., Shaltiel, R., Wigderson, A.: Computational analogues of entropy. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 200–215. Springer, Heidelberg (2003)Google Scholar
  3. [CLP]
    Chung, K.-M., Lui, E., Pass, R.: From weak to strong zero knowledge using a new non-black-box simulation technique (unpublished manuscript)Google Scholar
  4. [CT]
    Cover, T.M., Thomas, J.A.: Elements of information theory, 2nd edn. Wiley (2006)Google Scholar
  5. [DP]
    Dziembowski, S., Pietrzak, K.: Leakage-resilient cryptography. In: FOCS, pp. 293–302. IEEE Computer Society (2008)Google Scholar
  6. [FK]
    Frieze, A., Kannan, R.: Quick approximation to matrices and applications. Combinatorica 19(2), 175–220 (1999)MathSciNetCrossRefMATHGoogle Scholar
  7. [FR]
    Fuller, B., Reyzin, L.: Computational entropy and information leakage (2011), http://www.cs.bu.edu/fac/reyzin
  8. [FS]
    Freund, Y., Schapire, R.E.: Adaptive game playing using multiplicative weights. Games and Economic Behavior 29, 79–103 (1999)MathSciNetCrossRefMATHGoogle Scholar
  9. [GT]
    Green, B., Tao, T.: The primes contain arbitrarily long arithmetic progressions. Ann. of Math. 167(2), 481–547 (2008)Google Scholar
  10. [GW]
    Gentry, C., Wichs, D.: Separating succinct non-interactive arguments from all falsifiable assumptions. In: Fortnow, L., Vadhan, S.P. (eds.) STOC, pp. 99–108. ACM (2011)Google Scholar
  11. [HH]
    Haitner, I., Holenstein, T.: On the (Im)Possibility of key dependent encryption. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 202–219. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. [HHR]
    Haitner, I., Harnik, D., Reingold, O.: Efficient pseudorandom generators from exponentially hard one-way functions. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 228–239. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. [Hol1]
    Holenstein, T.: Key agreement from weak bit agreement. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), pp. 664–673 (2005)Google Scholar
  14. [Hol2]
    Holenstein, T.: Pseudorandom generators from one-way functions: A simple construction for any hardness. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 443–461. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. [HRV]
    Haitner, I., Reingold, O., Vadhan, S.: Efficiency improvements in constructing pseudorandom generators from one-way functions. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC), pp. 437–446 (2010)Google Scholar
  16. [HW]
    Herbster, M., Warmuth, M.: Tracking the best linear predictor. Journal of Machine Learning Research 1, 281–309 (2001)MathSciNetMATHGoogle Scholar
  17. [Imp]
    Impagliazzo, R.: Hard-core distributions for somewhat hard problems. In: Proceedings of the 36th Annual Symposium on Foundations of Computer Science (FOCS), pp. 538–545 (1995)Google Scholar
  18. [KS]
    Dziembowski, S., Pietrzak, K.: Leakage-resilient cryptography. In: FOCS, pp. 293–302. IEEE Computer Society (2008)Google Scholar
  19. [PJ]
    Pietrzak, K., Jetchev, D.: How to fake auxiliary input. In: ICITS 2012 Invited Talk (2012)Google Scholar
  20. [RTTV]
    Reingold, O., Trevisan, L., Tulsiani, M., Vadhan, S.: Dense subsets of pseudorandom sets. In: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008), October 26-28, pp. 76–85. IEEE (2008)Google Scholar
  21. [TTV]
    Trevisan, L., Tulsiani, M., Vadhan, S.: Regularity, boosting, and efficiently simulating every high-entropy distribution. In: Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC 2009), July 15-18, pp. 126–136 (2009); Preliminary version posted as ECCC TR08-103Google Scholar
  22. [TZ]
    Tao, T., Ziegler, T.: The primes contain arbitrarily long polynomial progressions. Acta Math. 201(2), 213–305 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. [VZ1]
    Vadhan, S., Zheng, C.J.: Characterizing pseudoentropy and simplifying pseudorandom generator constructions. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC 2012), May 19-22, pp. 817–836 (2012)Google Scholar
  24. [VZ]
    Vadhan, S.P., Zheng, C.J.: A uniform min-max theorem with applications in cryptography. To appear on the Cryptology ePrint Archive (in preparation, 2013)Google Scholar

Copyright information

© International Association for Cryptologic Research 2013

Authors and Affiliations

  • Salil Vadhan
    • 1
  • Colin Jia Zheng
    • 1
  1. 1.School of Engineering and Applied SciencesHarvard UniversityCambridgeUSA

Personalised recommendations