A New Approximate Min-Max Theorem with Applications in Cryptography

  • Maciej SkórskiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)


We propose a novel proof technique that can be applied to attack a broad class of problems in computational complexity, when switching the order of universal and existential quantifiers is helpful. Our approach combines the standard min-max theorem and convex approximation techniques, offering quantitative improvements over the standard way of using min-max theorems as well as more concise and elegant proofs.


Min-max theorems Convex approximation Cryptography 


  1. 1.
    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
  2. 2.
    Chung, K.-M., Kalai, Y.T., Liu, F.-H., Raz, R.: Memory delegation. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 151–168. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  3. 3.
    Docampo, D., Hush, D.R., Abdallah, C.T.: Constructive function approximation: theory and practice. In: Intelligent Methods in Signal Processing and Communications. Birkhauser Boston Inc. (1997)Google Scholar
  4. 4.
    Donahue, M.J., Darken, C., Gurvits, L., Sontag, E.: Rates of convex approximation in non-hilbert spaces. Constructive Approximation 13, 187–220 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fuller, B., O’Neill, A., Reyzin, L.: A unified approach to deterministic encryption (...). TCC 2012 (2012)Google Scholar
  6. 6.
    Hastad, J., Impagliazzo, R., Levin, L.A., Luby, M.: A pseudorandom generator from any one-way function. SIAM J. Comput. 28, 1364–1396 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Holenstein, T.: Key agreement from weak bit agreement. In: STOC 2005 (2005)Google Scholar
  8. 8.
    Impagliazzo, R.: Hard-core distributions for somewhat hard problems. In: FOCS 36 (1995)Google Scholar
  9. 9.
    Jetchev, D., Pietrzak, K.: How to fake auxiliary input. In: Lindell, Y. (ed.) TCC 2014. LNCS, vol. 8349, pp. 566–590. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  10. 10.
    Klivans, A.R., Servedio, R.A.: Boosting and hard-core set construction. Mach. Learn. 51(3), 217–238 (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Lu, C.-J., Tsai, S.-C., Wu, H.-L.: On the complexity of hard-core set constructions. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 183–194. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  12. 12.
    Neumann, J.: Zur theorie der gesellschaftsspiele. Math. Ann. 100, 295–320 (1928)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pietrzak, K.: Private communication, may (2015)Google Scholar
  14. 14.
    Reingold, O., Trevisan, L., Tulsiani, M., Vadhan, S.: Dense subsets of pseudorandom sets. FOCS 2008 (2008)Google Scholar
  15. 15.
    Skorski, M.: Metric pseudoentropy: Characterizations, transformations and applications. In: ICITS (2015)Google Scholar
  16. 16.
    Skorski, M.: Nonuniform indistinguishability and unpredictability hardcore lemmas: new proofs and applications to pseudoentropy. In: ICITS (2015)Google Scholar
  17. 17.
    Trevisan, L., Tulsiani, M., Vadhan, S.: Regularity, boosting, and efficiently simulating every high-entropy distribution. CCC 2009 (2008)Google Scholar
  18. 18.
    Vadhan, S., Zheng, C.J.: A uniform min-max theorem with applications in cryptography. In: Canetti, R., Garay, J.A. (eds.) CRYPTO 2013, Part I. LNCS, vol. 8042, pp. 93–110. Springer, Heidelberg (2013) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Cryptology and Data Security GroupUniversity of WarsawWarsawPoland

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