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Abstract

We consider (uniform) reductions from computing a function f to the task of distinguishing the output of some pseudorandom generator G from uniform. Impagliazzo and Wigderson [10] and Trevisan and Vadhan [24] exhibited such reductions for every function f in PSPACE. Moreover, their reductions are “black box,” showing how to use any distinguisher T, given as oracle, in order to compute f (regardless of the complexity of T). The reductions are also adaptive, but with the restriction that queries of the same length do not occur in different levels of adaptivity. Impagliazzo and Wigderson [10] also exhibited such reductions for every function f in EXP, but those reductions are not black-box, because they only work when the oracle T is computable by small circuits.

Our main results are that:

  • Nonadaptive black-box reductions as above can only exist for functions f in BPPNP (and thus are unlikely to exist for all of PSPACE).

  • Adaptive black-box reductions, with the same restriction on the adaptivity as above, can only exist for functions f in PSPACE (and thus are unlikely to exist for all of EXP).

Beyond shedding light on proof techniques in the area of hardness vs. randomness, our results (together with [10,24]) can be viewed in a more general context as identifying techniques that overcome limitations of black-box reductions, which may be useful elsewhere in complexity theory (and the foundations of cryptography).

Keywords

pseudorandom generators derandomization black-box reductions 

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References

  1. 1.
    Babai, L., Fortnow, L., Lund, C.: Non-deterministic exponential time has two-prover interactive protocols. In: Proceedings of the 31st Annual IEEE Symposium on Foundations of Computer Science, pp. 16–25 (1990)Google Scholar
  2. 2.
    Babai, L., Fortnow, L., Nisan, N., Wigderson, A.: BPP has subexponential simulation unless EXPTIME has publishable proofs. Computational Complexity 3, 307–318 (1993)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo-random bits. SIAM Journal on Computing 13(4), 850–864 (1984)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bogdanov, A., Trevisan, L.: On worst-case to average-case reductions for NP problems. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 308–317 (2003)Google Scholar
  5. 5.
    Gutfreund, D., Rothblum, G.: The complexity of local list decoding. Technical Report TR08-034, Electronic Colloquium on Computational Complexity (2008)Google Scholar
  6. 6.
    Gutfreund, D., Shaltiel, R., Ta-Shma, A.: If NP languages are hard in the worst-case then it is easy to find their hard instances. Computational Complexity 16(4), 412–441 (2007)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gutfreund, D., Ta-Shma, A.: Worst-case to average-case reductions revisited. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J. (eds.) RANDOM 2007. LNCS, vol. 4627, pp. 569–583. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Gutfreund, D., Vadhan, S.: Limitations of hardness vs. randomness under uniform reductions. Technical Report TR08-007, Electronic Colloquium on Computational Complexity (2008)Google Scholar
  9. 9.
    Impagliazzo, R., Kabanets, V., Wigderson, A.: In search of an easy witness: Exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences 65(4), 672–694 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Impagliazzo, R., Wigderson, A.: Randomness vs. time: de-randomization under a uniform assumption. Journal of Computer and System Sciences 63(4), 672–688 (2001)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Jerrum, M., Valiant, L., Vazirani, V.: Random generation of combinatorial structures from a uniform distribution. Theor. Comput. Sci. 43, 169–188 (1986)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. Computational Complexity 13(1-2), 1–46 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Karp, R.M., Lipton, R.J.: Some connections between nonuniform and uniform complexity classes. In: Proceedings of the 12th Annual ACM Symposium on Theory of Computing, pp. 302–309 (1980)Google Scholar
  14. 14.
    Klivans, A.R., van Melkebeek, D.: Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM Journal on Computing 31(5), 1501–1526 (2002)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lipton, R.: New directions in testing. In: Proceedings of DIMACS workshop on distributed computing and cryptography, vol. 2, pp. 191–202 (1991)Google Scholar
  16. 16.
    Nisan, N., Wigderson, A.: Hardness vs. randomness. Journal of Computer and System Sciences 49, 149–167 (1994)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Reingold, O., Trevisan, L., Vadhan, S.: Notions of reducibility between cryptographic primitives. In: Naor, M. (ed.) TCC 2004. LNCS, vol. 2951, pp. 1–20. Springer, Heidelberg (2004)Google Scholar
  18. 18.
    Santhanam, R.: Circuit lower bounds for arthur–merlin classes. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 275–283 (2007)Google Scholar
  19. 19.
    Sipser, M.: A complexity theoretic approach to randomness. In: Proceedings of the 15th Annual ACM Symposium on Theory of Computing, pp. 330–335 (1983)Google Scholar
  20. 20.
    Stockmeyer, L.: On approximation algorithms for \(\sharp\)P. SIAM Journal on Computing 14(4), 849–861 (1985)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Shaltiel, R., Umans, C.: Simple extractors for all min-entropies and a new pseudorandom generator. Journal of the ACM 52(2), 172–216 (2005)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Shaltiel, R., Viola, E.: Hardness amplification proofs require majority. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 589–598 (2008)Google Scholar
  23. 23.
    Trevisan, L.: Construction of extractors using pseudo-random generators. Journal of the ACM 48(4), 860–879 (2001)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Trevisan, L., Vadhan, S.: Pseudorandomness and average-case complexity via uniform reductions. Computational Complexity 16(4), 331–364 (2007)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Umans, C.: Pseudo-random generators for all hardnesses. Journal of Computer and System Sciences 67(2), 419–440 (2003)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Yao, A.C.: Theory and applications of trapdoor functions. In: Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science, pp. 80–91 (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Dan Gutfreund
    • 1
  • Salil Vadhan
    • 2
  1. 1.Department of Mathematics and CSAIL MIT  
  2. 2.School of Engineering and Applied SciencesHarvard University 

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