, Volume 55, Issue 1, pp 134–156 | Cite as

Reconstructive Dispersers and Hitting Set Generators



We give a generic construction of an optimal hitting set generator (HSG) from any good “reconstructive” disperser. Past constructions of optimal HSGs have been based on such disperser constructions, but have had to modify the construction in a complicated way to meet the stringent efficiency requirements of HSGs. The construction in this paper uses existing disperser constructions with the “easiest” parameter setting in a black-box fashion to give new constructions of optimal HSGs without any additional complications.

Our results show that a straightforward composition of the Nisan-Wigderson pseudorandom generator that is similar to the composition in works by Impagliazzo, Shaltiel and Wigderson in fact yields optimal HSGs (in contrast to the “near-optimal” HSGs constructed in those works). Our results also give optimal HSGs that do not use any form of hardness amplification or implicit list-decoding—like Trevisan’s extractor, the only ingredients are combinatorial designs and any good list-decodable error-correcting code.


Disperser Hitting set generator Derandomization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andreev, A.E., Clementi, A.E.F., Rolim, J.D.P.: A new general derandomization method. J. ACM 45(1), 179–213 (1998) MATHMathSciNetGoogle Scholar
  2. 2.
    Andreev, A.E., Clementi, A.E.F., Rolim, J.D.P., Trevisan, L.: Weak random sources, hitting sets, and BPP simulations. SIAM J. Comput. 28(6), 179–213 (1999) CrossRefMathSciNetGoogle Scholar
  3. 3.
    Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo-random bits. SIAM J. Comput. 13(4), 850–864 (1984) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buhrman, H., Fortnow, L.: One-sided versus two-sided error in probabilistic computation. In: Theoretical Aspects of Computer Science, 16th Annual Symposium, 1999 Google Scholar
  5. 5.
    Buhrman, H., Lee, T., van Melkebeek, D.: Language compression and pseudorandom generators. Comput. Complex. 14(3), 228–255 (2005) MATHCrossRefGoogle Scholar
  6. 6.
    Goldreich, O., Vadhan, S., Wigderson, A.: Simplified derandomization of BPP using a hitting set generator. Technical report TR00-004, Electronic Colloquium on Computational Complexity (January 2000) Google Scholar
  7. 7.
    Guruswami, V.: Better extractors for better codes? In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 436–444 (2004) Google Scholar
  8. 8.
    Guruswami, V., Sudan, M.: List decoding algorithms for certain concatenated codes. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, 2000 Google Scholar
  9. 9.
    Gutfreund, D., Shaltiel, R., Ta-Shma, A.: Uniform hardness vs. randomness tradeoffs for Arthur-Merlin games. Comput. Complex. 12(3–4), 85–130 (2003) MATHMathSciNetGoogle Scholar
  10. 10.
    Impagliazzo, R., Wigderson, A.: P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pp. 220–229 (1997) Google Scholar
  11. 11.
    Impagliazzo, R., Shaltiel, R., Wigderson, A.: Near-optimal conversion of hardness into pseudo-randomness. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, pp. 181–190 (1999) Google Scholar
  12. 12.
    Impagliazzo, R., Shaltiel, R., Wigderson, A.: Extractors and pseudo-random generators with optimal seed-length. In: Proceedings of the Thirty-second Annual ACM Symposium on the Theory of Computing, May 2000, pp. 21–23 Google Scholar
  13. 13.
    Impagliazzo, R., Shaltiel, R., Wigderson, A.: Reducing the seed length in the Nisan-Wigderson generator. Full version of [11, 12]. Manuscript. Combinatorica (2003, to appear) Google Scholar
  14. 14.
    Kabanets, V.: Derandomization: a brief overview. Bull. Eur. Assoc. Theor. Comput. Sci. 76, 88–103 (2002) MATHMathSciNetGoogle Scholar
  15. 15.
    Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complex. 13(1–2), 1–46 (2004) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Miltersen, P.B., Vinodchandran, N.V.: Derandomizing Arthur-Merlin games using hitting sets. Comput. Complex. 14(3), 256–279 (2005) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Nisan, N., Wigderson, A.: Hardness vs randomness. J. Comput. Syst. Sci. 49(2), 149–167 (1994) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Shaltiel, R., Umans, C.: Simple extractors for all min-entropies and a new pseudorandom generator. J. ACM 52(2), 172–216 (2005) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Sudan, M.: Decoding of Reed Solomon codes beyond the error-correction bound. J. Complex. 13, 180–193 (1997) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Sudan, M., Trevisan, L., Vadhan, S.: Pseudorandom generators without the XOR lemma. J. Comput. Syst. Sci. 62, 236–266 (2001) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Ta-Shma, A.: Storing information with extractors. Inf. Process. Lett. 83(5), 267–274 (2002) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Ta-Shma, A., Umans, C., Zuckerman, D.: Loss-less condensers, unbalanced expanders, and extractors. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 143–152 (2001) Google Scholar
  23. 23.
    Ta-Shma, A., Zuckerman, D.: Extractor codes. IEEE Trans. Inf. Theory 50(12), 3015–3025 (2004) CrossRefMathSciNetGoogle Scholar
  24. 24.
    Ta-Shma, A., Zuckerman, D., Safra, S.: Extractors from Reed-Muller codes. In: Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science, 2001 Google Scholar
  25. 25.
    Trevisan, L.: Extractors and pseudorandom generators. J. ACM 48(4), 860–879 (2002) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Umans, C.: Pseudo-random generators for all hardnesses. J. Comput. Syst. Sci. 67, 419–440 (2003) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    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 Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of Computer ScienceCalifornia Institute of TechnologyPasadenaUSA

Personalised recommendations