computational complexity

, Volume 16, Issue 4, pp 331–364

Pseudorandomness and Average-Case Complexity Via Uniform Reductions

Open AccessArticle

DOI: 10.1007/s00037-007-0233-x

Cite this article as:
Trevisan, L. & Vadhan, S. comput. complex. (2007) 16: 331. doi:10.1007/s00037-007-0233-x


Impagliazzo and Wigderson (1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP ≠ BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worst-case hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. In this paper:
  • We obtain an optimal worst-case to average-case connection for EXP: if EXP \(\nsubseteq\) BPTIME(t(n)), then EXP has problems that cannot be solved on a fraction \(1/2 + 1/t^{\prime}(n)\) of the inputs by BPTIME\((t^{\prime}(n))\) algorithms, for \(t^{\prime}= t^{\Omega(1)}\).

  • We exhibit a PSPACE-complete self-correctible and downward self-reducible problem. This slightly simplifies and strengthens the proof of Impagliazzo and Wigderson, which used a #P-complete problem with these properties.

  • We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via “black-box” uniform reductions.


Pseudorandomnessaverage-case complexityderandomizationinstance checkers

Subject classification.

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Copyright information

© Birkhaeuser 2007

Authors and Affiliations

  1. 1.Computer Science DivisionUniversity of California at BerkeleyBerkeleyUSA
  2. 2.School of Engineering and Applied SciencesHarvard UniversityCambridgeUSA