, Volume 16, Issue 4, pp 331-364,
Open Access This content is freely available online to anyone, anywhere at any time.
Date: 05 Dec 2007

Pseudorandomness and Average-Case Complexity Via Uniform Reductions

Abstract.

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.

Manuscript received 14 November 2006