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:
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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 \(t^{\prime}= t^{\Omega(1)}\).
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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.
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We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via “black-box” uniform reductions.
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Manuscript received 14 November 2006
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Trevisan, L., Vadhan, S. Pseudorandomness and Average-Case Complexity Via Uniform Reductions. comput. complex. 16, 331–364 (2007). https://doi.org/10.1007/s00037-007-0233-x
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DOI: https://doi.org/10.1007/s00037-007-0233-x