Pseudorandomness and Average-Case Complexity Via Uniform Reductions
Luca Trevisan, Salil Vadhan · Computational Complexity · 2007
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 $$ subseteq$$ 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.