Nearly Optimal Pseudorandomness from Hardness

Abstract

Existing proofs that deduce BPP = P from circuit lower bounds convert randomized algorithms into deterministic algorithms with a large polynomial slowdown. We convert randomized algorithms into deterministic ones with little slowdown. Specifically, assuming exponential lower bounds against randomized NP ∩ coNP circuits, formally known as randomized SVN circuits, we convert any randomized algorithm over inputs of length n running in time t ≥ n into a deterministic one running in time t^2+α for an arbitrarily small constant α > 0. Such a slowdown is nearly optimal for t close to n, since under standard complexity-theoretic assumptions, there are problems with an inherent quadratic derandomization slowdown. We also convert any randomized algorithm that errs rarely into a deterministic algorithm having a similar running time (with pre-processing). The latter derandomization result holds under weaker assumptions, of exponential lower bounds against deterministic SVN circuits.

Our results follow from a new, nearly optimal, explicit pseudorandom generator fooling circuits of size s with seed length (1+α)log s, under the assumption that there exists a function f ∈ E that requires randomized SVN circuits of size at least 2^(1-α′)n, where α = O(α)′. The construction uses, among other ideas, a new connection between pseudoentropy generators and locally list recoverable codes.