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Nearly Optimal Pseudorandomness from Hardness

Published:17 November 2022Publication History
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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 tn into a deterministic one running in time t2+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.

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        cover image Journal of the ACM
        Journal of the ACM  Volume 69, Issue 6
        December 2022
        302 pages
        ISSN:0004-5411
        EISSN:1557-735X
        DOI:10.1145/3570966
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        Publication History

        • Published: 17 November 2022
        • Online AM: 10 August 2022
        • Accepted: 2 August 2022
        • Revised: 17 June 2022
        • Received: 28 May 2021
        Published in jacm Volume 69, Issue 6

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