## 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

Original language | English |
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Article number | 43 |

Pages (from-to) | 1-55 |

Journal | Journal of the ACM |

Volume | 69 |

Issue number | 6 |

DOIs | |

State | Published - 17 Nov 2022 |

## Keywords

- Pseudorandom generators
- list recovery
- local list decoding
- pseudoentropy
- quantified derandomization

## ASJC Scopus subject areas

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence