## Abstract

Existing proofs that deduce text{BPP} = mathrm{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 single-valued nondeterministic (SVN) circuits, we convert any randomized algorithm over inputs of length n running in time t geq n to a deterministic one running in time t{2+ alpha} for an arbitrarily small constant alpha > 0. Such a slowdown is nearly optimal, as, under 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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Title of host publication | Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020 |

Publisher | Institute of Electrical and Electronics Engineers |

Pages | 1057-1068 |

Number of pages | 12 |

ISBN (Electronic) | 9781728196213 |

DOIs | |

State | Published - 19 Jan 2021 |

Externally published | Yes |

Event | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 - Virtual, Durham, United States Duration: 16 Nov 2020 → 19 Nov 2020 |

### Conference

Conference | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 |
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Country/Territory | United States |

City | Virtual, Durham |

Period | 16/11/20 → 19/11/20 |

## Keywords

- derandomization
- list-recoverable codes
- pseudo-entropy
- pseudorandom generators

## ASJC Scopus subject areas

- Computer Science (all)