## Abstract

The Exponential-Time Hypothesis (ETH) is a strengthening of the mathcal{P} neq mathcal{NP} conjecture, stating that 3-SAT on n variables cannot be solved in (uniform) time 2{epsilon cdot n}, for some epsilon > 0. In recent years, analogous hypotheses that are'exponentially-strong' forms of other classical complexity conjectures (such as mathcal{NP} not subseteq mathcal{BPP} or co mathcal{NP} not subseteq mathcal{NP}) have also been introduced, and have become widely influential. In this work, we focus on the interaction of exponential-time hypotheses with the fundamental and closely-related questions of derandomization and circuit lower bounds. We show that even relatively-mild variants of exponential-time hypotheses have far-reaching implications to derandomization, circuit lower bounds, and the connections between the two. Specifically, we prove that: 1)The Randomized Exponential-Time Hypothesis (rETH) implies that mathcal{BPP} can be simulated on'average-case' in deterministic (nearly-)polynomial-time (i.e., in time 2{tilde{O}(log(n))}=n{log ! log(n){O(1)}}). The derandomization relies on a conditional construction of a pseudorandom generator with near-exponential stretch (i.e., with seed length tilde{O}(log(n))); this significantly improves the state-of-the-art in uniform'hardness-to-randomness' results, which previously only yielded pseudorandom generators with sub-exponential stretch from such hypotheses. 2)The Non-Deterministic Exponential-Time Hypothesis (NETH) implies that derandomization of mathcal{BPP} is completely equivalent to circuit lower bounds against mathcal{E}, and in particular that pseudorandom generators are necessary for derandomization. In fact, we show that the foregoing equivalence follows from a very weak version of NETH, and we also show that this very weak version is necessary to prove a slightly stronger conclusion that we deduce from it. Lastly, we show that disproving certain exponential-time hypotheses requires proving breakthrough circuit lower bounds. In particular, if CireuitSAT for circuits over n bits of size poly(n) can be solved by probabilistic algorithms in time 2n/polylog(n), then mathcal{BP} epsilon does not have circuits of quasilinear size.

Original language | English |
---|---|

Title of host publication | Proceedings - 2020 IEEE 61st Annual Symposium on Foundations of Computer Science, FOCS 2020 |

Publisher | IEEE Computer Society |

Pages | 13-23 |

Number of pages | 11 |

ISBN (Electronic) | 9781728196213 |

DOIs | |

State | Published - Nov 2020 |

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 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
---|---|

Volume | 2020-November |

ISSN (Print) | 0272-5428 |

### Conference

Conference | 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020 |
---|---|

Country/Territory | United States |

City | Virtual, Durham |

Period | 16/11/20 → 19/11/20 |

### Bibliographical note

Funding Information:The work was initiated in the 2018 Complexity Workshop in Oberwolfach; the authors are grateful to the Mathematis-ches Forschungsinstitut Oberwolfach and to the organizers of the workshop for the productive and lovely work environment. Lijie Chen is supported by NSF CCF-1741615 and a Google Faculty Research Award. Ron Rothblum is supported in part by a Milgrom family grant, by the Israeli Science Foundation (Grant No. 1262/18), and the Technion Hiroshi Fujiwara cyber security research center and Israel cyber directorate. Roei Tell is supported by funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819702). Eylon Yogev is funded by the ISF grants 484/18, 1789/19, Len Blavatnik and the Blavatnik Foundation, and The Blavatnik Interdisciplinary Cyber Research Center at Tel Aviv University. Part of this work was done while the fourth author was visiting the Simons Institute for the Theory of Computing.

Publisher Copyright:

© 2020 IEEE.

## Keywords

- computational complexity