## Abstract

Liu and Pass (FOCS’20) recently demonstrated an equivalence between the existence of one-way functions (OWFs) and mild average-case hardness of the time-bounded Kolmogorov complexity problem. In this work, we establish a similar equivalence but to a different form of time-bounded Kolmogorov Complexity—namely, Levin’s notion of Kolmogorov Complexity—whose hardness is closely related to the problem of whether EXP≠ BPP. In more detail, let Kt(x) denote the Levin-Kolmogorov Complexity of the string x; that is, Kt(x)=minΠ∈{0,1}∗,t∈N{|Π|+⌈logt⌉:U(Π,1t)=x}, where U is a universal Turing machine, and U(Π, 1 ^{t}) denotes the output of the program Π after t steps, and let MKtP denote the language of pairs (x, k) having the property that Kt(x) ≤ k. We demonstrate that: MKtP∉ Heur_{neg}BPP (i.e., MKtP is infinitely-often two-sided error mildly average-case hard) iff infinitely-often OWFs exist.MKtP∉ Avg_{neg}BPP (i.e., MKtP is infinitely-often errorless mildly average-case hard) iff EXP≠ BPP. Thus, the only “gap” towards getting (infinitely-often) OWFs from the assumption that EXP≠ BPP is the seemingly “minor” technical gap between two-sided error and errorless average-case hardness of the MKtP problem. As a corollary of this result, we additionally demonstrate that any reduction from errorless to two-sided error average-case hardness for MKtP implies (unconditionally) that NP≠ P. We finally consider other alternative notions of Kolmogorov complexity—including space-bounded Kolmogorov complexity and conditional Kolmogorov complexity—and show how average-case hardness of problems related to them characterize log-space computable OWFs, or OWFs in NC^{0}.

Original language | English |
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Title of host publication | Advances in Cryptology – CRYPTO 2021 - 41st Annual International Cryptology Conference, CRYPTO 2021, Proceedings |

Editors | Tal Malkin, Chris Peikert |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 11-40 |

Number of pages | 30 |

ISBN (Print) | 9783030842413 |

DOIs | |

State | Published - 2021 |

Externally published | Yes |

Event | 41st Annual International Cryptology Conference, CRYPTO 2021 - Virtual, Online Duration: 16 Aug 2021 → 20 Aug 2021 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12825 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 41st Annual International Cryptology Conference, CRYPTO 2021 |
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City | Virtual, Online |

Period | 16/08/21 → 20/08/21 |

### Funding

Funders | Funder number |
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National Science Foundation | SATC-1704788, RI-1703846 |

Air Force Office of Scientific Research | FA9550-18-1-0267 |

Defense Advanced Research Projects Agency | HR00110C0086 |