## Abstract

The class TFNP is the search analog of NP with the additional guarantee that any instance has a solution. TFNP has attracted extensive attention due to its natural syntactic subclasses that capture the computational complexity of important search problems from algorithmic game theory, combinatorial optimization and computational topology. Thus, one of the main research objectives in the context of TFNP is to search for efficient algorithms for its subclasses, and at the same time proving hardness results where efficient algorithms cannot exist. Currently, no problem in TFNP is known to be hard under assumptions such as NP hardness, the existence of one-way functions, or even public-key cryptography. The only known hardness results are based on less general assumptions such as the existence of collision-resistant hash functions, one-way permutations less established cryptographic primitives (e.g., program obfuscation or functional encryption). Several works explained this status by showing various barriers to proving hardness of TFNP. In particular, it has been shown that hardness of TFNP hardness cannot be based on worst-case NP hardness, unless NP = coNP. Therefore, we ask the following question: What is the weakest assumption sufficient for showing hardness in TFNP? In this work, we answer this question and show that hard-on-Average TFNP problems can be based on the weak assumption that there exists a hard-on-Average language in NP. In particular, this includes the assumption of the existence of one-way functions. In terms of techniques, we show an interesting interplay between problems in TFNP, derandomization techniques, and zeroknowledge proofs.

Original language | English |
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Title of host publication | 8th Innovations in Theoretical Computer Science Conference, ITCS 2017 |

Editors | Christos H. Papadimitriou |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770293 |

DOIs | |

State | Published - 1 Nov 2017 |

Externally published | Yes |

Event | 8th Innovations in Theoretical Computer Science Conference, ITCS 2017 - Berkeley, United States Duration: 9 Jan 2017 → 11 Jan 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 67 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 8th Innovations in Theoretical Computer Science Conference, ITCS 2017 |
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Country/Territory | United States |

City | Berkeley |

Period | 9/01/17 → 11/01/17 |

### Bibliographical note

Funding Information:∗ Supported in part by a grant from the I-CORE Program of the Planning and Budgeting Committee, the Israel Science Foundation and BSF.

### Funding

∗ Supported in part by a grant from the I-CORE Program of the Planning and Budgeting Committee, the Israel Science Foundation and BSF.

Funders | Funder number |
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United States-Israel Binational Science Foundation | |

Israel Science Foundation | |

Israeli Centers for Research Excellence |

## Keywords

- TFNP
- average-case hardness
- derandomization
- one-way functions
- zeroknowledge