The Journey from NP to TFNP Hardness

Pavel Hubácek, Moni Naor, Eylon Yogev

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 languageEnglish
Title of host publication8th Innovations in Theoretical Computer Science Conference, ITCS 2017
EditorsChristos H. Papadimitriou
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959770293
StatePublished - 1 Nov 2017
Externally publishedYes
Event8th Innovations in Theoretical Computer Science Conference, ITCS 2017 - Berkeley, United States
Duration: 9 Jan 201711 Jan 2017

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference8th Innovations in Theoretical Computer Science Conference, ITCS 2017
Country/TerritoryUnited States


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


Dive into the research topics of 'The Journey from NP to TFNP Hardness'. Together they form a unique fingerprint.

Cite this