### On Average-Case Hardness in TFNP from One-Way Functions

Pavel Hubáček, Chethan Kamath, Karel Král, and Veronika Slívová

##### Abstract

The complexity class TFNP consists of all NP search problems that are total in the sense that a solution is guaranteed to exist for all instances. Over the years, this class has proved to illuminate surprising connections among several diverse subfields of mathematics like combinatorics, computational topology, and algorithmic game theory. More recently, we are starting to better understand its interplay with cryptography. We know that certain cryptographic primitives (e.g. one-way permutations, collision-resistant hash functions, or indistinguishability obfuscation) imply average-case hardness in TFNP and its important subclasses. However, its relationship with the most basic cryptographic primitive -- i.e., one-way functions (OWFs) -- still remains unresolved. Under an additional complexity theoretic assumption, OWFs imply hardness in TFNP (Hubacek, Naor, and Yogev, ITCS 2017). It is also known that average-case hardness in most structured subclasses of TFNP does not imply any form of cryptographic hardness in a black-box way (Rosen, Segev, and Shahaf, TCC 2017) and, thus, one-way functions might be sufficient. Specifically, no negative result which would rule out basing average-case hardness in TFNP solely on OWFs is currently known. In this work, we further explore the interplay between TFNP and OWFs and give the first negative results. As our main result, we show that there cannot exist constructions of average-case (and, in fact, even worst-case) hard TFNP problem from OWFs with a certain type of simple black-box security reductions. The class of reductions we rule out is, however, rich enough to capture many of the currently known cryptographic hardness results for TFNP. Our results are established using the framework of black-box separations (Impagliazzo and Rudich, STOC 1989) and involve a novel application of the reconstruction paradigm (Gennaro and Trevisan, FOCS 2000).

##### Metadata
Available format(s)
Category
Foundations
Publication info
A major revision of an IACR publication in TCC 2020
Keywords
TFNPPPADaverage-case hardnessone-way functionsblack-box separations
Contact author(s)
hubacek @ iuuk mff cuni cz
History
2020-09-28: revised
2020-09-25: received
See all versions
Short URL
https://ia.cr/2020/1162
License

CC BY

BibTeX

@misc{cryptoeprint:2020/1162,
author = {Pavel Hubáček and Chethan Kamath and Karel Král and Veronika Slívová},
title = {On Average-Case Hardness in TFNP from One-Way Functions},
howpublished = {Cryptology ePrint Archive, Paper 2020/1162},
year = {2020},
note = {\url{https://eprint.iacr.org/2020/1162}},
url = {https://eprint.iacr.org/2020/1162}
}

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