On One-way Functions and Sparse Languages

Yanyi Liu; Rafael Pass

Paper 2021/890

On One-way Functions and Sparse Languages

Yanyi Liu, Cornell Tech

Rafael Pass, Cornell Tech, Tel-Aviv University

Abstract

We show equivalence between the existence of one-way functions and the existence of a \emph{sparse} language that is hard-on-average w.r.t. some efficiently samplable ``high-entropy'' distribution. In more detail, the following are equivalent: - The existence of a $S(\cdot)$-sparse language $L$ that is hard-on-average with respect to some samplable distribution with Shannon entropy $h(\cdot)$ such that $h(n)-\log(S(n)) \geq 4\log n$; - The existence of a $S(\cdot)$-sparse language $L \in \NP$, that is hard-on-average with respect to some samplable distribution with Shannon entropy $h(\cdot)$ such that $h(n)-\log(S(n)) \geq n/3$; - The existence of one-way functions. Our results are inspired by, and generalize, the recent elegant paper by Ilango, Ren and Santhanam (ECCC'21), which presents similar characterizations for concrete sparse languages.

Metadata

Available format(s): PDF
Category: Foundations
Publication info: Preprint.
Keywords: one-way functions average-case complexity
Contact author(s): yl2866 @ cornell edu
rafael @ cs cornell edu
History: 2023-02-10: revised; 2021-06-29: received; See all versions
Short URL: https://ia.cr/2021/890
License: CC BY

BibTeX

@misc{cryptoeprint:2021/890,
      author = {Yanyi Liu and Rafael Pass},
      title = {On One-way Functions and Sparse Languages},
      howpublished = {Cryptology ePrint Archive, Paper 2021/890},
      year = {2021},
      note = {\url{https://eprint.iacr.org/2021/890}},
      url = {https://eprint.iacr.org/2021/890}
}