On One-Way Functions and Sparse Languages

Yanyi Liu*, Rafael Pass

*Corresponding author for this work

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

Abstract

We show equivalence between the existence of one-way functions and the existence of a 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(· ) -sparse language L that is hard-on-average with respect to some samplable distribution with Shannon entropy h(· ) such that h(n) - log (S(n) ) ≥ 4 log n ;The existence of a S(· ) -sparse language L∈ NP, that is hard-on-average with respect to some samplable distribution with Shannon entropy h(· ) such that h(n) - log (S(n) ) ≥ n/ 3 ;The existence of one-way functions. where a language L is said to be S(· ) -sparse if | L∩ { 0, 1 }n| ≤ S(n) for all n∈ N. Our results are inspired by, and generalize, results from the elegant recent paper by Ilango, Ren and Santhanam (IRS, STOC’22), which presents similar connections for specific sparse languages.

Original languageEnglish
Title of host publicationTheory of Cryptography - 21st International Conference, TCC 2023, Proceedings
EditorsGuy Rothblum, Hoeteck Wee
PublisherSpringer Science and Business Media Deutschland GmbH
Pages219-237
Number of pages19
ISBN (Print)9783031486142
DOIs
StatePublished - 2023
Event21st International conference on Theory of Cryptography Conference, TCC 2023 - Taipei, Taiwan, Province of China
Duration: 29 Nov 20232 Dec 2023

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume14369 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference21st International conference on Theory of Cryptography Conference, TCC 2023
Country/TerritoryTaiwan, Province of China
CityTaipei
Period29/11/232/12/23

Funding

FundersFunder number
Algorand Centres of Excellence
Algorand Foundation
National Science FoundationCNS 2149305, CNS-2128519, RI-1703846
Air Force Office of Scientific ResearchFA9550-18-1-0267, FA9550-23-1-0312
Defense Advanced Research Projects AgencyHR00110C0086

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