On the Possibility of Basing Cryptography on EXP≠ BPP

Yanyi Liu*, Rafael Pass

*Corresponding author for this work

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

Abstract

Liu and Pass (FOCS’20) recently demonstrated an equivalence between the existence of one-way functions (OWFs) and mild average-case hardness of the time-bounded Kolmogorov complexity problem. In this work, we establish a similar equivalence but to a different form of time-bounded Kolmogorov Complexity—namely, Levin’s notion of Kolmogorov Complexity—whose hardness is closely related to the problem of whether EXP≠ BPP. In more detail, let Kt(x) denote the Levin-Kolmogorov Complexity of the string x; that is, Kt(x)=minΠ∈{0,1}∗,t∈N{|Π|+⌈logt⌉:U(Π,1t)=x}, where U is a universal Turing machine, and U(Π, 1 t) denotes the output of the program Π after t steps, and let MKtP denote the language of pairs (x, k) having the property that Kt(x) ≤ k. We demonstrate that: MKtP∉ HeurnegBPP (i.e., MKtP is infinitely-often two-sided error mildly average-case hard) iff infinitely-often OWFs exist.MKtP∉ AvgnegBPP (i.e., MKtP is infinitely-often errorless mildly average-case hard) iff EXP≠ BPP. Thus, the only “gap” towards getting (infinitely-often) OWFs from the assumption that EXP≠ BPP is the seemingly “minor” technical gap between two-sided error and errorless average-case hardness of the MKtP problem. As a corollary of this result, we additionally demonstrate that any reduction from errorless to two-sided error average-case hardness for MKtP implies (unconditionally) that NP≠ P. We finally consider other alternative notions of Kolmogorov complexity—including space-bounded Kolmogorov complexity and conditional Kolmogorov complexity—and show how average-case hardness of problems related to them characterize log-space computable OWFs, or OWFs in NC0.

Original languageEnglish
Title of host publicationAdvances in Cryptology – CRYPTO 2021 - 41st Annual International Cryptology Conference, CRYPTO 2021, Proceedings
EditorsTal Malkin, Chris Peikert
PublisherSpringer Science and Business Media Deutschland GmbH
Pages11-40
Number of pages30
ISBN (Print)9783030842413
DOIs
StatePublished - 2021
Externally publishedYes
Event41st Annual International Cryptology Conference, CRYPTO 2021 - Virtual, Online
Duration: 16 Aug 202120 Aug 2021

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume12825 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference41st Annual International Cryptology Conference, CRYPTO 2021
CityVirtual, Online
Period16/08/2120/08/21

Funding

FundersFunder number
National Science FoundationSATC-1704788, RI-1703846
Air Force Office of Scientific ResearchFA9550-18-1-0267
Defense Advanced Research Projects AgencyHR00110C0086

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