Cryptography from Planted Graphs: Security with Logarithmic-Size Messages

Damiano Abram, Amos Beimel, Yuval Ishai, Eyal Kushilevitz, Varun Narayanan

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

1 Scopus citations


We study the following broad question about cryptographic primitives: is it possible to achieve security against arbitrary poly(n) -time adversary with O(log n) -size messages? It is common knowledge that the answer is “no” unless information-theoretic security is possible. In this work, we revisit this question by considering the setting of cryptography with public information and computational security. We obtain the following main results, assuming variants of well-studied intractability assumptions: A private simultaneous messages (PSM) protocol for every f: [ n] × [ n] → { 0, 1 } with (1 + ϵ) log n -bit messages, beating the known lower bound on information-theoretic PSM protocols. We apply this towards non-interactive secure 3-party computation with similar message size in the preprocessing model, improving over previous 2-round protocols.A secret-sharing scheme for any “forbidden-graph” access structure on n nodes with O(log n) share size.On the negative side, we show that computational threshold secret-sharing schemes with public information require share size Ω(log log n). For arbitrary access structures, we show that computational security does not help with 1-bit shares. The above positive results guarantee that any adversary of size no ( log n ) achieves an n- Ω ( 1 ) distinguishing advantage. We show how to make the advantage negligible by slightly increasing the asymptotic message size, still improving over all known constructions. The security of our constructions is based on the conjectured hardness of variants of the planted clique problem, which was extensively studied in the algorithms, statistical inference, and complexity theory communities. Our work provides the first applications of such assumptions to improving the efficiency of mainstream cryptographic primitives, gives evidence for the necessity of such assumptions, and suggests new questions in this domain that may be of independent interest.

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
Number of pages30
ISBN (Print)9783031486142
StatePublished - 1 Jan 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


Conference21st International conference on Theory of Cryptography Conference, TCC 2023
Country/TerritoryTaiwan, Province of China

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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