Paper 2024/1497

Low-degree Security of the Planted Random Subgraph Problem

Andrej Bogdanov, University of Ottawa
Chris Jones, Bocconi University
Alon Rosen, Bocconi University
Ilias Zadik, Yale University
Abstract

The planted random subgraph detection conjecture of Abram et al. (TCC 2023) asserts the pseudorandomness of a pair of graphs $(H, G)$, where $G$ is an Erdos-Renyi random graph on $n$ vertices, and $H$ is a random induced subgraph of $G$ on $k$ vertices. Assuming the hardness of distinguishing these two distributions (with two leaked vertices), Abram et al. construct communication-efficient, computationally secure (1) 2-party private simultaneous messages (PSM) and (2) secret sharing for forbidden graph structures. We prove the low-degree hardness of detecting planted random subgraphs all the way up to $k\leq n^{1 - \Omega(1)}$. This improves over Abram et al.'s analysis for $k \leq n^{1/2 - \Omega(1)}$. The hardness extends to $r$-uniform hypergraphs for constant $r$. Our analysis is tight in the distinguisher's degree, its advantage, and in the number of leaked vertices. Extending the constructions of Abram et al, we apply the conjecture towards (1) communication-optimal multiparty PSM protocols for random functions and (2) bit secret sharing with share size $(1 + \epsilon)\log n$ for any $\epsilon > 0$ in which arbitrary minimal coalitions of up to $r$ parties can reconstruct and secrecy holds against all unqualified subsets of up to $\ell = o(\epsilon \log n)^{1/(r-1)}$ parties.

Metadata
Available format(s)
PDF
Category
Foundations
Publication info
Published by the IACR in TCC 2024
Keywords
Low-degree securityplanted cliquesecret sharingprivate simultaneous messages
Contact author(s)
abogdano @ uottawa ca
chris jones @ unibocconi it
alon rosen @ unibocconi it
ilias zadik @ yale edu
History
2024-09-30: approved
2024-09-24: received
See all versions
Short URL
https://ia.cr/2024/1497
License
Creative Commons Attribution-NonCommercial
CC BY-NC

BibTeX

@misc{cryptoeprint:2024/1497,
      author = {Andrej Bogdanov and Chris Jones and Alon Rosen and Ilias Zadik},
      title = {Low-degree Security of the Planted Random Subgraph Problem},
      howpublished = {Cryptology {ePrint} Archive, Paper 2024/1497},
      year = {2024},
      url = {https://eprint.iacr.org/2024/1497}
}
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