International Association for Cryptologic Research

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Low-degree Security of the Planted Random Subgraph Problem

Authors:
Andrej Bogdanov , University of Ottawa
Chris Jones , Bocconi University
Alon Rosen , Bocconi University and Reichman University
Ilias Zadik , Yale University
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Presentation: Slides
Conference: TCC 2024
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 low-degree hardness of detecting planted random subgraphs all the way up to k≤n1−Ω(1). This improves over Abram et al.'s analysis for k≤n1/2−Ω(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 that are secure even against multiple random evaluations and (2) bit secret sharing with share size (1+ϵ)log⁡n for any ϵ>0 in which arbitrary coalitions of up to r parties can reconstruct and secrecy holds against all unqualified subsets of up to ℓ=o(ϵlog⁡n)1/(r−1) parties.
BibTeX
@inproceedings{tcc-2024-34793,
  title={Low-degree Security of the Planted Random Subgraph Problem},
  publisher={Springer-Verlag},
  author={Andrej Bogdanov and Chris Jones and Alon Rosen and Ilias Zadik},
  year=2024
}