Low-degree Security of the Planted Random Subgraph Problem

TL;DR

This paper proves the low-degree hardness of detecting planted random subgraphs up to nearly linear size, extending previous bounds and applying these results to secure multiparty protocols and secret sharing schemes.

Contribution

It advances the understanding of the planted subgraph problem's hardness by extending the degree bounds and applies this to develop communication-efficient secure protocols.

Findings

Hardness extends to subgraphs with size up to n^{1 - Ω(1)}.

Analysis is tight in degree, advantage, and leaked vertices.

Applications include communication-optimal multiparty protocols and efficient secret sharing schemes.

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.