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.

翻译：Abram等人（TCC 2023）提出的植入随机子图检测猜想断言了一对图$(H, G)$的伪随机性，其中$G$是$n$个顶点上的Erdos-Renyi随机图，$H$是$G$在$k$个顶点上的随机诱导子图。基于区分这两种分布（在泄露两个顶点的情况下）的困难性假设，Abram等人构建了通信高效、计算安全的（1）两方私有同步消息（PSM）协议和（2）针对禁止图结构的秘密共享方案。我们证明了检测植入随机子图的低度困难性可一直推广至$k\leq n^{1 - \Omega(1)}$，这改进了Abram等人针对$k \leq n^{1/2 - \Omega(1)}$的分析结果。该困难性可进一步推广至常数$r$的$r$-一致超图。我们的分析在区分器的阶数、优势度以及泄露顶点数量方面均是紧致的。通过扩展Abram等人的构造，我们将该猜想应用于（1）随机函数的通信最优多方PSM协议，以及（2）份额大小为$(1 + \epsilon)\log n$（对任意$\epsilon > 0$）的比特秘密共享方案，其中任意至多$r$方的最小联盟均可重构秘密，且保密性可抵御所有至多$\ell = o(\epsilon \log n)^{1/(r-1)}$方的非授权子集。