Many deployments of differential privacy in industry are in the local model, where each party releases its private information via a differentially private randomizer. We study triangle counting in the noninteractive and interactive local model with edge differential privacy (that, intuitively, requires that the outputs of the algorithm on graphs that differ in one edge be indistinguishable). In this model, each party's local view consists of the adjacency list of one vertex. In the noninteractive model, we prove that additive $\Omega(n^2)$ error is necessary, where $n$ is the number of nodes. This lower bound is our main technical contribution. It uses a reconstruction attack with a new class of linear queries and a novel mix-and-match strategy of running the local randomizers with different completions of their adjacency lists. It matches the additive error of the algorithm based on Randomized Response, proposed by Imola, Murakami and Chaudhuri (USENIX2021) and analyzed by Imola, Murakami and Chaudhuri (CCS2022) for constant $\varepsilon$. We use a different postprocessing of Randomized Response and provide tight bounds on the variance of the resulting algorithm. In the interactive setting, we prove a lower bound of $\Omega(n^{3/2})$ on the additive error. Previously, no hardness results were known for interactive, edge-private algorithms in the local model, except for those that follow trivially from the results for the central model. Our work significantly improves on the state of the art in differentially private graph analysis in the local model.

翻译：许多差分隐私在工业界的部署采用局部模型，其中各方通过差分隐私随机化器发布其私有信息。我们研究非交互式和交互式局部模型中具有边差分隐私的三角计数（直观上要求算法在仅有一条边不同的图上的输出不可区分）。在此模型中，各方的局部视图由单个顶点的邻接表组成。在非交互式模型中，我们证明加性误差至少为$\Omega(n^2)$（其中$n$为节点数）是必要的。这一下界是我们的主要技术贡献。它利用一类新的线性查询和一种新颖的"混合匹配"策略，通过运行不同邻接表完成情况下的局部随机化器来实现重构攻击。该下界与Imola、Murakami和Chaudhuri（USENIX2021）提出、并由Imola、Murakami和Chaudhuri（CCS2022）在常数$\varepsilon$下分析的基于随机响应的算法的加性误差相匹配。我们对随机响应采用不同的后处理方式，并给出了所得算法方差的紧界。在交互式设置中，我们证明加性误差的下界为$\Omega(n^{3/2})$。此前，除平凡地从中心模型结果推导出的结论外，局部模型中交互式边私有算法尚无已知的困难性结果。我们的工作显著改进了局部模型中差分隐私图分析的最新成果。