A recent line of work initiated by Chiesa and Gur and further developed by Herman and Rothblum investigates the sample and communication complexity of verifying properties of distributions with the assistance of a powerful, knowledgeable, but untrusted prover. In this work, we initiate the study of differentially private (DP) distribution property testing. After all, if we do not trust the prover to help us with verification, why should we trust it with our sensitive sample? We map a landscape of DP prover-aided proofs of properties of distributions. In the non-private case it is known that one-round (two message) private-coin protocols can have substantially lower complexity than public-coin AM protocols, but in the private case, the possibility for improvement depends on the parameter regime and privacy model. Drawing on connections to replicability and techniques for amplification, we show: (1) There exists a reduction from any one-round $(\varepsilon,δ)$-DP private-coin interactive proof to a one-round public-coin DP interactive proof with the same privacy parameters, for the parameter regime $\varepsilon=O(1/\sqrt{n})$ and $δ=O(1/n^{5/2})$, and with the same sample and communication complexities. (2) If the verifier's message in the private-coin interactive proof is $O(1/\sqrt{\log n})$ locally DP -- a far more relaxed privacy parameter regime in a different model -- then applying one additional transformation again yields a one-round public-coin protocol with the same privacy bound and the same sample and computational complexities. (3) However, when the privacy guarantee is very relaxed ($\varepsilon\inΩ(\log n)$), private coins indeed reduce complexity. We also obtain a Merlin-Arthur (one-message) proof for privately testing whether samples are drawn from a product distribution, and prove that its sample complexity is optimal.

翻译：由Chiesa和Gur开创、Herman和Rothblum进一步发展的一系列近期工作，研究了在强大、博学但不可信证明者的协助下，验证分布性质所需的样本与通信复杂度。在本文中，我们首次开展了差分隐私（DP）分布性质测试的研究。毕竟，如果我们不信任证明者帮助我们进行验证，为何要信任它处理我们的敏感样本呢？我们描绘了DP证明者辅助的分布性质证明的图景。在非隐私情形中，已知单轮（两消息）私密币协议可能具有远低于公开币AM协议的复杂度，但在隐私情形中，改进的可能性取决于参数区间与隐私模型。利用可复现性与放大技术之间的联系，我们证明：(1) 对于参数区间$\varepsilon=O(1/\sqrt{n})$且$δ=O(1/n^{5/2})$，存在从任意单轮$(\varepsilon,δ)$-DP私密币交互式证明到具有相同隐私参数的单轮公开币DP交互式证明的归约，且样本与通信复杂度保持不变。(2) 若私密币交互式证明中验证者的消息是$O(1/\sqrt{\log n})$的局部DP——这对应于另一种模型中宽松得多的隐私参数区间——则应用一次额外变换后，同样可得到具有相同隐私界、样本与计算复杂度的单轮公开币协议。(3) 然而，当隐私保证非常宽松时（$\varepsilon\inΩ(\log n)$），私密币确实降低了复杂度。我们还得到了用于隐私测试样本是否来自乘积分布的Merlin-Arthur（单消息）证明，并证明了其样本复杂度是最优的。