Fingerprinting arguments, first introduced by Bun, Ullman, and Vadhan (STOC 2014), are the most widely used method for establishing lower bounds on the sample complexity or error of approximately differentially private (DP) algorithms. Still, there are many problems in differential privacy for which we don't know suitable lower bounds, and even for problems that we do, the lower bounds are not smooth, and usually become vacuous when the error is larger than some threshold. In this work, we present a simple method to generate hard instances by applying a padding-and-permuting transformation to a fingerprinting code. We illustrate the applicability of this method by providing new lower bounds in various settings: 1. A tight lower bound for DP averaging in the low-accuracy regime, which in particular implies a new lower bound for the private 1-cluster problem introduced by Nissim, Stemmer, and Vadhan (PODS 2016). 2. A lower bound on the additive error of DP algorithms for approximate k-means clustering, as a function of the multiplicative error, which is tight for a constant multiplication error. 3. A lower bound for estimating the top singular vector of a matrix under DP in low-accuracy regimes, which is a special case of DP subspace estimation studied by Singhal and Steinke (NeurIPS 2021). Our main technique is to apply a padding-and-permuting transformation to a fingerprinting code. However, rather than proving our results using a black-box access to an existing fingerprinting code (e.g., Tardos' code), we develop a new fingerprinting lemma that is stronger than those of Dwork et al. (FOCS 2015) and Bun et al. (SODA 2017), and prove our lower bounds directly from the lemma. Our lemma, in particular, gives a simpler fingerprinting code construction with optimal rate (up to polylogarithmic factors) that is of independent interest.

翻译：指纹论证由Bun、Ullman和Vadhan（STOC 2014）首次提出，是建立近似差分隐私算法样本复杂度或误差下界最广泛使用的方法。然而，差分隐私领域仍存在许多我们尚未掌握合适下界的问题，即便已知下界的问题，这些下界也往往不具平滑性，且当误差超过某个阈值时通常会失效。本文提出一种简单方法：通过对指纹编码施加填充与置换变换来生成困难实例。我们通过在不同场景中提供新的下界来展示该方法的适用性：1. 低精度机制下差分隐私平均的紧下界，该结果特别蕴含了Nissim、Stemmer和Vadhan（PODS 2016）提出的隐私1-聚类问题的新下界；2. 近似k-means聚类的差分隐私算法加性误差下界（作为乘性误差的函数），该下界在常数值乘性误差下是紧的；3. 低精度机制下差分隐私矩阵主奇异向量估计的下界，该问题是Singhal和Steinke（NeurIPS 2021）研究的差分隐私子空间估计特例。我们的核心技术是对指纹编码施加填充与置换变换。然而，不同于通过黑盒调用现有指纹编码（如Tardos编码）来证明结果，我们发展了一个比Dwork等人（FOCS 2015）和Bun等人（SODA 2017）更强的引理，并直接从该引理出发证明下界。特别地，该引理能构造出具有最优速率（至多相差多对数因子）且更简洁的指纹编码，这一成果本身具有独立研究价值。