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：指纹论证方法由Bun、Ullman和Vadhan（STOC 2014）首次提出，是建立近似差分隐私（DP）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问题缺乏合适的下界，即使已知下界的问题，这些下界通常也不平滑，且当误差超过某个阈值时往往变得无效。本文提出一种简单方法，通过对指纹编码进行填充-置换变换来生成困难实例。我们通过在不同场景下建立新的下界展示了该方法的适用性：1. 低精度条件下DP平均问题的紧下界，这特别地隐含了Nissim、Stemmer和Vadhan（PODS 2016）提出的私有1-聚类问题的新下界；2. DP算法用于近似k-means聚类时加法误差关于乘法误差的下界，该下界对常数乘法误差情形是紧的；3. 低精度条件下DP估计矩阵主奇异向量的下界，这是Singhal和Steinke（NeurIPS 2021）研究的DP子空间估计的特例。我们的主要技术是对指纹编码应用填充-置换变换。然而，我们并非通过黑箱调用现有指纹编码（例如Tardos编码）来证明结果，而是建立了比Dwork等人（FOCS 2015）和Bun等人（SODA 2017）更强的新的指纹引理，并直接从该引理推导出下界。该引理特别地给出了具有最优速率（达到多对数因子）的更简洁的指纹编码构造，这一结果本身具有独立价值。