In this paper, we consider the $k$-approximate pattern matching problem under differential privacy, where the goal is to report or count all substrings of a given string $S$ which have a Hamming distance at most $k$ to a pattern $P$, or decide whether such a substring exists. In our definition of privacy, individual positions of the string $S$ are protected. To be able to answer queries under differential privacy, we allow some slack on $k$, i.e. we allow reporting or counting substrings of $S$ with a distance at most $(1+\gamma)k+\alpha$ to $P$, for a multiplicative error $\gamma$ and an additive error $\alpha$. We analyze which values of $\alpha$ and $\gamma$ are necessary or sufficient to solve the $k$-approximate pattern matching problem while satisfying $\epsilon$-differential privacy. Let $n$ denote the length of $S$. We give 1) an $\epsilon$-differentially private algorithm with an additive error of $O(\epsilon^{-1}\log n)$ and no multiplicative error for the existence variant; 2) an $\epsilon$-differentially private algorithm with an additive error $O(\epsilon^{-1}\max(k,\log n)\cdot\log n)$ for the counting variant; 3) an $\epsilon$-differentially private algorithm with an additive error of $O(\epsilon^{-1}\log n)$ and multiplicative error $O(1)$ for the reporting variant for a special class of patterns. The error bounds hold with high probability. All of these algorithms return a witness, that is, if there exists a substring of $S$ with distance at most $k$ to $P$, then the algorithm returns a substring of $S$ with distance at most $(1+\gamma)k+\alpha$ to $P$. Further, we complement these results by a lower bound, showing that any algorithm for the existence variant which also returns a witness must have an additive error of $\Omega(\epsilon^{-1}\log n)$ with constant probability.

翻译：本文研究差分隐私约束下的$k$-近似模式匹配问题，目标是在给定字符串$S$中报告或计数所有与模式$P$的汉明距离不超过$k$的子串，或判定是否存在这样的子串。在隐私定义中，我们保护字符串$S$的各个位置信息。为在差分隐私条件下回答查询，允许在$k$上引入松弛量，即允许报告或计数$S$中与$P$距离不超过$(1+\gamma)k+\alpha$的子串（其中$\gamma$为乘性误差，$\alpha$为加性误差）。我们分析了在满足$\epsilon$-差分隐私条件下，解决$k$-近似模式匹配问题所需的$\alpha$和$\gamma$取值条件。设$n$表示$S$的长度，本文给出：(1) 对存在性变体，提出一个具有$O(\epsilon^{-1}\log n)$加性误差且无乘性误差的$\epsilon$-差分隐私算法；(2) 对计数变体，提出一个加性误差为$O(\epsilon^{-1}\max(k,\log n)\cdot\log n)$的$\epsilon$-差分隐私算法；(3) 对特殊类别模式报告变体，提出一个加性误差为$O(\epsilon^{-1}\log n)$且乘性误差为$O(1)$的$\epsilon$-差分隐私算法。误差界以高概率成立。所有算法均返回见证结果：即若$S$中存在与$P$距离不超过$k$的子串，则算法返回$S$中与$P$距离不超过$(1+\gamma)k+\alpha$的子串。此外，我们通过下界证明补充上述结果：任何返回见证结果的存在性变体算法，其加性误差必须以常概率至少达到$\Omega(\epsilon^{-1}\log n)$。