On Differentially Private String Distances

TL;DR

A novel adaptation of the randomized response technique as a bit flipping procedure, applied to the sketched strings, which results in strong privacy guarantees and time- and space-efficient data structures for differentially private tasks on Hamming and edit distance.

Abstract

Given a database of bit strings $A_1,\ldots,A_m\in \{0,1\}^n$, a fundamental data structure task is to estimate the distances between a given query $B\in \{0,1\}^n$ with all the strings in the database. In addition, one might further want to ensure the integrity of the database by releasing these distance statistics in a secure manner. In this work, we propose differentially private (DP) data structures for this type of tasks, with a focus on Hamming and edit distance. On top of the strong privacy guarantees, our data structures are also time- and space-efficient. In particular, our data structure is $ε$-DP against any sequence of queries of arbitrary length, and for any query $B$ such that the maximum distance to any string in the database is at most $k$, we output $m$ distance estimates. Moreover, - For Hamming distance, our data structure answers any query in $\widetilde O(mk+n)$ time and each estimate deviates from the true distance by at most $\widetilde O(k/e^{ε/\log k})$; - For edit distance, our data structure answers any query in $\widetilde O(mk^2+n)$ time and each estimate deviates from the true distance by at most $\widetilde O(k/e^{ε/(\log k \log n)})$. For moderate $k$, both data structures support sublinear query operations. We obtain these results via a novel adaptation of the randomized response technique as a bit flipping procedure, applied to the sketched strings.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1: Chebyshev's Inequality
• Lemma 3.2: Hoeffding's Inequality
• Definition 3.3: $\epsilon$-Differential Privacy
• Definition 3.4: $\epsilon$-DP Data Structure
• Lemma 3.5: Post-Processing
• Theorem 4.1
• Lemma 4.2
• proof