Locality Sensitive Hashing with Extended Differential Privacy

Extended differential privacy, a generalization of standard differential privacy (DP) using a general metric rather than the Hamming metric, has been widely studied to provide rigorous privacy guarantees while keeping high utility. However, existing works on extended DP focus on a specific metric such as the Euclidean metric, the $l_1$ metric, and the Earth Mover's metric, and cannot be applied to other metrics. Consequently, existing extended DP mechanisms are limited to a small number of applications such as location-based services and document processing. In this paper, we propose a mechanism providing extended DP with a wide range of metrics. Our mechanism is based on locality sensitive hashing (LSH) and randomized response, and can be applied to a wide variety of metrics including the angular distance (or cosine) metric, Jaccard metric, Earth Mover's metric, and $l_p$ metric. Moreover, our mechanism works well for personal data in a high-dimensional space. We theoretically analyze the privacy properties of our mechanism, introducing new versions of concentrated and probabilistic extended DP to explain the guarantees provided. Finally, we apply our mechanism to friend matching based on high-dimensional personal data with an angular distance metric in the local model. We show that existing local DP mechanisms such as the RAPPOR do not work in this application. We also show through experiments that our mechanism makes possible friend matching with rigorous privacy guarantees and high utility.