Mitigating Leakage in Secure Cloud-Hosted Data Structures: Volume-Hiding for Multi-Maps via Hashing

Volume leakage has recently been identified as a major threat to the security of cryptographic cloud-based data structures by Kellaris \em et al. [CCS'16] (see also the attacks in Grubbs \em et al. [CCS'18] and Lacharité \em et al. [S&P'18]). In this work, we focus on volume-hiding implementations of \em encrypted multi-maps as first considered by Kamara and Moataz [Eurocrypt'19]. Encrypted multi-maps consist of outsourcing the storage of a multi-map to an untrusted server, such as a cloud storage system, while maintaining the ability to perform private queries. Volume-hiding encrypted multi-maps ensure that the number of responses (volume) for any query remains hidden from the adversarial server. As a result, volume-hiding schemes can prevent leakage attacks that leverage the adversary's knowledge of the number of query responses to compromise privacy. We present both conceptual and algorithmic contributions towards volume-hiding encrypted multi-maps. We introduce the first formal definition of volume-hiding leakage functions. In terms of design, we present the first volume-hiding encrypted multi-map dprfMM whose storage and query complexity are both asymptotically optimal. Furthermore, we experimentally show that our construction is practically efficient. Our server storage is smaller than the best previous construction while we improve query complexity by a factor of 10-16x. In addition, we introduce the notion of differentially private volume-hiding leakage functions which strikes a better, tunable balance between privacy and efficiency. To accompany our new notion, we present a differentially private volume-hiding encrypted multi-map dpMM whose query complexity is the volume of the queried key plus an additional logarithmic factor. This is a significant improvement compared to all previous volume-hiding schemes whose query overhead was the maximum volume of any key. In natural settings, our construction improves the average query overhead by a factor of 150-240x over the previous best volume-hiding construction even when considering small privacy budget of ε=0.2.