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In this work, we present the first database reconstruction attacks against response-hiding private range search schemes on encrypted databases of arbitrary dimensions. Falzon et al. (VLDB 2022) present a number of range-supporting schemes on arbitrary dimensions exhibiting different security and efficiency trade-offs. Additionally, they characterize a form of leakage, structure pattern leakage, also present in many one-dimensional schemes e.g., Demertzis et al. (SIGMOD 2016) and Faber et al. (ESORICS 2015). We present the first systematic study of this leakage and attack a broad collection of schemes, including schemes that allow the responses to contain false-positives (often considered the gold standard in security). We characterize the information theoretic limitations of a passive persistent adversary. Our work shows that for range queries, structure pattern leakage can be as vulnerable to attacks as access pattern leakage. We give a comprehensive evaluation of our attacks with a complexity analysis, a prototype implementation, and an experimental assessment on real-world datasets. 

In the past few years, we have seen multiple attacks on one-dimensional databases that support range queries. These attacks achieve full database reconstruction by exploiting access pattern leakage along with known query distribution or search pattern leakage. We are the first to go beyond one dimension, exploring this threat in two dimensions. We unveil an intrinsic limitation of reconstruction attacks by showing that there can be an exponential number of distinct databases that produce equivalent leakage. Next, we present a full database reconstruction attack. Our algorithm runs in polynomial time and returns a poly-size encoding of all databases consistent with the given leakage profile. We implement our algorithm and observe real-world databases that admit a large number of equivalent databases, which aligns with our theoretical results.