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We revisit the problem of finding B-block-long collisions in Merkle-Damg˚ard Hash Functions in the auxiliary-input random oracle model, in which an attacker gets a piece of S-bit advice about the random oracle and makes T oracle queries. Akshima, Cash, Drucker and Wee (CRYPTO 2020), based on the work of Coretti, Dodis, Guo and Steinberger (EUROCRYPT 2018), showed a simple attack for 2 ≤ B ≤ T (with respect to a random salt). The attack achieves advantage Ω( e ST B/2 n + T 2/2 n) where n is the output length of the random oracle. They conjectured that this attack is optimal. However, this so-called STB conjecture was only proved for B ≈ T and B = 2. Very recently, Ghoshal and Komargodski (CRYPTO 22) confirmed STB conjecture for all constant values of B, and provided an Oe(S 4T B2/2 n + T 2/2 n) bound for all choices of B. In this work, we prove an Oe((ST B/2 n)· max{1, ST2/2 n}+T 2/2 n) bound for every 2 < B < T. Our bound confirms the STB conjecture for ST2 ≤ 2 n, and is optimal up to a factor of S for ST2 > 2 n (note as T 2 is always at most 2n, otherwise finding a collision is trivial by the birthday attack). Our result subsumes all previous upper bounds for all ranges of parameters except for B = Oe(1) and ST2 > 2 n. We obtain our results by adopting and refining the technique of Chung, Guo, Liu, and Qian (FOCS 2020). Our approach yields more modular proofs and sheds light on how to bypass the limitations of prior techniques. Along the way, we obtain a considerably simpler and illuminating proof for B = 2, recovering the main result of Akshima, Cash, Drucker and Wee. 

We build the first sub-linear (in fact, potentially constant-time) public-key searchable encryption system: server can publish a public key PK. anybody can build an encrypted index for document D under PK. client holding the index can obtain a token 𝑧𝑤 from the server to check if a keyword w belongs to D. search using 𝑧𝑤 is almost as fast (e.g., sub-linear) as the non-private search. server granting the token does not learn anything about the document D, beyond the keyword w. yet, the token 𝑧𝑤 is specific to the pair (D, w): the client does not learn if other keywords 𝑤′≠𝑤 belong to D, or if w belongs to other, freshly indexed documents 𝐷′. server cannot fool the client by giving a wrong token 𝑧𝑤. We call such a primitive Encapsulated Search Index (ESI). Our ESI scheme can be made (t, n)-distributed among n servers in the best possible way: non-interactive, verifiable, and resilient to any coalition of up to (𝑡−1) malicious servers. We also introduce the notion of delegatable ESI and show how to extend our construction to this setting. Our solution — including public indexing, sub-linear search, delegation, and distributed token generation — is deployed as a commercial application by a real-world company. 

We investigate a simple but overlooked folklore approach for searching encrypted documents held at an untrusted service: Just stash an index (with unstructured encryption) at the service and download it for updating and searching. This approach is simple to deploy, enables rich search support beyond unsorted keyword lookup, requires no persistent client state, and (intuitively at least) provides excellent security com- pared with approaches like dynamic searchable symmetric encryption (DSSE). This work first shows that implementing this construct securely is more subtle than it appears, and that naive implementations with commodity indexes are insecure due to the leakage of the byte-length of the encoded index. We then develop a set of techniques for encoding indexes, called size-locking, that eliminates this leakage. Our key idea is to fix the size of indexes to depend only on features that are safe to leak. We further develop techniques for securely partitioning indexes into smaller pieces that are downloaded, trading leakage for large increases in performance in a mea- sured way. We implement our systems and evaluate that they provide search quality matching plaintext systems, support for stateless clients, and resistance to damaging injection attacks. 

We introduce a new technique for indexing joins in encrypted SQL databases called partially precomputed joins which achieves lower leakage and bandwidth than those used in prior constructions. These techniques are incorporated into state-of-the-art structured encryption schemes for SQL data, yielding a hybrid indexing scheme with both partially and fully precomputed join indexes. We then introduce the idea of leakage-aware query planning by giving a heuristic that helps the client decide, at query time, which index to use so as to minimize leakage and stay below a given bandwidth budget. We conclude by simulating our constructions on real datasets, showing that our heuristic is accurate and that partially-precomputed joins perform well in practice. 

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.