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Cuckoo Hashing in Cryptography: Optimal Parameters, Robustness and Applications

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

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Abstract

Cuckoo hashing is a powerful primitive that enables storing items using small space with efficient querying. At a high level, cuckoo hashing maps n items into b entries storing at most \(\ell \) items such that each item is placed into one of k randomly chosen entries. Additionally, there is an overflow stash that can store at most s items. Many cryptographic primitives rely upon cuckoo hashing to privately embed and query data where it is integral to ensure small failure probability when constructing cuckoo hashing tables as it directly relates to the privacy guarantees.

As our main result, we present a more query-efficient cuckoo hashing construction using more hash functions. For construction failure probability \(\epsilon \), the query overhead of our scheme is \(O(1 + \sqrt{\log (1/\epsilon )/\log n})\). Our scheme has quadratically smaller query overhead than prior works for any target failure probability \(\epsilon \). We also prove lower bounds matching our construction. Our improvements come from a new understanding of the locality of cuckoo hashing failures for small sets of items.

We also initiate the study of robust cuckoo hashing where the input set may be chosen with knowledge of the hash functions. We present a cuckoo hashing scheme using more hash functions with query overhead \(\tilde{O}(\log \lambda )\) that is robust against \(\textsf{poly}(\lambda )\) adversaries. Furthermore, we present lower bounds showing that this construction is tight and that extending previous approaches of large stashes or entries cannot obtain robustness except with \(\varOmega (n)\) query overhead.

As applications of our results, we obtain improved constructions for batch codes and PIR. In particular, we present the most efficient explicit batch code and blackbox reduction from single-query PIR to batch PIR.

The full version of this paper may be found at [66].

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Acknowledgements

The author would like to thank Mo (Helen) Zhou for feedback on earlier manuscripts and Daniel Noble for pointing out an error and fix in a proof. This research was supported in part by the Algorand Centres of Excellence programme managed by Algorand Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are solely those of the authors.

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    Kevin Yeo

  2. Columbia University, New York City, USA

    Kevin Yeo

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    Helena Handschuh

  2. Brown University, Providence, RI, USA

    Anna Lysyanskaya

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Yeo, K. (2023). Cuckoo Hashing in Cryptography: Optimal Parameters, Robustness and Applications. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14084. Springer, Cham. https://doi.org/10.1007/978-3-031-38551-3_7

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