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Hintless Single-Server Private Information Retrieval

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Advances in Cryptology – CRYPTO 2024 (CRYPTO 2024)

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Abstract

We present two new constructions for private information retrieval (PIR) in the classical setting where the clients do not need to do any preprocessing or store any database dependent information, and the server does not need to store any client-dependent information.

Our first construction (\({\textsf{Hintless}\textsf{PIR}}\)) eliminates the client preprocessing step from the recent LWE-based SimplePIR (Henzinger et al., USENIX Security 2023) by outsourcing the “hint” related computation to the server, leveraging a new concept of homomorphic encryption with composable preprocessing. We realize this concept with RLWE encryption schemes, and by leveraging the composibility of this technique we are able to preprocess almost all the expensive parts of the homomorphic computation and reuse them across multiple protocol executions. As a concrete application, we propose highly efficient matrix vector multiplication that allows us to build \({\textsf{Hintless}\textsf{PIR}}\). For a database of size 8 GB, \({\textsf{Hintless}\textsf{PIR}}\) achieves throughput about 6.37 GB/s without requiring transmission of any client or server state. We additionally formalize the matrix vector multiplication protocol as a novel primitive that we call \(\textsf{LinPIR}\), which may be of independent interest.

In our second construction (\(\textsf{Tensor}\textsf{PIR}\)) we reduce the communication of \({\textsf{Hintless}\textsf{PIR}}\) from square root to cubic root in the database size. We show how to use RLWE encryption with preprocessing to outsource LWE decryption for ciphertexts generated by homomorphic multiplications. This allows the server to do more complex processing using a more compact query under LWE.

We implement and benchmark \({\textsf{Hintless}\textsf{PIR}}\) which achieves better concrete costs than \(\textsf{Tensor}\textsf{PIR}\) for a large set of databases of interest. We show that it improves the communication of recent preprocessing constructions when clients do not have large numbers of queries or the database updates frequently. The computation cost for removing the hint is small and decreases as the database becomes larger, and it is always more efficient than other constructions with client hints such as Spiral PIR (Menon and Wu, S&P 2022). In the setting of anonymous queries we also improve on Spiral’s communication.

M. Schultz-Wu—Work performed at Google.

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Notes

  1. 1.

    Without preprocessing.

  2. 2.

    While in principle one can dynamically update the hint, this has many practical difficulties. We direct readers to the application of SimplePIR to Certificate Transparency Auditing [33, Section 7.2], where the SimplePIR authors preferred to have clients cache “stale” queries and periodically redownload the entire hint, rather than leverage their dynamic hint update mechanisms [33, Appendices C.3 and E.3].

  3. 3.

    In particular, we are able to get considerable (though non-asymptotic) speedups in server processing time, which was our primary goal in this work. \({\textsf{NTTless}\textsf{PIR}}\) used in isolation significantly improves the server preprocessing of SimplePIR (and schemes that build on it) from \(O(mN)\) to \(O(m\log n)\), and is likely of independent interest in applications where minimizing this quantity is of primary importance.

  4. 4.

    Our optimization is even compatible with a SimplePIR-type “hint” to reduce our per-query bandwidth. As it has a smaller impact (\(2\times \) reduction) in our setting than that of SimplePIR (\(2^{10}\times \) reduction), we instead omit it to achieve our goal of no database-dependent state on our clients.

  5. 5.

    This is the idea at the basis of the “amortized LWE” cryptosystem of [43], as well as the property SimplePIR leverages to securely allow all clients to use the same hint.

  6. 6.

    Of larger impact is that clients have to resample any encrypted key material they use. We minimize the use of such key material in our protocol to reduce this cost.

  7. 7.

    There has been another recent practically-fast single-server PIR scheme, namely Piano [52]. This scheme requires the entire database be streamed to a memory-constrained client in a preprocessing step. As we are not modelling memory-constraints on clients, in our setting this protocol is essentially equivalent to the trivial PIR scheme that transmits the whole database to the client in a preprocessing step, so we will not formally compare our work to theirs.

  8. 8.

    In our current implementation, this constant factor is somewhat large—\(\approx 33\times \) larger. We discuss several optimizations that would reduce this to \(\approx 9\times \) larger in the full version, though they are not currently implemented.

  9. 9.

    Its server preprocessing is identical to SimplePIR, so the gap between its cost and \({\textsf{Hintless}\textsf{PIR}}\)’s costs similarly vanishes as \(m\rightarrow \infty \).

  10. 10.

    For this database size, Tiptoe PIR’s bandwidth is \(3\times \) larger than the trivial PIR scheme of transmitting the entire database. Our bandwidth is \(\approx 1/16\) of the cost of this trivial protocol.

  11. 11.

    In general p can be a prime power, but in our application we only use the case where p is a prime number.

  12. 12.

    For simplicity, we assume the number of rows of \(H\) is equal to the RLWE ring degree \(n\).

  13. 13.

    We reassure the reader that this restriction on \(n_{\textsf{cols}}\) will be unimportant to our main application of this scheme, namely removing the hint from \({\textsf{LWE}\textsf{PIR}}\) in Sect. 5.

  14. 14.

    https://github.com/google/hintless_pir.

  15. 15.

    We set N higher than the one proposed in [33] according to latest lattice attack estimates.

  16. 16.

    https://www.speedtest.net/global-index/united-states.

  17. 17.

    For Spiral, we could not find parameters that natively support this database on the public Spiral implementation; so we shard the database into smaller ones and estimated Spiral’s performance when running sequentially on the sub-databases.

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  1. Google, Mountain View, USA

    Baiyu Li & Mariana Raykova

  2. UCSD, San Diego, USA

    Daniele Micciancio & Mark Schultz-Wu

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  1. Boston University, Boston, MA, USA

    Leonid Reyzin

  2. University of Waterloo, Waterloo, ON, Canada

    Douglas Stebila

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Li, B., Micciancio, D., Raykova, M., Schultz-Wu, M. (2024). Hintless Single-Server Private Information Retrieval. In: Reyzin, L., Stebila, D. (eds) Advances in Cryptology – CRYPTO 2024. CRYPTO 2024. Lecture Notes in Computer Science, vol 14928. Springer, Cham. https://doi.org/10.1007/978-3-031-68400-5_6

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