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On the Plausibility of Fully Homomorphic Encryption for RAMs

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Advances in Cryptology – CRYPTO 2019 (CRYPTO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11692))

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Abstract

We initiate the study of fully homomorphic encryption for RAMs (RAM-FHE). This is a public-key encryption scheme where, given an encryption of a large database D, anybody can efficiently compute an encryption of P(D) for an arbitrary RAM program P. The running time over the encrypted data should be as close as possible to the worst case running time of P, which may be sub-linear in the data size.

A central difficulty in constructing a RAM-FHE scheme is hiding the sequence of memory addresses accessed by P. This is particularly problematic because an adversary may homomorphically evaluate many programs over the same ciphertext, therefore effectively “rewinding” any mechanism for making memory accesses oblivious.

We identify a necessary prerequisite towards constructing RAM-FHE that we call rewindable oblivious RAM (rewindable ORAM), which provides security even in this strong adversarial setting. We show how to construct rewindable ORAM using symmetric-key doubly efficient PIR (SK-DEPIR) (Canetti-Holmgren-Richelson, Boyle-Ishai-Pass-Wootters: TCC ’17). We then show how to use rewindable ORAM, along with virtual black-box (VBB) obfuscation for specific circuits, to construct RAM-FHE. The latter primitive can be heuristically instantiated using existing indistinguishability obfuscation candidates. Overall, we obtain a RAM-FHE scheme where the multiplicative overhead in running time is polylogarithmic in the database size N. Our basic scheme is single-hop, but we also extend it to obtain multi-hop RAM-FHE with overhead \(N^\epsilon \) for arbitrarily small \(\epsilon >0\).

We view our work as the first evidence that RAM-FHE is likely to exist.

Justin Holmgren is supported in part by the Simons Collaboration on Algorithms and Geometry and by NSF grant CCF-1714779. This research was done in part while affiliated with MIT, supported in part by the NSF MACS project CNS-1413920. Mor Weiss is supported in part by ISF grants 1861/16 and 1399/17, and AFOSR Award FA9550-17-1-0069. Daniel Wichs and Ariel Hamlin are supported by NSF grants CNS-1314722, CNS-1413964, CNS-1750795 and the Alfred P. Sloan Research Fellowship.

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Notes

  1. 1.

    Furthermore, it is possible to replace VBB obfuscation by a small stateless hardware token, resulting in a RAM-FHE scheme where ciphertexts contain such tokens, which appears to still be non-trivial. We note that VBB was similarly used to construct a public-key DEPIR scheme [BIPW17].

  2. 2.

    For example, the adversary can observe the sequential ORAM execution of programs \(P_1,P_2,P_3\), then rewind the client/server state to the point immediately after \(P_1\)’s execution and observe the execution of a different program \(P'_2\), etc.

  3. 3.

    We note that as in [OS97], reshuffles can be “spread-out” over many operations to achieve low worst-case complexity.

  4. 4.

    We note that if an a-priori bound on the scratch tape size is known during encryption, then in the single-hop setting the scratch tape can be included as part of the encrypted database, since any updates to the database during execution are anyway lost when the execution ends.

  5. 5.

    More formally, there is a discrepancy since the access pattern of homomorphic evaluation, though revealing nothing about D, may reveal something about P. To prevent this, we can append an encryption secret key \({\mathsf {sk}}\) to the database D, and execute a program \(\widetilde{P}\) in which P’s code is encrypted under \({\mathsf {sk}}\), where \(\widetilde{P}\) first decrypts P and then executes it over D. This way, the access pattern of the FHE evaluation cannot reveal anything about neither P nor D.

  6. 6.

    Using a technique of Ostrovsky and Shoup [OS97], these operations can be spread-out over multiple \(\texttt {write}\) operations. We analyze the scheme below assuming the reshuffle operations are indeed spread-out across all \(\texttt {write}\) operations.

  7. 7.

    In fact, in our construction \(\mathsf {Eval}\) and \(\mathsf {Dec}\) are deterministic.

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Authors and Affiliations

  1. Khoury College of Computer Sciences, Northeastern University, Boston, MA, USA

    Ariel Hamlin & Daniel Wichs

  2. Department of Computer Science, Princeton University, Princeton, NJ, USA

    Justin Holmgren

  3. Department of Computer Science, IDC Herzliya, Herzliya, Israel

    Mor Weiss

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  1. Ariel Hamlin
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  2. Justin Holmgren
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  3. Mor Weiss
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  4. Daniel Wichs
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Correspondence to Ariel Hamlin or Justin Holmgren .

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Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

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Hamlin, A., Holmgren, J., Weiss, M., Wichs, D. (2019). On the Plausibility of Fully Homomorphic Encryption for RAMs. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11692. Springer, Cham. https://doi.org/10.1007/978-3-030-26948-7_21

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