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Sub-logarithmic Distributed Oblivious RAM with Small Block Size

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Public-Key Cryptography – PKC 2019 (PKC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11442))

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Abstract

Oblivious RAM (ORAM) is a cryptographic primitive that allows a client to securely execute RAM programs over data that is stored in an untrusted server. Distributed Oblivious RAM is a variant of ORAM, where the data is stored in \(m>1\) servers. Extensive research over the last few decades have succeeded to reduce the bandwidth overhead of ORAM schemes, both in the single-server and the multi-server setting, from \(O(\sqrt{N})\) to O(1). However, all known protocols that achieve a sub-logarithmic overhead either require heavy server-side computation (e.g. homomorphic encryption), or a large block size of at least \(\varOmega (\log ^3 N)\).

In this paper, we present a family of distributed ORAM constructions that follow the hierarchical approach of Goldreich and Ostrovsky [17]. We enhance known techniques, and develop new ones, to take better advantage of the existence of multiple servers. By plugging efficient known hashing schemes in our constructions, we get the following results:

  1. 1.

    For any number \(m\ge 2\) of servers, we show an m-server ORAM scheme with \(O(\log N/\log \log N)\) overhead, and block size \(\varOmega (\log ^2 N)\). This scheme is private even against an \((m-1)\)-server collusion.

  2. 2.

    A three-server ORAM construction with \(O(\omega (1)\cdot \log N/\log \log N)\) overhead and a block size almost logarithmic, i.e. \(\varOmega (\log ^{1+\epsilon }N)\).

We also investigate a model where the servers are allowed to perform a linear amount of light local computations, and show that constant overhead is achievable in this model, through a simple four-server ORAM protocol. From theoretical viewpoint, this is the first ORAM scheme with asymptotic constant overhead, and polylogarithmic block size, that does not use homomorphic encryption. Practically speaking, although we do not provide an implementation of the suggested construction, evidence from related work (e.g. [12]) confirms that despite the linear computational overhead, our construction is practical, in particular when applied to secure computation.

A full version is available on arXiv.org e-Print archive as arXiv:1802.05145 [cs.CR]. Research supported by ISF grant 1709/14, BSF grant 2012378, NSF-BSF grant 2015782, and a grant from the Ministry of Science and Technology, Israel, and the Department of Science and Technology, Government of India.

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Notes

  1. 1.

    To the best of our knowledge, the only tree-based ORAM that bypasses recursion, due to Wang et al. [19], works in a different model where linear server work is allowed (see preceding discussion).

  2. 2.

    Up to polylogarithmic factors.

  3. 3.

    Implicitly stated parameters may be omitted for brevity.

  4. 4.

    In the scheme of [22], the shared stash is ‘virtualized’, and is re-inserted into the hierarchy. We roll-back this optimization in preparation to our constructions.

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Acknowledgments

We thank Yuval Ishai, Rafail Ostrovsky and Benny Pinkas for useful comments.

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Authors and Affiliations

  1. Computer Science Department, Technion, 32000, Haifa, Israel

    Eyal Kushilevitz & Tamer Mour

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  1. Eyal Kushilevitz
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Correspondence to Tamer Mour .

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  1. SKLOIS, Institute of Information Engineering, Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

  2. Security Research Laboratories, NEC Corporation, Kawasaki, Japan

    Kazue Sako

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Kushilevitz, E., Mour, T. (2019). Sub-logarithmic Distributed Oblivious RAM with Small Block Size. In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11442. Springer, Cham. https://doi.org/10.1007/978-3-030-17253-4_1

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