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Is There an Oblivious RAM Lower Bound for Online Reads?

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Abstract

Oblivious RAM (ORAM), introduced by Goldreich (STOC 1987) and Ostrovsky (STOC 1990), can be used to read and write to memory in a way that hides which locations are being accessed. The best known ORAM schemes have an \(O(\log n)\) overhead per access, where \(n\) is the data size. The work of Goldreich and Ostrovsky (JACM 1996) gave a lower bound, showing that this is optimal for ORAM schemes that operate in a “balls and bins” model, where memory blocks can only be shuffled between different locations but not manipulated otherwise (and the server is used solely as remote storage). The lower bound even extends to weaker settings such as offline ORAM, where all of the accesses to be performed need to be specified ahead of time, and read-only ORAM, which only allows reads but not writes. But can we get lower bounds for general ORAM, beyond “balls and bins”? The work of Boyle and Naor (ITCS 2016) shows that this is unlikely in the offline setting. In particular, they construct an offline ORAM with \(o(\log n)\) overhead assuming the existence of small sorting circuits. Although we do not have instantiations of the latter, ruling them out would require proving new circuit lower bounds. On the other hand, the recent work of Larsen and Nielsen (CRYPTO 2018) shows that there indeed is an \(\Omega (\log n)\) lower bound for general online ORAM. This still leaves the question open for online read-only ORAM or for read/write ORAM where we want very small overhead for the read operations. In this work, we show that a lower bound in these settings is also unlikely. In particular, our main result is a construction of online ORAM, in which the server is used solely as remote storage, where reads (but not writes) have an \(o(\log n)\) overhead, assuming the existence of small sorting circuits as well as very good locally decodable codes (LDCs). Although we do not have instantiations of either of these with the required parameters, ruling them out is beyond current lower bounds.

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Notes

  1. The work of [2] was published after the conference version of this paper [58].

  2. Similar to previous works (e.g., [50, 51, 53]), we assume words are of at least logarithmic size.

  3. This is reminiscent of a construction of [44], which also instantiated the levels of a hierarchical ORAM with a primitive guaranteeing read privacy (specifically, they use PIR). However, our goals, and the details of our construction, differs significantly from [44].

  4. In [9], the blocks consist solely of the tag, but the algorithm is usually run when tags are concatenated with memory blocks (which are carried as a “payload”, and the complexity increases accordingly). We choose to explicitly include the data portion in the block.

  5. In this context, we note that recently (and after the conference version of this paper [58] was published), Farhadi et al. [18] showed an interesting connection between oblivious-access sort algorithms and seemingly unrelated problems in the field of network coding.

  6. In particular, the accesses performed during \(\textsf {Setup}\) are not included in \(\textsf {AP}\), i.e., it includes only the accesses performed during the \(\textsf {Read}\) and \(\textsf {Write}\) executions.

  7. Recall that the client memory stores blocks of size \({\mathsf {B}}\). Jumping ahead, for the setting discussed in the theorem statement such blocks are large enough to store the entire client memory needed for the metadata ORAM.

  8. Here, we assume \(\lambda \ge \log {M/2k}\), which is the number of bits needed to represent the counter.

  9. We could have similarly defined this notion as an extension of ORAM schemes (that support \(\textsf {write}\) operations), but since we only use this property for RO-ORAM schemes, we choose to define it for this (more restricted) setting.

  10. The construction can use any RO-ORAM scheme, but the \(\textsf {read}\) overhead is at least the overhead of the RO-ORAM scheme. Therefore, to obtain \(o\left( \log n\right) \) overhead, we need to instantiate the ORAM with our RO-ORAM scheme.

  11. We note that several ORAM schemes (such as tree-based ORAM schemes, and in particular the ORAM of Theorem 3.7), though described for logical memories given as arrays, can actually support logical memories given as map data structures.

  12. This assumption is without loss of generality since for the block sizes we consider, concatenating the address to the block would cause at most a constant multiplicative increase in the block size.

  13. More accurately, blocks from level i cause one access to \(\mathcal {DB}^i\) and one access to \(\mathcal {DB}^{i+1}\), but these operations have the same complexity since they entail reading or writing a size-\({\mathsf {B}}\) block.

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Acknowledgements

We thank the anonymous Journal of Cryptology reviewers for their comments, which helped us improve the paper. This research was supported by NSF Grants CNS-1314722, CNS-1413964, CNS-1750795 and the Alfred P. Sloan Research Fellowship. The first author was supported in part by The Eric and Wendy Schmidt Postdoctoral Grant for Women in Mathematical and Computing Sciences.

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  1. Faculty of Engineering, Bar-Ilan University, Ramat-Gan, Israel

    Mor Weiss

  2. Department of Computer Science, Northeastern University, Boston, MA, USA

    Daniel Wichs

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Weiss, M., Wichs, D. Is There an Oblivious RAM Lower Bound for Online Reads?. J Cryptol 34, 18 (2021). https://doi.org/10.1007/s00145-021-09392-1

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