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ORAMs in a Quantum World

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Post-Quantum Cryptography (PQCrypto 2017)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 10346))

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Abstract

We study the security of Oblivious Random Access Machines (ORAM) in the quantum world. First we introduce a new formal treatment of ORAMs, which is at the same time elegant and simpler than the known formalization by Goldreich and Ostrovsky. Then we define a new security model for ORAMs, based on a strong, adaptive, game-based security definition, which we show to be at least as strong as other existing notions in the literature. We extend such security notion to the post-quantum setting in the natural way, i.e., by considering classical ORAMs resistant against quantum adversaries. We show a standard quantum attack against an insecure instantiation of PathORAM, one of the most efficient general ORAM constructions to date, introduced by Stefanov et al. On the other hand, we show that PathORAM is post-quantum secure if instantiated using post-quantum underlying primitives. Furthermore, we initiate the study of quantum ORAMs (QORAMs), that is, ORAM constructions meant to be executed between quantum parties acting on arbitrary quantum data. We address many problems arising when formalizing QORAM security through a novel technique of independent interest (which we call safe extractor), modeling a quantum adversary able to extract information from a quantum system in a computationally undetectable way. Finally, we provide a secure QORAM construction (based on PathORAM and a quantum encryption scheme introduced by Alagic et al.) which has the interesting property of making read and write operations inherently equivalent.

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Notes

  1. 1.

    One might wonder why the definition of ORAM should depend on an encryption scheme, and why not on other cryptographic primitives, such as PRNGs or hash functions. The reason is that not all ORAM constructions use such primitives (cf. [19, 29, 36]), while the encryption of the database is a minimal requirement for security, and present in all known ORAM constructions to date. Such semantic artifice is therefore not restrictive in practice.

  2. 2.

    Due to its size, the position map has to be stored recursively to smaller PathORAMs as in [33]. For ease of exposition (and without loss of generality), we will assume here that the position map is stored locally.

  3. 3.

    Our PathQORAM construction is secure by using a merely post-quantum secure PRNG. However notice that, in a quantum scenario such as the one we consider, quantum mechanics allows to generate truly random numbers.

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Acknowledgments

We are grateful to the anonymous reviewers for insightful comments, and to Marc Fischlin and Christian Schaffner for many fruitful discussions. This work has been funded by CYSEC, CRISP, and the DFG as part of projects S4 and S5 within the CRC 1119 CROSSING. Tommaso Gagliardoni is supported by the EU ERC PERCY, grant agreement no. 32131.

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Authors and Affiliations

  1. IBM Zurich Research Labs, Rüschlikon, Switzerland

    Tommaso Gagliardoni

  2. Technische Universität Darmstadt, Darmstadt, Germany

    Nikolaos P. Karvelas & Stefan Katzenbeisser

Authors
  1. Tommaso Gagliardoni
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  2. Nikolaos P. Karvelas
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  3. Stefan Katzenbeisser
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Correspondence to Nikolaos P. Karvelas .

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Editors and Affiliations

  1. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Tanja Lange

  2. Kyushu University , Fukuoka, Japan

    Tsuyoshi Takagi

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Gagliardoni, T., Karvelas, N.P., Katzenbeisser, S. (2017). ORAMs in a Quantum World. In: Lange, T., Takagi, T. (eds) Post-Quantum Cryptography . PQCrypto 2017. Lecture Notes in Computer Science(), vol 10346. Springer, Cham. https://doi.org/10.1007/978-3-319-59879-6_23

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  • DOI: https://doi.org/10.1007/978-3-319-59879-6_23

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