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4-Round Luby-Rackoff Construction is a qPRP

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Advances in Cryptology – ASIACRYPT 2019 (ASIACRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11921))

Abstract

The Luby-Rackoff construction, or the Feistel construction, is one of the most important approaches to construct secure block ciphers from secure pseudorandom functions. The 3- and 4-round Luby-Rackoff constructions are proven to be secure against chosen-plaintext attacks (CPAs) and chosen-ciphertext attacks (CCAs), respectively, in the classical setting. However, Kuwakado and Morii showed that a quantum superposed chosen-plaintext attack (qCPA) can distinguish the 3-round Luby-Rackoff construction from a random permutation in polynomial time. In addition, Ito et al. recently showed a quantum superposed chosen-ciphertext attack (qCCA) that distinguishes the 4-round Luby-Rackoff construction. Since Kuwakado and Morii showed the result, a problem of much interest has been how many rounds are sufficient to achieve provable security against quantum query attacks. This paper answers to this fundamental question by showing that 4-rounds suffice against qCPAs. Concretely, we prove that the 4-round Luby-Rackoff construction is secure up to \(O(2^{n/12})\) quantum queries. We also give a query upper bound for the problem of distinguishing the 4-round Luby-Rackoff construction from a random permutation by showing a distinguishing qCPA with \(O(2^{n/6})\) quantum queries. Our result is the first to demonstrate the security of a typical block-cipher construction against quantum query attacks, without any algebraic assumptions. To give security proofs, we use an alternative formalization of Zhandry’s compressed oracle technique.

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Notes

  1. 1.

    Strictly speaking, the attack by Kuwakado and Morii works only when all round functions are keyed permutations. Kaplan et al. [18] showed that the attack works for more general cases.

  2. 2.

    Note that the condition in which the round function of the sponge construction is one-way is unusual in the context of classical symmetric-key provable security.

  3. 3.

    Here we do not mean that our model captures all reasonable stateful quantum oracles. We use our model of stateful quantum oracles just for intermediate arguments to prove our main results, and the claims of the main results are described in the typical model of stateless oracles.

  4. 4.

    Note that the Hadamard operator \(H^{\otimes n}\) corresponds to the Fourier transformation over the group \(\left( \mathbb {Z} / 2 \mathbb {Z} \right) ^{\oplus n}\).

  5. 5.

    Note that this three-step procedure is a quoted verbatim from the original paper [37] of version 20180814:183812, except that the symbol \(y'\) and 0 are used instead of \(\alpha _x\) and \(0^n\), respectively, in the original procedure.

  6. 6.

    Here we have to truncate outputs of \(\mathcal {O}\) without destroying quantum states, which is pointed out to be non-trivial in the quantum setting [18]. However, this “truncation” issue can be overcome by using a technique described in [15].

  7. 7.

    See Section H in this paper’s full version [14] for the reason that closing the gap is important.

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Acknowledgments

The authors thank Qipeng Liu and anonymous reviewers for pointing out an issue of Proposition 5 in a previous version of this paper.

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Authors and Affiliations

  1. NTT Secure Platform Laboratories, Tokyo, Japan

    Akinori Hosoyamada

  2. Nagoya University, Nagoya, Japan

    Akinori Hosoyamada & Tetsu Iwata

Authors
  1. Akinori Hosoyamada
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  2. Tetsu Iwata
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Corresponding author

Correspondence to Akinori Hosoyamada .

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Editors and Affiliations

  1. University of Auckland, Auckland, New Zealand

    Steven D. Galbraith

  2. Security Fundamentals Lab, NICT, Tokyo, Japan

    Shiho Moriai

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Hosoyamada, A., Iwata, T. (2019). 4-Round Luby-Rackoff Construction is a qPRP. In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11921. Springer, Cham. https://doi.org/10.1007/978-3-030-34578-5_6

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  • DOI: https://doi.org/10.1007/978-3-030-34578-5_6

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