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Communication Lower Bounds of Key-Agreement Protocols via Density Increment Arguments

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Theory of Cryptography (TCC 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14371))

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Abstract

Constructing key-agreement protocols in the random oracle model (ROM) is a viable method to assess the feasibility of developing public-key cryptography within Minicrypt. Unfortunately, as shown by Impagliazzo and Rudich (STOC 1989) and Barak and Mahmoody (Crypto 2009), such protocols can only guarantee limited security: any \(\ell \)-query protocol can be attacked by an \(O(\ell ^2)\)-query adversary. This quadratic gap matches the key-agreement protocol proposed by Merkle (CACM 78), known as Merkle’s Puzzles.

Besides query complexity, the communication complexity of key-agreement protocols in the ROM is also an interesting question in the realm of find-grained cryptography, even though only limited security is achievable. Haitner et al. (ITCS 2019) first observed that in Merkle’s Puzzles, to obtain secrecy against an eavesdropper with \(O(\ell ^2)\) queries, the honest parties must exchange \(\varOmega (\ell )\) bits. Therefore, they conjectured that high communication complexity is unavoidable, any \(\ell \)-query protocols with c bits of communication could be attacked by an \(O(c\cdot \ell )\)-query adversary. This, if true, will suggest that Merkle’s Puzzle is also optimal regarding communication complexity. Building upon techniques from communication complexity, Haitner et al. (ITCS 2019) confirmed this conjecture for two types of key agreement protocols with certain natural properties.

This work affirms the above conjecture for all non-adaptive protocols with perfect completeness. Our proof uses a novel idea called density increment argument. This method could be of independent interest as it differs from previous communication lower bounds techniques (and bypasses some technical barriers).

Supported by NSF CAREER award 2141536.

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Notes

  1. 1.

    We drop low order terms such as \(\log N\) and \(\log M\) here.

  2. 2.

    This is not exactly the same as [BMG09] due to some technical challenges in [BMG09].

  3. 3.

    More precisely, ‘almost all’ means if we sample an entry (x, y) according to the probability that it appears in real execution (conditioned on \(\tau , f_E\)), we have \(\mathcal {M}(x, y) = b\) with high probability.

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Acknowledgements

We thank Noam Mazor for presenting us with this intriguing question and for giving the ingenious construction of a key-agreement protocol with perfect completeness. We are grateful to the anonymous reviewers for their insightful comments and suggestions, which have significantly enhanced the presentation of this work.

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Authors and Affiliations

  1. Thomas Lord Department of Computer Science, University of Southern California, Los Angeles, CA, 90007, USA

    Mi-Ying Huang, Xinyu Mao, Guangxu Yang & Jiapeng Zhang

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  1. Mi-Ying Huang
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Correspondence to Mi-Ying Huang .

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  1. Apple, Cupertino, CA, USA

    Guy Rothblum

  2. NTT Research, Sunnyvale, CA, USA

    Hoeteck Wee

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Huang, MY., Mao, X., Yang, G., Zhang, J. (2023). Communication Lower Bounds of Key-Agreement Protocols via Density Increment Arguments. In: Rothblum, G., Wee, H. (eds) Theory of Cryptography. TCC 2023. Lecture Notes in Computer Science, vol 14371. Springer, Cham. https://doi.org/10.1007/978-3-031-48621-0_7

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