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Full Quantum Equivalence of Group Action DLog and CDH, and More

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Advances in Cryptology – ASIACRYPT 2022 (ASIACRYPT 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13791))

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Abstract

Cryptographic group actions are a relaxation of standard cryptographic groups that have less structure. This lack of structure allows them to be plausibly quantum resistant despite Shor’s algorithm, while still having a number of applications. The most famous example of group actions are built from isogenies on elliptic curves.

Our main result is that CDH for abelian group actions is quantumly equivalent to discrete log. Galbraith et al. (Mathematical Cryptology) previously showed perfectly solving CDH to be equivalent to discrete log quantumly; our result works for any non-negligible advantage. We also explore several other questions about group action and isogeny protocols.

Proving the equivalence of breaking the Diffie-Hellman protocol and computing discrete-log is one of the oldest problems in public key cryptography.

Boneh and Lipton [BL96]

A portion of this work was done when the author was employed by Fujitsu Research.

The full version of this paper is available at https://eprint.iacr.org/2022/1135.

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Notes

  1. 1.

    A regular group action is a group action that, for every \(x_{1}, x_{2} \in X\), there exists a unique element \(g \in G\) such that \(x_{1} = g \star x_{2}\).

  2. 2.

    A few very recent works [CD22, MM22, Rob22] break a certain isogeny-based protocol called SIDH. SIDH, however, is just one of a number of isogeny protocols, and in particular it is not a group action. For a slightly more in depth discussion about different isogeny protocols, see Sect. 2.5.

  3. 3.

    We note that our result does not directly apply to restricted effective group actions (REGAs) like CSIDH [CLM+18] and explain this in more detail later.

  4. 4.

    We defer a formal definition of this problem to the body of the paper. It is shown in [BLP+13] that 1D-SIS, for certain parameter settings, is equivalent to the “standard” LWE problem.

  5. 5.

    The adversary could be quantum but is restricted to classical queries to the group and group action oracles.

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  1. Linux Foundation, San Francisco, USA

    Hart Montgomery

  2. NTT Research and Princeton University, Princeton, USA

    Mark Zhandry

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  1. Indian Institute of Technology Madras, Chennai, India

    Shweta Agrawal

  2. Chinese Academy of Sciences, Beijing, China

    Dongdai Lin

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Montgomery, H., Zhandry, M. (2022). Full Quantum Equivalence of Group Action DLog and CDH, and More. In: Agrawal, S., Lin, D. (eds) Advances in Cryptology – ASIACRYPT 2022. ASIACRYPT 2022. Lecture Notes in Computer Science, vol 13791. Springer, Cham. https://doi.org/10.1007/978-3-031-22963-3_1

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