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International Conference on Cryptology and Information Security in Latin America

LATINCRYPT 2023: Progress in Cryptology – LATINCRYPT 2023 pp 192–211Cite as

On the Discrete Logarithm Problem in the Ideal Class Group of Multiquadratic Fields

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 14168)

Abstract

In this work we show that the discrete logarithm problem in the ideal class group of the multiquadratic field \(K = \mathbb {Q}(\sqrt{d_1}, \ldots , \sqrt{d_n})\) of degree \(m = 2^n\) can be solved in classical time \(e^{\widetilde{\mathcal {O}}(\max (\log {m},\sqrt{\log {D}}))}\) using an adaptation of Pohlig-Hellman approach, where \(D = d_1 \cdot \ldots \cdot d_n\). This complexity is for the case when the factorization of the target ideal norm is not given. Thanks to our implementation, we provide numerical examples of discrete logarithm computation in real and imaginary number fields.

Keywords

  • multiquadratic field
  • ideal class group
  • norm relation
  • discrete logarithm problem
  • complexity

The research was funded by the Russian Science Foundation (project No. 22-41-04411, https://rscf.ru/en/project/22-41-04411/).

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Notes

  1. 1.

    Source code is available here: https://github.com/novoselov-sa/mqCLDL.

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Authors and Affiliations

  1. Immanuel Kant Baltic Federal University, A. Nevskogo str., 14, Kaliningrad, 236016, Russia

    S. A. Novoselov

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  1. S. A. Novoselov
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Correspondence to S. A. Novoselov .

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Editors and Affiliations

  1. Technology Innovation Institute, Abu Dhabi, United Arab Emirates

    Abdelrahaman Aly

  2. NTT Social Informatics Laboratories, Tokyo, Japan

    Mehdi Tibouchi

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Novoselov, S.A. (2023). On the Discrete Logarithm Problem in the Ideal Class Group of Multiquadratic Fields. In: Aly, A., Tibouchi, M. (eds) Progress in Cryptology – LATINCRYPT 2023. LATINCRYPT 2023. Lecture Notes in Computer Science, vol 14168. Springer, Cham. https://doi.org/10.1007/978-3-031-44469-2_10

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  • DOI: https://doi.org/10.1007/978-3-031-44469-2_10

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