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Cryptanalysis of Symmetric Primitives over Rings and a Key Recovery Attack on Rubato

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

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Abstract

Symmetric primitives are a cornerstone of cryptography, and have traditionally been defined over fields, where cryptanalysis is now well understood. However, a few symmetric primitives defined over rings \(\mathbb Z _q\) for a composite number \(q\) have recently been proposed, a setting where security is much less studied. In this paper we focus on studying established algebraic attacks typically defined over fields and the extent of their applicability to symmetric primitives defined over the ring of integers modulo a composite \(q\). Based on our analysis, we present an attack on full Rubato, a family of symmetric ciphers proposed by Ha et al. at Eurocrypt 2022 designed to be used in a transciphering framework for approximate fully homomorphic encryption. We show that at least 25\(\%\) of the possible choices for \(q\) satisfy certain conditions that lead to a successful key recovery attack with complexity significantly lower than the claimed security level for five of the six ciphers in the Rubato family.

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Notes

  1. 1.

    Note that the subspace \(\{x\cdot p + x \in \mathbb Z_{p^2} \mid \forall x\in \mathbb F_p\}\) is invariant if \(S_1=S_2\). However, it is possible to break such invariant subspace via a proper choices of round constants (see e.g. [55, 56] for details).

  2. 2.

    For completeness, we present an equivalent version of Rubato in [39, Appendix G].

  3. 3.

    The code can be found at https://github.com/Simula-UiB/RubatoAttack.

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Acknowledgments

Lorenzo Grassi is supported by the German Research Foundation (DFG) within the framework of the Excellence Strategy of the Federal Government and the States - EXC 2092 CaSa - 39078197.

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Authors and Affiliations

  1. Ruhr University Bochum, Bochum, Germany

    Lorenzo Grassi

  2. Simula UiB, Bergen, Norway

    Irati Manterola Ayala, Martha Norberg Hovd, Morten Øygarden & Håvard Raddum

  3. Telecom Paris, Institut Polytechnique de Paris, Palaiseau, France

    Qingju Wang

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Correspondence to Lorenzo Grassi , Irati Manterola Ayala , Martha Norberg Hovd , Morten Øygarden , Håvard Raddum or Qingju Wang .

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  2. Brown University, Providence, RI, USA

    Anna Lysyanskaya

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Grassi, L., Manterola Ayala, I., Hovd, M.N., Øygarden, M., Raddum, H., Wang, Q. (2023). Cryptanalysis of Symmetric Primitives over Rings and a Key Recovery Attack on Rubato. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14083. Springer, Cham. https://doi.org/10.1007/978-3-031-38548-3_11

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