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Internal Symmetries and Linear Properties: Full-permutation Distinguishers and Improved Collisions on Gimli

Abstract

\(\mathsf {Gimli}\) is a family of cryptographic primitives (both a hash function and an AEAD scheme) that has been selected for the second round of the NIST competition for standardizing new lightweight designs. The candidate \(\mathsf {Gimli}\) is based on the permutation \(\mathsf {Gimli}\), which was presented at CHES 2017. In this paper, we study the security of both the permutation and the constructions that are based on it. We exploit the slow diffusion in \(\mathsf {Gimli}\) and its internal symmetries to build, for the first time, a distinguisher on the full permutation of complexity \(2^{64}\). We also provide a practical distinguisher on 23 out of the full 24 rounds of \(\mathsf {Gimli}\) that has been implemented. Next, we give (full state) collision and semi-free start collision attacks on \(\mathsf {Gimli}\)-Hash, reaching, respectively, up to 12 and 18 rounds. On the practical side, we compute a collision on 8-round \(\mathsf {Gimli}\)-Hash. In the quantum setting, these attacks reach 2 more rounds. Finally, we perform the first study of linear trails in \(\mathsf {Gimli}\), and we find a linear distinguisher on the full permutation.

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Notes

  1. This behavior appears because the linear layer of \(\mathsf {Gimli}\) is round dependent.

  2. https://project.inria.fr/quasymodo/files/2020/05/gimli_cryptanalysis_eprint.tar.gz.

  3. Note that the formulas given page 15 of the specification of \(\mathsf {Gimli}\) are erroneous. In the line \(z_n' \leftarrow z_n + y_n + (x_{n-3} \wedge z_{n-3})\), \(z_{n-3}\) should be replaced by \(y_{n-3}\) and \(x_j \wedge z_j\) must be replaced by \(x_j \wedge y_j\) in the subsequent formulas.

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Acknowledgements

The authors would like to thank all the members of the cryptanalysis party meetings, for many useful comments and discussions, in particular many thanks to Anne Canteaut, Virginie Lallemand and Thomas Fuhr for many interesting discussions over previous versions of this work. Thanks to Donghoon Chang for finding some mistakes and inaccuracies, including an error in a 32-round version of our distinguisher. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 714294 - acronym QUASYModo).

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Correspondence to André Schrottenloher.

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Communicated by Tetsu Iwata

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This article is an extended version of the paper “New Results on Gimli: Full-Permutation Distinguishers and Improved Collisions” which appeared in the proceedings of ASIACRYPT 2020 [18].

This work was carried out while André Schrottenloher was at Inria.

Appendix

Appendix

SP-Box Inverse

The SP-Box is a bijective operation, but its inverse is difficult to write (and it is never used).

  1. 1.

    Swap x and z

  2. 2.

    Perform:Footnote 3

    $$\begin{aligned} x_0&\leftarrow x_0' \\ y_0&\leftarrow y_0' + x_0' \\ z_0&\leftarrow z_0' + x_0' + y_0' \\ x_1&\leftarrow x_1' + z_0 \\ y_1&\leftarrow y_1' + x_1' + z_0 + (x_0 \vee z_0) \\ z_1&\leftarrow z_1' + y_1' + x_1' + z_0 + (x_0 \vee z_0) \\ x_2&\leftarrow x_2' + z_1 + (y_0 \wedge z_0) \\ y_2&\leftarrow y_2' + x_2' + z_1 + (y_0 \wedge z_0) + (x_1 \vee z_1) \\ z_2&\leftarrow z_2' + y_2' + x_2' + z_1 + (y_0 \wedge z_0) + (x_1 \vee z_1) \\ \forall 3 \le i \le 32, x_i&\leftarrow x_i' + z_{i-1} + (y_{i-2} \wedge z_{i-2}) \\ y_i&\leftarrow y_i' + x_i + (x_{i-1} \vee z_{i-1}) \\ z_i&\leftarrow z_i' + y_i + (x_{i-3} \wedge y_{i-3}) \end{aligned}$$
  3. 3.

    Rotate x and y: \(x_i = x_{i+24 \mod 32}\) and \(y_i = y_{i+9 \mod 32}\)

Gimli-Hash

figure e

Representation of Full Gimli

See Fig 10.

Fig. 10
figure 10

A representation of full Gimli

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Flórez-Gutiérrez, A., Leurent, G., Naya-Plasencia, M. et al. Internal Symmetries and Linear Properties: Full-permutation Distinguishers and Improved Collisions on Gimli. J Cryptol 34, 45 (2021). https://doi.org/10.1007/s00145-021-09413-z

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  • DOI: https://doi.org/10.1007/s00145-021-09413-z

Keywords

  • \(\mathsf {Gimli}\)
  • Symmetries
  • Symmetric cryptanalysis
  • Full-round distinguisher
  • Collision attacks
  • Linear approximations