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Block Cipher Invariants as Eigenvectors of Correlation Matrices

Abstract

A new approach to invariant subspaces and nonlinear invariants is developed. This results in both theoretical insights and practical attacks on block ciphers. It is shown that, with minor modifications to some of the round constants, Midori-64 has a nonlinear invariant with \(2^{96} + 2^{64}\) corresponding weak keys. Furthermore, this invariant corresponds to a linear hull with maximal correlation. By combining the new invariant with integral cryptanalysis, a practical key-recovery attack on ten rounds of unmodified Midori-64 is obtained. The attack works for \(2^{96}\) weak keys and irrespective of the choice of round constants. The data complexity is \(1.25 \cdot 2^{21}\) chosen plaintexts, and the computational cost is dominated by \(2^{56}\) block cipher calls. The validity of the attack is verified by means of experiments.

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Notes

  1. 1.

    Such functions may be called defective probability mass functions [17].

  2. 2.

    It is not hard to see that it will be linearly independent from any previously computed eigenvectors.

  3. 3.

    A transformation such as \(C^P\) may be called a braiding map.

  4. 4.

    A Sage implementation is available online at https://homes.esat.kuleuven.be/~tbeyne/invariants/algorithm_1.html.

  5. 5.

    If the zero-sum property can be used, this actually yields a five-round property.

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Acknowledgements

I acknowledge the anonymous referees for their comments and corrections. In addition, I thank Tomer Ashur and Yunwen Liu for discussions related to this work. Finally, I am especially grateful to Vincent Rijmen for his comments on a draft version of this paper and for his support.

Author information

Correspondence to Tim Beyne.

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Publisher's Note

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This work was supported by the Research Council KU Leuven: C16/18/004. The author is supported by a PhD Fellowship from the Research Foundation Flanders (FWO). A preliminary version of this paper was published at ASIACRYPT 2018 [6]. The full version of this work is available on ePrint [7].

Communicated by Kaisa Nyberg.

Appendices

List of Invariants Produced by Algorithm 1

See Table 5.

Table 5 Invariants for two rounds of (modified) Midori-64, as obtained using Algorithm 1

Test Code for Nonlinear Invariant from Sect. 5.3

The following code was tested using Sage 8.1.

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Beyne, T. Block Cipher Invariants as Eigenvectors of Correlation Matrices. J Cryptol (2020). https://doi.org/10.1007/s00145-020-09344-1

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Keywords

  • Invariant subspace attack
  • Nonlinear invariant attack
  • Linear cryptanalysis
  • Integral cryptanalysis
  • Correlation matrices
  • Midori-64