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A Geometric Approach to Linear Cryptanalysis

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Advances in Cryptology – ASIACRYPT 2021 (ASIACRYPT 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13090))

Abstract

A new interpretation of linear cryptanalysis is proposed. This ‘geometric approach’ unifies all common variants of linear cryptanalysis, reveals links between various properties, and suggests additional generalizations. For example, new insights into invariants corresponding to non-real eigenvalues of correlation matrices and a generalization of the link between zero-correlation and integral attacks are obtained. Geometric intuition leads to a fixed-key motivation for the piling-up principle, which is illustrated by explaining and generalizing previous results relating invariants and linear approximations. Rank-one approximations are proposed to analyze cell-oriented ciphers, and used to resolve an open problem posed by Beierle, Canteaut and Leander at FSE 2019. In particular, it is shown how such approximations can be analyzed automatically using Riemannian optimization.

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Notes

  1. 1.

    The present framework only describes zero-sum properties.

  2. 2.

    For the case of multidimensional zero-correlation approximations with ‘coupled masks’, apply Theorem 4.2 to the function \(x \mapsto (x, \mathsf {F}{}(x))\) to obtain their result.

  3. 3.

    If \(B_i\) is a matrix whose columns form a basis for \(V_i\), then the matrix-representation of \(\langle V_{i + 1}, V_i \rangle _{\mathsf {F}_i}\) with respect to these bases is \((B_{i + 1}^* B_{i + 1})^{-1} B_{i + 1}^* C^{\mathsf {F}_i} B_{i}(B_{i}^* B_{i})^{-1} \). Note the normalization factors for non-orthonormal bases.

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Acknowledgments

This work is based on my master’s thesis “Linear Cryptanalysis in the Weak-Key Model” [6]. I’m grateful to Vincent Rijmen for supervising this thesis, and for comments on a draft of this paper. I also thank Gregor Leander and Christof Beierle for interesting discussions about this work at Ruhr-University Bochum. The author is supported by a PhD Fellowship from the Research Foundation – Flanders (FWO).

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  1. Imec-COSIC, ESAT, KU Leuven, Leuven, Belgium

    Tim Beyne

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  1. NTT Corporation, Tokyo, Japan

    Mehdi Tibouchi

  2. Nanyang Technological University, Singapore, Singapore

    Huaxiong Wang

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Beyne, T. (2021). A Geometric Approach to Linear Cryptanalysis. In: Tibouchi, M., Wang, H. (eds) Advances in Cryptology – ASIACRYPT 2021. ASIACRYPT 2021. Lecture Notes in Computer Science(), vol 13090. Springer, Cham. https://doi.org/10.1007/978-3-030-92062-3_2

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