Abstract
Designing a block cipher or cryptographic permutation can be approached in many different ways. One such approach, popularized by AES, consists in grouping the bits along the S-box boundaries, e.g., in bytes, and in consistently processing them in these groups. This aligned approach leads to hierarchical structures like superboxes that make it possible to reason about the differential and linear propagation properties using combinatorial arguments. In contrast, an unaligned approach avoids any such grouping in the design of transformations. However, without hierarchical structure, sophisticated computer programs are required to investigate the differential and linear propagation properties of the primitive. In this paper, we formalize this notion of alignment and study four primitives that are exponents of different design strategies. We propose a way to analyze the interactions between the linear and the nonlinear layers w.r.t. the differential and linear propagation, and we use it to systematically compare the four primitives using non-trivial computer experiments. We show that alignment naturally leads to different forms of clustering, e.g., of active bits in boxes, of two-round trails in activity patterns, and of trails in differentials and linear approximations.
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Acknowledgements
We thank Bart Mennink for helpful comments. Moreover, we would like to thank the anonymous reviewers of an earlier version of this paper for their useful feedback. Joan Daemen and Daniël Kuijsters are supported by the European Research Council under the ERC advanced grant agreement under grant ERC-2017-ADG Nr. 788980 ESCADA. This work is partially supported by the French National Research Agency in the framework of the Investissements d’avenir programme (ANR-15-IDEX-02).
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Bordes, N., Daemen, J., Kuijsters, D., Van Assche, G. (2021). Thinking Outside the Superbox. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12827. Springer, Cham. https://doi.org/10.1007/978-3-030-84252-9_12
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