Abstract
The study of symmetric structures based on quasigroups is relatively new and certain gaps can be found in the literature. In this paper, we want to fill one of these gaps. More precisely, in this work we study substitution permutation networks based on quasigroups that make use of permutation layers that are non-linear relative to the quasigroup operation. We prove that for quasigroups isotopic with a group \(\mathbb {G}\), the complexity of mounting a differential attack against this type of substitution permutation network is the same as attacking another symmetric structure based on \(\mathbb {G}\). The resulting structure is interesting and new, and we hope that it will form the basis for future secure block ciphers.
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Notes
- 1.
Comprised of several substitution boxes (s-boxes) with small block length.
- 2.
The trapdoor consists in knowing the group operation that weakens the structure.
- 3.
From the point of view of differential attacks.
- 4.
Restricted to quasigroups isotopic to commutative groups.
- 5.
- 6.
Left quasigroup operation: \(\tilde{k}_i^1 \otimes \tilde{p}_i^1 \Vert \ldots \Vert \tilde{k}_i^8 \otimes \tilde{p}_i^8\).
- 7.
This condition is implied by the fact that the permutation is not linear.
- 8.
This condition implies that the sets \(G = \mathbb {Z}_{2^{b'}}\) and \((\mathbb {Z}_{2^b})^{b'/b}\) are isomorphic.
- 9.
From a differential point of view.
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Teşeleanu, G. (2023). The Security of Quasigroups Based Substitution Permutation Networks. In: Bella, G., Doinea, M., Janicke, H. (eds) Innovative Security Solutions for Information Technology and Communications. SecITC 2022. Lecture Notes in Computer Science, vol 13809. Springer, Cham. https://doi.org/10.1007/978-3-031-32636-3_18
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