Abstract
In this paper we, for the first time, study the question under which circumstances decomposing a round function of a Substitution-Permutation Network is possible uniquely. More precisely, we provide necessary and sufficient criteria for the non-linear layer on when a decomposition is unique. Our results in particular imply that, when cryptographically strong S-boxes are used, the decomposition is indeed unique. We then apply our findings to the notion of alignment, pointing out that the previous definition allows for primitives that are both aligned and unaligned simultaneously.
As a second result, we present experimental data that shows that alignment might only have limited impact. For this, we compare aligned and unaligned versions of the cipher PRESENT.
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Notes
- 1.
- 2.
Thankfully, for the case of counting S-boxes those requirements lead to trivial bounds for the probability/absolute correlation of a differential/linear trail, no matter the decomposition.
- 3.
The arguments are mainly based on the rational canonical form of the linear layer and this form might change when using linear equivalent S-boxes and modifying the linear layer accordingly.
- 4.
In other words, the direct sum enables us to express every element \(x \in W\) as \(\sum _i x_i\) for unique \(x_i \in U_i\). Hence, \(\pi _j^U\) is the map** defined by \(x = \sum _i x_i \mapsto x_j\).
- 5.
More precisely, we allow the subspaces to be equal to \(\mathbb {F}_2^{n}\) but not to be \(\{0\}\).
- 6.
This version is equivalent to the one from [8] since \(T(U_i) = U_{\tau (i)}\) means that T is the composition of a \(\Pi _N\)-Shuffle and a \(\Pi _N\) aligned linear function. As the composition of a \(\Pi _{N'}\) aligned and a \(\Pi _N\) aligned function is obviously \(\Pi _{N'}\) aligned if \(\Pi _N \le \Pi _{N'}\), it is then enough to check that M is \(\Pi _{N'}\) aligned, with \(\Pi _{N'}\) being non-trivial. Since it is only important that \(\Pi _{N'}\) is non-trivial, this can be done by checking if \(M\left( \bigoplus _{i \in J} U_i\right) = \bigoplus _{i \in J} U_i\) and \(M\left( \bigoplus _{i \notin J} U_i\right) = \bigoplus _{i \notin J} U_i\) for some \(J \subset \{1,\dots ,m\}\), i. e. checking for all possible \(\Pi _{N'}\) with two boxes.
- 7.
The code we used to make these experiments is available at: https://doi.org/10.5281/zenodo.7660387.
- 8.
Note that in the case of trivial intersections, we have that \(\pi _i^U \circ \pi _j^W \circ \pi _{l \ne i}^U = \pi _j^W \circ \pi _i^U \circ \pi _{l \ne i}^U = 0\), which means that \(F \circ \pi _i^U \circ \pi _j^W \circ \pi _{l \ne i}^U + F(0) = 0\).
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Acknowledgment
This work was funded by the by the project Analysis and Protection of Lightweight Cryptographic Algorithms (432878529) and by DFG (German Research Foudation), under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972.
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Lambin, B., Leander, G., Neumann, P. (2023). Pitfalls and Shortcomings for Decompositions and Alignment. In: Hazay, C., Stam, M. (eds) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol 14007. Springer, Cham. https://doi.org/10.1007/978-3-031-30634-1_11
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