Geodesic
On preserving the structure of a subspace by a vectorial Boolean function
Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 23-26

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We consider the following property of a function $F: \mathbb{F}_2^{n} \to \mathbb{F}_2^{m}$: $F$ preserves the structure of an affine subspace $U \subseteq \mathbb{F}_2^{n}$ if $F(U) = \{F(x) : x \in U\}$ is an affine subspace of $\mathbb{F}_2^{m}$. The connection between this property and the existence of component functions of $F$ whose restriction to the subspace is constant is established. Estimations for the nonlinearity and the order of differential uniformity of such $F$ are provided. We also prove that the set of dimensions of affine subspaces whose structure is preserved by the multiplicative inversion function is the smallest among all one-to-one monomial functions.
Keywords: affine subspaces, invariant subspaces, nonlinearity, differential uniformity, APN functions
Mots-clés : monomial functions. 
@article{PDMA_2023_16_a5,
     author = {N. A. Kolomeets},
     title = {On preserving the structure of a subspace by a vectorial {Boolean} function},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {23--26},
     publisher = {mathdoc},
     number = {16},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2023_16_a5/}
}
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N. A. Kolomeets. On preserving the structure of a subspace by a vectorial Boolean function. Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 23-26. http://geodesic.mathdoc.fr/item/PDMA_2023_16_a5/