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. 
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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/

[1] Leander G., Abdelraheem M. A., AlKhzaimi H., and Zenner E., “A cryptanalysis of PRINTcipher: The invariant subspace attack”, LNCS, 6841, 2011, 206–221 | MR | Zbl

[2] Trifonov D. I., Fomin D. B., “Ob invariantnykh podprostranstvakh v XSL-shifrakh”, Prikladnaya diskretnaya matematika, 2021, no. 54, 58–76 | Zbl

[3] Todo Y., Leander G., and Sasaki Y., “Nonlinear invariant attack: practical attack on full SCREAM, iSCREAM, and Midori64”, LNCS, 10032, 2016, 3–33 | MR | Zbl

[4] Burov D. A., “O suschestvovanii nelineinykh invariantov spetsialnogo vida dlya raundovykh preobrazovanii XSL-algoritmov”, Diskretnaya matematika, 33:2 (2021), 31–45 | DOI | MR

[5] Nyberg K., “Differentially uniform mappings for cryptography”, LNCS, 765, 1994, 245–265 | MR

[6] Carlet S., “Open questions on nonlinearity and on APN functions”, LNCS, 9061, 2015, 83–107 | MR | Zbl

[7] Kolomeec N. and Bykov D., On the Image of an Affine Subspace under the Inverse Function within a Finite Field, 2022, arXiv: 2206.14980

[8] Kolomeets N. A., Bykov D. A., “Ob invariantnykh podprostranstvakh funktsii, affinno ekvivalentnykh obrascheniyu elementov konechnogo polya”, Prikladnaya diskretnaya matematika. Prilozhenie, 2022, no. 15, 5–8

[9] Charpin P., “Normal Boolean functions”, J. Complexity, 20:2–3 (2004), 245–265 | DOI | MR | Zbl

[10] Gorodilova A. A., “Kharakterizatsiya pochti sovershenno nelineinykh funktsii cherez podfunktsii”, Diskretnaya matematika, 27:3 (2015), 3–16 | DOI

[11] Canteaut A., Carlet S., Charpin P., and Fontaine S., “On cryptographic properties of the cosets of R(1, m)”, IEEE Trans. Inform. Theory, 47 (2001), 1494–1513 | DOI | MR | Zbl

[12] Carlet S. and Feukoua S., “Three parameters of Boolean functions related to their constancy on affine spaces”, Adv. Math. Commun., 14:4 (2020), 651–676 | DOI | MR | Zbl