Geodesic
Properties of a~bent function construction by a~subspace of an arbitrary dimension
Prikladnaya Diskretnaya Matematika. Supplement, no. 11 (2018), pp. 41-43.

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Let $f$ be a bent function in $2k$ variables, $L$ be an affine subspace of $\mathbb F_2^{2k}$, and $\mathrm{Ind}_L$ be a Boolean function with values $1$ on $L$. Here, we study the properties of the function $f\oplus\mathrm{Ind}_L$. Particularly, we give some necessary and sufficient conditions under which the increase or decrease of the dimension of $L$ by $1$ doesn't change the property bent of $f\oplus\mathrm{Ind}_L$. We prove that if the function $f(x_1,\dots,x_{2k})\oplus x_{2k+1}x_{2k+2}\oplus\mathrm{Ind}_U$ is a bent function and $U$ is an affine subspace, then the function $f\oplus\mathrm{Ind}_L$ is a bent function for some affine subspace $L$ of dimension $\operatorname{dim}U-1$ or $\operatorname{dim}U-2$. An example of bent function $f$ in $10$ variables for which $f\oplus\mathrm{Ind}_L$ is a bent function for only $\operatorname{dim}L\in\{9,10\}$ is provided.
Keywords: Boolean functions, bent functions, subspaces, affinity. 
@article{PDMA_2018_11_a11,
     author = {N. A. Kolomeec},
     title = {Properties of a~bent function construction by a~subspace of an arbitrary dimension},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {41--43},
     publisher = {mathdoc},
     number = {11},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2018_11_a11/}
}
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N. A. Kolomeec. Properties of a~bent function construction by a~subspace of an arbitrary dimension. Prikladnaya Diskretnaya Matematika. Supplement, no. 11 (2018), pp. 41-43. http://geodesic.mathdoc.fr/item/PDMA_2018_11_a11/

[1] Rothaus O., “On bent functions”, J. Combin. Theory. Ser. A, 20:3 (1976), 300–305 | DOI | MR | Zbl

[2] Logachev O. A., Salnikov A. A., Smyshlyaev S. V., Yaschenko V. V., Bulevy funktsii v teorii kodirovaniya i kriptologii, 2-e izd., MTsNMO, M., 2012, 584 pp.

[3] Tokareva N. N., Bent Functions, Results and Applications to Cryptography, Acad. Press Elsevier, 2015 | MR | Zbl

[4] Carlet C., “Two new classes of bent functions”, LNCS, 765, 1994, 77–101 | MR | Zbl

[5] Leander G., McGuire G., “Construction of bent functions from near-bent functions”, J. Combin. Theory. Ser. A, 116:4 (2009), 960–970 | DOI | MR | Zbl

[6] Canteaut A., Daum M., Dobbertin H., and Leander G., “Finding nonnormal bent functions”, Discr. Appl. Math., 154:2, 202–218 | DOI | MR | Zbl