Geodesic
On derivatives of Boolean bent functions
Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 57-58 Cet article a éte moissonné depuis la source Math-Net.Ru

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Bent function can be defined as a Boolean function $f(x)$ in $n$ variables ($n$ is even) such that for any nonzero vector $y$ its derivative $D_yf(x)=f(x)\oplus f(x\oplus y)$ is balanced, that is, it takes values $0$ and $1$ equally often. Whether every balanced function is a derivative of some bent function or not is an open problem. In this paper, special case of this problem is studied. It is proven that every non-constant affine function in $n$ variables, $n\geqslant4$, $n$ is even, is a derivative of $(2^{n-1}-1)|\mathcal{B}_{n-2}|^2$ bent functions, where $|\mathcal{B}_{n-2}|$ is the number of bent functions in $n-2$ variables. New iterative lower bounds for the number of bent functions are presented.
Keywords: Boolean functions, bent functions, derivatives of bent function, lower bounds for the number of bent functions. 
@article{PDMA_2021_14_a11,
     author = {A. S. Shaporenko},
     title = {On derivatives of {Boolean} bent functions},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {57--58},
     year = {2021},
     number = {14},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2021_14_a11/}
}
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A. S. Shaporenko. On derivatives of Boolean bent functions. Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 57-58. http://geodesic.mathdoc.fr/item/PDMA_2021_14_a11/

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