Geodesic
On the decomposition of a vectorial Boolean function into a composition of two functions
Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 31-32

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In the paper, we prove that if a vectorial Boolean function $F$ in $n$ variables, $\deg(F)=d>2$, is decomposable, then the function $ F '= A_2 \circ F \circ A_1 $, where $ A_1, A_2 $ are arbitrary affine $ (n, n) $-permutations, is also decomposable; and if $F(x)=G(H(x))$, $\max\{\deg(F),\deg(H)\}=d'$, function $H$ is invertible and $ \deg (H ^ {- 1}) \leq d'$, then the function $ F^{''} = F + A_0 $ is decomposable for any affine function $A_0$. The construction of a decomposable vectorial Boolean function of the third degree in an arbitrary number of variables is presented. A computational experiment showed that all vectorial Boolean functions of the third degree in three variables are decomposable.
Keywords: vectorial Boolean function, threshold implementation.
Mots-clés : decomposition 
@article{PDMA_2020_13_a7,
     author = {G. M. Pintus},
     title = {On the decomposition of a vectorial {Boolean} function into a composition of two functions},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {31--32},
     publisher = {mathdoc},
     number = {13},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2020_13_a7/}
}
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G. M. Pintus. On the decomposition of a vectorial Boolean function into a composition of two functions. Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 31-32. http://geodesic.mathdoc.fr/item/PDMA_2020_13_a7/