Geodesic
On one-to-one property of a vectorial Boolean function of the special type
Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 40-41.

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$\mathrm{S}$-boxes are widely used in cryptography. In particular, they form important components of SP and Feistel networks. Mathematically, $\mathrm{S}$-box is a vectorial Boolean function $F:\mathbb{F}_{2}^{n} \to \mathbb{F}_{2}^{m}$ that should satisfy several cryptographic properties. Usually $n=m$. We study one-to-one property of a vectorial Boolean function constructed in a special way on the base of a Boolean function and a permutation on $n$ elements. The number of all one-to-one functions of this type is calculated.
Keywords: Boolean function, vectorial Boolean function, $\mathrm{S}$-box. 
@article{PDMA_2020_13_a11,
     author = {M. M. Zapolskiy and N. N. Tokareva},
     title = {On one-to-one property of a vectorial {Boolean} function of the special type},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {40--41},
     publisher = {mathdoc},
     number = {13},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2020_13_a11/}
}
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M. M. Zapolskiy; N. N. Tokareva. On one-to-one property of a vectorial Boolean function of the special type. Prikladnaya Diskretnaya Matematika. Supplement, no. 13 (2020), pp. 40-41. http://geodesic.mathdoc.fr/item/PDMA_2020_13_a11/