About the components of some classes of invertible vectorial Boolean functions

I. A. Pankratova

Prikladnaya Diskretnaya Matematika. Supplement, no. 12 (2019), pp. 66-68.

Résumé

In the class of invertible vectorial Boolean functions in $n$ variables with coordinate functions depending on all variables, we consider the subclasses $\mathcal{K}_{n}$ and $\mathcal{K}'_{n}$, the functions in which are obtained using $n$ independent transpositions, respectively, from the identity permutation and from the permutation, each coordinate function of which essentially depends on some one variable. It is shown that, for any $F=(f_1\ldots f_n)\in\mathcal{K}_{n}\cup\mathcal{K}'_{n}$ and $i=1,\ldots,n$, the coordinate function $f_i$ has a single linear variable, the component function $vF$ has no nonessential and linear variables for each vector $v\in{\mathbb F}_2^n$ weight of which is greater than $1$, the nonlinearity, the degree, and the component algebraic immunity are $2$, $n-1$, and $2$ respectively.

Détail

Keywords: vectorial Boolean functions, invertible functions, nonlinearity, component algebraic immunity.

Bibliographie

[1] Pankratova I. A., “Construction of invertible vectorial Boolean functions with coordinates depending on given number of variables”, Materialy Mezhdunar. nauch. kongressa po informatike: Informatsionnye sistemy i tekhnologii (Respublika Belarus, Minsk, 24–27 okt. 2016), BGU, Minsk, 2016, 519–521

[2] Karpova L. A., Pankratova I. A., “Svoistva koordinatnykh funktsii odnogo klassa podstanovok na $\mathbb{F}_2^n$”, Prikladnaya diskretnaya matematika. Prilozhenie, 2017, no. 10, 38–40

[3] Carlet C., Vectorial Boolean Functions for Cryptography, Cambridge University Press, Cambridge, 2010, 93 pp. | Zbl

[4] Pankratova I. A., “Svoistva komponent nekotorykh klassov vektornykh bulevykh funktsii”, Prikladnaya diskretnaya matematika, 2019, no. 44, 5–11