Geodesic
Construction of a substitution on~$\mathbb{F}_2^n$ based on a single Boolean function
Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 29-31

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The following construction of a vector Boolean function is considered: $F(x)=\big(f(x),f(\pi(x)),f(\pi^2(x)),\ldots, f(\pi^{n-1}(x))\big)$, where $\pi\in\mathbb{S}_n$, $f$ is a $n$-dimensional Boolean function. Some necessary conditions for $F$ to be a bijection are proved, namely: $f$ must be balanced, $f(0^n)\neq f(1^n)$, $\pi$ must be full cycle substitution, $f\neq\mathrm{const}$ on any cycle of substitution $\pi'$, where $\pi'(a_1\ldots a_n)=(a_{\pi(1)}\ldots a_{\pi(n)})$ for all $a_1\ldots a_n\in\mathbb{F}_2^n$.
Mots-clés : bijection
Keywords: vector Boolean function. 
@article{PDMA_2023_16_a7,
     author = {I. A. Pankratova and A. A. Medvedev},
     title = {Construction of a substitution on~$\mathbb{F}_2^n$ based on a single {Boolean} function},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {29--31},
     publisher = {mathdoc},
     number = {16},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2023_16_a7/}
}
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I. A. Pankratova; A. A. Medvedev. Construction of a substitution on~$\mathbb{F}_2^n$ based on a single Boolean function. Prikladnaya Diskretnaya Matematika. Supplement, no. 16 (2023), pp. 29-31. http://geodesic.mathdoc.fr/item/PDMA_2023_16_a7/