Geodesic
On the construction of invertible vector Boolean functions
Prikladnaya Diskretnaya Matematika. Supplement, no. 17 (2024), pp. 40-44

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The following construction of a vector Boolean function is considered: $G(x)=\big(f(x),f(\pi(x)),f(\pi^2(x)),\ldots,f(\pi^{n-1}(x))\big)$, where $\pi(i)=i\bmod n+1$, $f$ is a $n$-dimensional Boolean function. An algorithm for constructing such a function with the invertibility property has been proposed; its completeness and correctness have been proven; the number $t(n)$ of invertible functions of type $G$ has been calculated: $t(n)=\prod\limits_{d|n}p(d)! d^{p(d)}$, where $p(d)$ is the number of binary Lyndon words of length $d$. If $\pi$ is an arbitrary full-cycle substitution (not necessarily a cyclic shift), then the number of invertible functions of type $G$ is $(n-1)!$ times greater.
Keywords: vector Boolean functions, invertible functions, cyclically equivalent vectors, Lyndon words. 
@article{PDMA_2024_17_a9,
     author = {I. A. Pankratova and P. R. Garchukova},
     title = {On the construction of invertible vector {Boolean} functions},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {40--44},
     publisher = {mathdoc},
     number = {17},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2024_17_a9/}
}
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I. A. Pankratova; P. R. Garchukova. On the construction of invertible vector Boolean functions. Prikladnaya Diskretnaya Matematika. Supplement, no. 17 (2024), pp. 40-44. http://geodesic.mathdoc.fr/item/PDMA_2024_17_a9/