Geodesic
$\mathrm{S}$-blocks with maximum component algebraic immunity on a small number of variables
Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 40-42

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Let $\pi$ be a permutation on $ n $ elements, $f$ be a Boolean function in $n$ variables. Define a vector Boolean function $F_\pi:\mathbb{F}_2^n\rightarrow\mathbb{F}_2^n$ as $F_\pi(x) = (f(x), f(\pi(x)), \cdots, f (\pi^{n-1}(x))))$. In this paper, we study the component algebraic immunity of the vector Boolean function $F_\pi$ as a function of the Boolean function $f$ and the permutation $\pi$ for $n = 3, 4, 5$. We obtain complete sets of Boolean and, partly, vector Boolean functions with maximum algebraic immunity in $3, 4$ and $5$ variables. If the function $F_\pi$ has maximum algebraic immunity, then the permutation $\pi$ is full cycle.
Keywords: Boolean function, vector Boolean function, algebraic immunity, component algebraic immunity. 
@article{PDMA_2021_14_a5,
     author = {D. A. Zyubina and N. N. Tokareva},
     title = {$\mathrm{S}$-blocks with maximum component algebraic immunity on a small number of variables},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {40--42},
     publisher = {mathdoc},
     number = {14},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2021_14_a5/}
}
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D. A. Zyubina; N. N. Tokareva. $\mathrm{S}$-blocks with maximum component algebraic immunity on a small number of variables. Prikladnaya Diskretnaya Matematika. Supplement, no. 14 (2021), pp. 40-42. http://geodesic.mathdoc.fr/item/PDMA_2021_14_a5/