$\mathrm{S}$-blocks of a special type with a small number of variables

When constructing block ciphers, it is necessary to use vector Boolean functions with special cryptographic properties as $\mathrm{S}$-blocks for the cipher's resistance to various types of cryptanalysis. In this paper, we investigate the following $\mathrm{S}$-block construction: let $\pi$ be a permutation on $n$ elements, $\pi^i$ $i$-multiple application $\pi,$ and $f$ a Boolean function in $n$ variables. Define a vectorial Boolean function $F_{\pi}\colon\mathbb{Z}_2^n \to \mathbb{Z}_2^n$ as $F_{\pi}(x) = (f(x), f(\pi(x)), \ldots , f(\pi_{n-1}(x))).$ We study cryptographic properties of $F_{\pi}$ such as high nonlinearity, balancedness, and low differential $\delta$-uniformity in dependence on properties of $f$ and $\pi$ for small $n.$ Complete sets of Boolean functions $f$ and vector Boolean functions $F_{\pi}$ in a small number of variables with maximum algebraic immunity are also obtained. Bibliogr. 16.