S-Blocks of Special Type with Few Variables
D. A. Zyubina, Natalia Tokareva · Journal of Applied and Industrial Mathematics · 2023
When constructing block ciphers, it is necessary to use vector Boolean functions with special cryptographic properties as S-blocks for the cipher’s resistance to various types of cryptanalysis. In this paper, we investigate the following S-block construction: let $$ \pi $$ be a permutation on $$ n $$ elements, let $$ \pi ^i $$ be the $$ i $$ -fold application of the permutation $$ \pi $$ , and let $$ f $$ be a Boolean function of $$ n $$ variables. Define a vector 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 the cryptographic properties of $$ F_{\pi } $$ such as high nonlinearity, balancedness, and low differential $$ \delta $$ -uniformity in the dependence on the properties of $$ f $$ and $$ \pi $$ for small $$ n $$ . Complete sets of Boolean functions $$ f $$ and vector Boolean functions $$ F_{\pi } $$ of few variables with maximum algebraic immunity are also obtained.