S-Blocks of Special Type with Few Variables

Abstract

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.