On possibility to construct algebraic immune S-boxes by choosing coordinate Boolean functions

Vectorial Boolean functions, or S-boxes, are the main nonlinear components of symmetric ciphers, and their properties ensure the cipher’s resistance to various types of cryptanalysis. \protect\break S-box can be presented as a set of Boolean functions called coordinate functions. One good way of constructing S-boxes is to carefully choose these coordinate Boolean functions with necessary cryptographic properties. We continue the study of the set of Boolean functions in a small number of variables with optimal algebraic and correlation immunity orders. The possibility of using these functions as coordinate functions of S-box resistant to algebraic cryptanalysis has been verified programmatically. However, these Boolean functions cannot be used to construct a permutation on $\mathbb{Z}^4_2$ as well as S-box with optimal component algebraic immunity using only a single Boolean function and a permutation.