Correlation-immune functions with optimal algebraic immunity

Boolean functions are the main components of symmetric ciphers, and their properties ensure the cipher's resistance to various types of cryptanalysis. An important problem is to combine several cryptographic properties in one function, since the properties may contradict each other. Also, an interesting way to build Boolean functions is an iterative construction, i.e., constructing functions in a larger number of variables based on functions in a smaller number while preserving cryptographic properties. In this paper, we intersect sets of functions with maximal algebraic immunity and functions with the maximal order of correlation immunity equal to one, of a small number of variables. There are no correlation-immune Boolean functions in 3 variables with maximal algebraic immunity. There are 392 functions in 4 variables with the maximal order of correlation immunity 1 and maximal algebraic immunity, and for the case of 5 variables there are 96 768 such functions. For functions in 4 variables, a classification is obtained based on their Hamming weight and the type of their geometric representation. The construction of functions in 6 variables has been studied on the basis of functions in 4 variables, in which each vertex of the Boolean cube $\mathbb{E}^{4}$ is replaced by a face of dimension 2 containing elements of the support of the 6-variable function only if the original vertex belonged to the support. It has been programmatically verified that this construction preserves the indices of algebraic and correlation immunity.