Construction of balanced functions with high nonlinearity and other cryptographic properties

We present an iterative construction that can be used to construct balanced functions with high nonlinearity. Using this construction, we obtained Boolean functions in an even number $n\geqslant 18$ of variables which have no linear structures with nonlinearity $2^{n-1}-(2^{{n}/{2}-1}+2^{{n}/{2}-3}+2^{{n}/{2}-5}+2^{{n}/{2}-7})$. Additional conditions are given under which the functions obtained using the construction will be correlation immune. We also present results concerning “bent sum decomposition problem”.