Improving the algebraic immunity of resilient and nonlinear functions and constructing bent functions

The currently known constructions of Boolean functions with high nonlinearities, high algebraic degrees and high resiliency orders do not seem to permit achieving sufficiently high algebraic immunities. We introduce a construction of Boolean functions, which builds a new function from three known ones. Assuming that the three functions have some resiliency order, nonlinearity and algebraic degree, as well as their sum modulo 2, the constructed function has the same resiliency order and can have the same nonlinearity, but has potentially better algebraic degree and algebraic immunity. The set of classical constructions together with this new one (and with a simpler derived one, having the same advantages) permit now to obtain functions achieving all necessary criteria for being used in the pseudo-random generators in stream ciphers. We also apply this construction to obtain bent functions from known ones.