A method for obtaining lower bounds on the higher order nonlinearity of Boolean function

Obtainment of exact value or high lower bound on the r-th order nonlinearity of Boolean function is a very complicated problem (especial if r > 1). In a number of papers lower bounds on the r-th order nonlinearity of Boolean function via its algebraic immunity were obtain for different r. This bounds is rather high for function with maximum near maximum possible algebraic immunity. In this paper we prove theorem, which try to obtain rather high lower bound on the r-th order nonlinearity for many functions with small algebraic immunity.