On nonlinearity of Boolean functions generated by the generalized Dobbertin construction

We propose a generalization of Dobbertin's 1995 construction for balanced highly nonlinear Boolean functions. The Walsh–Hadamard spectrum of the proposed functions is studied. An exact upper bound for the spectral radius (lower bound for nonlinearity) is achieved. We also introduce a method for constructing a balanced function of $2n$ variables and spectral radius $2^n + 2^k R$ using a balanced function of $n-k$ variables and spectral radius $R$. Bibliogr. 20.