Construction of highly non-linear cubic homogeneous Boolean functions on GF2n+l (2)

The work studies highly nonlinear Boolean functions in GF2n+1(2), i.e. for the dimensions where bent functions do not exist. We prove that for every n > 2 there exist homogeneous Boolean functions on GF(2)2n+1 with non-linearity greater than or equal to 22n — 2n and without linear structures.