Constructions of several special classes of cubic bent functions outside the completed Maiorana-McFarland class

We show that the direct sum, under more relaxed conditions compared to those of Polujan and Pott (2020), can generate bent functions provably outside the completed Maiorana-McFarland class (${\mathrm{MM}}^{\#}$). We also show that the indirect sum method of generating bent functions, by imposing certain conditions (which are completely absent if only the bentness of the resulting function is required) on the initial bent functions, can be employed in the design of bent functions outside ${\mathrm{MM}}^{\#}$. Furthermore, applying this method to suitably chosen bent functions we construct several generic classes of homogeneous cubic bent functions (considered as a difficult problem) that might possess additional properties (namely without affine derivatives and/or outside ${\mathrm{MM}}^{\#}$). Our results significantly improve upon the best known instances of this type of bent functions given by Polujan and Pott (2020), and additionally we provide a solution to an open problem presented in their paper.