Designing bent functions using the Maiorana-McFarland secondary construction

Given a bent function of dimension $$s-r$$ , we apply the Maiorana-McFarland secondary construction to produce a new bent function of dimension $$s+r$$ . Throughout this process, we consider affine spaces, since bent functions can also be constructed on these spaces. The bent functions defined on affine spaces ultimately serve as building blocks for generating bent functions of higher dimension. Furthermore, by applying the case $$r=1$$ iteratively, we obtain the algebraic expression of a bent function. We then demonstrate that the two bent functions that we have obtained are distinct. In almost all constructions, the bent functions we obtain are balanced when the domain is restricted to vectors with an even Hamming weight. Finally, to illustrate these concepts, we present an example algorithm focusing on the case where $$r=2$$ , as well as another algorithm that employs $$r=1$$ twice.