Homogeneous Bent Functions, Invariants, and Designs

We establish a new connection between invariant theory and the theory of bent functions. This enables us to construct Boolean functions with a prescribed symmetry group action. Besides the quadratic bent functions the only other previously known homogeneous bent functions are six variable degree three functions. We show that these bent functions arise as invariants under an action of the symmetric group on four letters and determine the stabilizer which turns out to be the simple group of order 168. We apply the machinery of invariant theory in order to construct homogeneous bent functions of degree three in 8, 10, and 12 variables. This approach gives a great computational advantage over the unstructured search problem and yields Boolean functions which have a concise description in terms of certain designs and graphs. We consider the question of linear equivalence of the constructed bent functions and study the properties of the associated elementary abelian difference sets.