On Homogeneous Bent Functions

A new surprising connection between invariant theory and the theory of bent functions is established. This enables us to construct Boolean function having a prescribed symmetry given by a group action. Besides the quadratic bent functions the only other 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. Extending to more variables we apply the machinery of invariant theory to construct previously unknown homogeneous bent functions of degree three in 8 and 10 variables. This approach gives a great computational advantage over the unstructured search problem. We finally consider the question of linear equivalence of the constructed bent functions.