Additive subgroups and inverses in finite fields

The inversion map in a finite field plays a role in various cryptographic algorithms, one of the reasons being its 'high nonlinearity'.  The set of inverses of the nonzero elements of an additive subgroup of a finite field might be expected to be a 'highly nonlinear' subset, aside from trivial cases. In this talk I will provide concrete expressions to this expectation, in the form of certain bounds. Some interesting geometric configurations arise as exceptional cases.