Two dimensional PN functions

There is a differential operator ∂ mapping 1D functions φ:G→C to 2D functions ∂φ: G×G→C which are coboundaries, the simplest form of cocycle. Perfect nonlinear (PN) 1D functions determine coboundaries with balanced partial derivatives. We use this property to define 2D PN and differentially k-uniform functions. We list the known PN permutations of GF(p^a) as specific 2D PN coboundaries and show ∂ has an inverse for these PN functions. There are many more families of 2D PN cocycles on GF(p^a) than those arising as coboundaries, even when p=2 (no 1D PN functions for p=2 exist; APN is the best possible). These ideas can be extended to include APN and differentially k-uniform 2D cocycles.