Two dimensional PN functions

Abstract

There is a differential operator /spl part/ mapping 1D functions /spl phi/:G/spl rarr/C to 2D functions /spl part//spl phi/: G/spl times/G/spl rarr/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/sup a/) as specific 2D PN coboundaries and show /spl part/ has an inverse for these PN functions. There are many more families of 2D PN cocycles on GF(p/sup 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.