Differentially 2-Uniform Cocycles — The Binary Case

Abstract

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. Differentially k-uniform 1D functions determine coboundaries with the same distribution. Extending the idea of differential uniformity to cocycles gives a unified perspective from which to approach existence and construction problems for highly nonlinear functions, sought for their resistance to differential cryptanalysis. We describe two constructions of 2D differentially 2-uniform (APN) cocycles over GF(2^a), of which one gives 1D binary APN functions.