Quadratic Zero-Difference Balanced Functions, APN Functions and Strongly Regular Graphs

Let F be a function from Fpn to itself and δ a positive integer. F is called zero-difference δ-balanced if the equation F (x+a)−F (x) = 0 has exactly δ solutions for all nonzero a ∈ Fpn. As a particular case, all known quadratic planar functions are zero-difference 1-balanced; and some quadratic APN functions over F2n are zero-difference 2-balanced. In this paper, we study the relationship between this notion and differential uniformity; we show that all quadratic zero-difference δ-balanced functions are differentially δ-uniform and we investigate in particular such functions with the form F = G(xd), where gcd(d, pn − 1) = δ+ 1 and where the restriction of G to the set of all nonzero (δ + 1)-th powers in Fpn is an injection. We introduce new families of zero-difference pt-balanced functions. More interestingly, we show that the image set of such functions is a regular partial difference set, and hence yields strongly regular graphs; this generalizes the constructions of strongly regular graphs using planar functions by Weng et al. Using recently discovered quadratic APN functions on F28, we obtain 15 new (256, 85, 24, 30) negative Latin square type strongly regular graphs