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 zerodifference 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 zerodifference 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(x), where gcd(d, p − 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 p-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.