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

Let $F$ be a function from $\mathbb{F}_{p^n}$ 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 non-zero $a\in\mathbb{F}_{p^n}$. As a particular case, all known quadratic planar functions are zero-difference 1-balanced; and some quadratic APN functions over $\mathbb{F}_{2^n}$ 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 $F=G(x^d)$, where $\gcd(d,p^n-1)=δ+1$ and where the restriction of $G$ to the set of all non-zero $(δ+1)$-th powers in $\mathbb{F}_{p^n}$ is an injection. We introduce new families of zero-difference $p^t$-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 $\mathbb{F}_{2^8}$, we obtain $15$ new $(256, 85, 24, 30)$ negative Latin square type strongly regular graphs.