Quadratic zero-difference balanced functions, APN functions and strongly regular graphs

Abstract

Let $F$ be a function from $\mathbb {F}_{p^n}$ to itself and $\delta$ a positive integer. $F$ is called zero-difference $\delta$-balanced if the equation $F(x+a)-F(x)=0$ has exactly $\delta$ solutions for all nonzero $a\in \mathbb {F}_{p^n}$. 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 $\delta$-balanced functions are differentially $\delta$-uniform and we investigate in particular such functions with the form $F=G(x^d)$, where $\gcd (d,p^n-1)=\delta +1$ and where the restriction of $G$ to the set of all nonzero $(\delta +1)$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.

Appendix

The generating polynomial of the field $\mathbb {F}_{2^8}$ over $\mathbb {F}_2$ is $x^8+x^4+x^3+x^2+1$ (see Table 2).

Table 2 Quadratic APN functions on $\mathbb {F}_{2^8}$ with the form $G(x^3)$ and $G|_{C_3}$ is an injection