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

TL;DR

This paper explores zero-difference balanced functions, their relation to differential uniformity, and constructs new strongly regular graphs from these functions, including new graphs derived from quadratic APN functions.

Contribution

It introduces new families of zero-difference p^t-balanced functions and links them to strongly regular graphs, extending previous constructions using planar functions.

Findings

All quadratic zero-difference δ-balanced functions are differentially δ-uniform.

Constructed new zero-difference p^t-balanced functions with image sets forming regular partial difference sets.

Generated 15 new strongly regular graphs from quadratic APN functions on ^8.

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 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 $\delta$-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 non-zero $(\delta +1)$-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.