Bivariate functions with low $c$-differential uniformity

TL;DR

This paper investigates the $c$-differential uniformity of bivariate functions over finite fields, providing new constructions of functions with low $c$-differential uniformity, including many P$c$N and AP$c$N functions.

Contribution

It introduces novel methods to construct bivariate functions with low $c$-differential uniformity, expanding the class of functions with desirable cryptographic properties.

Findings

Constructed several classes of bivariate functions with low $c$-differential uniformity.

Produced many P$c$N and AP$c$N functions from the new constructions.

Provided theoretical analysis linking function choices to differential properties.

Abstract

Starting with the multiplication of elements in $\mathbb{F}_{q}^2$ which is consistent with that over $\mathbb{F}_{q^2}$, where $q$ is a prime power, via some identification of the two environments, we investigate the $c$-differential uniformity for bivariate functions $F(x,y)=(G(x,y),H(x,y))$. By carefully choosing the functions $G(x,y)$ and $H(x,y)$, we present several constructions of bivariate functions with low $c$-differential uniformity. Many P$c$N and AP$c$N functions can be produced from our constructions.