Bivariate functions with low $c$-differential uniformity

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.

Low $c$-differential uniformity, perfect and almost perfect $c$-nonlinearity, the bivariate function