  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.
