  Recently, a new concept called multiplicative differential cryptanalysis and
the corresponding $c$-differential uniformity were introduced by Ellingsen et
al.~\cite{Ellingsen2020}, and then some low differential uniformity functions
were constructed. In this paper, we further study the constructions of perfect
$c$-nonlinear (PcN) power functions. First, we give a necessary and sufficient
condition for the Gold function to be PcN and a conjecture on all power
functions to be PcN over $\gf(2^m)$. Second, several classes of PcN power
functions are obtained over finite fields of odd characteristic for $c=-1$ and
our theorems generalize some results in~\cite{Bartoli,Hasan,Zha2020}. Finally,
the $c$-differential spectrum of a class of almost perfect $c$-nonlinear (APcN)
power functions is determined.
