  In this paper we define a new (output) multiplicative differential, and the
corresponding $c$-differential uniformity. With this new concept, even for
characteristic $2$, there are perfect $c$-nonlinear (PcN) functions. We first
characterize the $c$-differential uniformity of a function in terms of its
Walsh transform. We further look at some of the known perfect nonlinear (PN)
and show that only one remains a PcN function, under a different condition on
the parameters. In fact, the $p$-ary Gold PN function increases its
$c$-differential uniformity significantly, under some conditions on the
parameters. We then precisely characterize the $c$-differential uniformity of
the inverse function (in any dimension and characteristic), relevant for the
Rijndael (and Advanced Encryption Standard) block cipher.
