C -Differentials, Multiplicative Uniformity, and (Almost) Perfect c -Nonlinearity

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) functions 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.