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.