C-differentials, multiplicative uniformity and (almost) perfect c-nonlinearity

09/09/2019

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by   Pal Ellingsen, et al.

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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.