$C$-differentials, multiplicative uniformity and (almost) perfect $c$-nonlinearity
This work addresses cryptographic security by defining a new differential measure for functions, which is incremental but relevant for analyzing block ciphers like AES.
The paper introduces a new multiplicative differential concept called $c$-differential uniformity, which enables perfect $c$-nonlinear (PcN) functions even in characteristic 2, and characterizes this uniformity for functions like the inverse function used in AES, showing that only one known perfect nonlinear function remains PcN under specific conditions.
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.