Power Functions over Finite Fields with Low c-Differential Uniformity
Abstract
Very recently, a new concept called multiplicative differential (and the corresponding c-differential uniformity) was introduced by Ellingsen et al in [C-differentials, multiplicative uniformity and (almost) perfect c-nonlinearity, IEEE Trans. Inform. Theory, 2020] which is motivated from practical differential cryptanalysis. Unlike classical perfect nonlinear functions, there are perfect c-nonlinear functions even for characteristic two. The objective of this paper is to study power function F(x)=xd over finite fields with low c-differential uniformity. Some power functions are shown to be perfect c-nonlinear or almost perfect c-nonlinear. Notably, we completely determine the c-differential uniformity of almost perfect nonlinear functions with the well-known Gold exponent. We also give an affirmative solution to a recent conjecture proposed by Bartoli and Timpanella in 2019 related to an exceptional quasi-planar power function.