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 \textit{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 $F(x)=x^d$ 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.

Cited by in corpus (6)

• Investigations on c-(almost) perfect nonlinear functions
• New P$c$N and AP$c$N functions over finite fields
• A class of APcN power functions over finite fields of even characteristic
• Several classes of PcN power functions over finite fields
• An extension of the avalanche criterion in the context of c-differentials
• On $-1$-differential uniformity of ternary APN power functions