Cryptographic functions with low differential uni-formity have wide applications in cryptography and cyber security. In this paper, we further investigate c-differential uniformity proposed by Ellingsen et al. (IEEE Transactions on Information Theory. 2020, 66(9): 5781–5789). Specifically, two classes of power functions with low (-1)-differential uniformity over finite fields with odd characteristic are obtained. In addition, a class of permutation polynomials and $\mathrm{P}_{-1}\mathrm{N}$ or $\text{AP}_{-1}\mathrm{N}$ functions over $\mathbb{F}_{3^{n}}$ are proposed. These functions with low $c$-differential uniformity are utilized to resist against $c$-differential attacks in cyberspace from those with the usual low differential uniformity.