Permutation polynomials and their differential properties over residue class rings

This paper mainly focuses on permutation polynomials over the residue class ring ZNZN, where N>3N>3 is composite. We have proved that for the polynomial f(x)=a1x+a2x2+⋯+akxkf(x)=a1x+a2x2+⋯+akxk with integral coefficients, f(x)modNf(x)modN permutes ZNZN if and only if f(x)modNf(x)modN permutes SμSμ for all μ∣Nμ∣N, where Sμ={0