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This paper mainly focuses on permutation polynomials over the residue class ring Z"N, where N3 is composite. We have proved that for the polynomial f(x)=a"1x+a"2x^2+...+a"kx^k with integral coefficients, f(x)modN permutes Z"N if and only if f(x)modN permutes S"@m for all @m|N, where S"@m={0N and S"N={0}. Based on it, we give a lower bound of the differential uniformities for such permutation polynomials, that is, @d(f)=N#S"a, where a is the biggest nontrivial divisor of N. Especially, f(x) cannot be APN permutations over the residue class ring Z"N. It is also proved that f(x)modN and (f(x)+x)modN cannot permute Z"N at the same time when N is even.