Cohen proved that except for finite q as exceptional values,there are some primi- tive elements (roots) ξ of GF(q) such that aξ+b can be used to represent a nonzero cubic power residue.

This paper deals with the relation between dth power residues and primitive roots for the residue class ring Z_p~α, and proved that polynomals ax~d+b can be used to represent some primitive roots, provided that p is sufficiently large while d is relatively small, where a and b are units, and d is a divisor of p—1 .

In this new system,we use still power residue function to cryptograph,but periodicity of the cryptogram function has been removed.

If 〈x〉 is pronormal in NG(P) for every element x of P∩Gp-N with order p or 4(when p=2),where p is any prime divisor of |H|,P is a Sylow p-subgroup of H and Gp-N is the p-nilpotent residual of G,then G is supersolvable.

If every element of P∩G~(p-N) with order p or 4(when p=2) lies in Z(N_G(P)),where P∈Syl_p(N) and G~(p-N) is the p-nilpotent residual of G,then G is p-nilpotent.