DICKSON POLYNOMIALS OF THE SECOND KIND THAT PERMUTE Z(m)

In this paper, we investigate the permutation property of the Dickson polynomials En(x, a) of the second kind over Z(m). Due to a known result, it suffices to consider permutation polynomials E-n(x, a) over Z(pt), where p is a prime and t is a positive integer. We identify all permutation polynomials of E-n(x, a) over Z(pt) for (I) p = 2 and (II) p is odd and a is a square over Z(p). For odd p and nonsquares a in Z(p), we determine a large class (if not all) of permutation polynomials E-n(x, a) over Z(pt). A conjecture is also presented in this paper. If this conjecture is true, then all Dickson permutation polynomials E-n(x, a) of the second kind over Z(m) are determined