Two new classes of permutation polynomials over finite fields are presented: (i) f (x) = (1? x? x~2)x ~(3e+1)/2 ?1? x+ x~2|over where e is a positive even integer; (ii) gn,p(x) =∑ ~n/p ≤l≤ ~n/p?1 n/l (n?l l/(p?1) ×x~(n?l(p?1)) over Fpe where e is a positive integer such that if p=2, and n=(p-1)p~m+p~(0e)+p~(1e)+...+p~((p-1)e), (m-1,e)=1. The permutation polynomial in (i) answers an open question about reversed Dickson polynomials.
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