Permutation and complete permutation polynomials

Polynomials of type xq+2+bxxq+2+bx over the field Fq2Fq2 and of type xq2+q+2+bxxq2+q+2+bx over Fq3Fq3, where q=pm>2q=pm>2 is a power of a prime p   are considered. All cases when these polynomials are permutation polynomials are classified. Therefore, all cases when the polynomials b−1xq+2b−1xq+2 over Fq2Fq2 and b−1xq2+q+2b−1xq2+q+2 over Fq3Fq3 are the complete permutation polynomials are enumerated.