The cycle structure of a class of permutation polynomials

In this paper, we study the cycle structure of a permutation polynomial of the form f ( x ) = x ( q + 1 ) s 1 ( x + x q ) s 2 + x s 3 over F q 2 , where ( s 1 , s 2 , s 3 ) ∈ { ( q 2 - 2 , 3 , q ) , ( 1 , q 2 - 2 , 1 ) , ( 1 , q 2 - 1 , 2 ) , ( 1 , 2 , 4 ) } and q is even. By calculating the sum of all elements in each cycle of a fraction polynomial x x 3 + x 2 + 1 or a linearized polynomial x 2 e + x 2 + x with e ≥ 0 , the cycle structure of f ( x ) over F q 2 in the first three cases, that is, ( s 1 , s 2 , s 3 ) = { ( q 2 - 2 , 3 , q ) , ( 1 , q 2 - 2 , 1 ) or ( 1 , q 2 - 1 , 2 ) } , is characterized. For the case ( s 1 , s 2 , s 3 ) = ( 1 , 2 , 4 ) , we give the cycle structure of f ( x ) over F q 2 for q = 2 2 k with a positive integer k . For q = 2 2 p k with an odd prime p , it needs more techniques to determine the cycle structure of f ( x ). We only give its cycle length.