Complete Mapping Polynomials over Finite Field F16

A polynomial f(x) over F q , the finite field with qelements, is called a complete mapping polynomial if the two mappings F q i¾?F q respectively defined by f(x) and f(x) + xare one-to-one. In this correspondence, complete mapping polynomials over F 16 are considered. The nonexistence of the complete mapping polynomial of degree 9 and the existence of the ones of degree 8 and 11 are proved; the result that the reduced degree of complete mapping polynomials over F 16 are 1, 4, 8, 10, 11, 12, 13 is presented; and by searching with computer, the degree distribution of complete mapping polynomials over the field is given.