Complete mapping polynomials over finite field F16

Abstract

A polynomial f(x) over Fq, the finite field with q elements, is called a complete mapping polynomial if the two mappings Fq → Fq respectively defined by f(x) and f(x) + x are one-to-one. In this correspondence, complete mapping polynomials over F16 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 F16 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. © Springer-Verlag Berlin Heidelberg 2007.