Nonexistence of a special kind of orthomorphic permutation polynomials over finite fields with characteristic 2
Zhihui Li · Journal of Shaanxi Normal University · 2008
Nonexistence of a special kind of orthomorphic permutation polynomials over finite fields with characteristic 2 is studied.By the distributive law of degrees for multiplying polynomials and some technic of expression for base-m number,the sufficient conditions for nonexistence of orthomorphic permutation polynomials of degree 2d-1 is either n(mod d)≡0,1,or the coefficient of the term with degree 2r-1 of this polynomial is zero whenever n(mod d)≡r(1rd),where 1dlog2n.Furthermore,the necessary conditions that orthomorphic permutation polynomials of degree 2d exsit are: when n(mod d)≡0,1,the coefficient of the term with degree 2d-1 of this polynomial is zero;or when n(mod d)≡r(1rd,1dlog2n),and the coefficient of the term with degree 2r-1 of that is not equal to zero,the coefficient of term with degree 2d-1 of that is zero.By using these results,an enumeration of all orthomorphic permutation polynomials of degree 4 over the finite field F2n is given.