CONSTRUCTING PERMUTATION POLYNOMIALS OVER FINITE FIELDS

摘要

In this paper, we construct several new permutation polynomials over finite fields. First, using the linearised polynomials, we construct the permutation polynomial of the form ∑_(i=1)~k(L_i(x) + γ_i)h_iB(x)) over F_q~m, where L_i(x) and B(x) are linearised polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalise a result of Marcos by constructing permutation polynomials of the forms xh(λ_j(x)) and xh(μ_j(x)), where λ_j(x)is the jth elementary symmetric polynomial of x,x~q,..., x_q~(m-1) and μ_j(x) = Tr_(F_q~m/F_q)(x~j).This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator,we construct the permutation polynomial of the form L_1(x)+ L_2(γ)h(f(x)) over F_q~m,which extends a result of Kyureghyan.