Constructing permutation polynomials over finite fields
Keywords:
Permutation polynomial, linearized polynomial, linear translator, elementary symmetric polynomialAbstract
In this paper, we construct several new permutation polynomials over finite fields. First, using the linearized polynomials, we construct the permutation polynomial of the form \(\sum_{i=1}^k(L_{i}(x)+\gamma_i)h_i(B(x))\) over \({\bf F}_{q^{m}}\), where \(L_i(x)\) and \(B(x)\) are linearized polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalise a result of Marcos by constructing permutation polynomials of the forms \(x h(\lambda_{j}(x))\) and \(xh(\mu_{j}(x))\), where \(\lambda_{j}(x)\) is the \(j\)-th elementary symmetric polynomial of \(x, x^{q}, ..., x^{q^{m-1}}\) and \(\mu_{j}(x)={Tr}_{{\bf F}_{q^{m}}/{\bf F}_{q}}(x^{j})\). This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form \(L_1(x)+L_{2}(\gamma)h(f(x))\) over \({\bf F}_{q^{m}}\), which extends the result of Kyureghyan. DOI: 10.1017/S0004972713000646Published
2014-03-25
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