几类特殊形式的置换多项式

SEVERAL SPECIAL TYPES OF PERMUTATION POLYNOMIALS

有限域上的置换多项式在密码学,编码理论和序列设计等领域有着广泛应用,但目前已知的置换多项式的构造还很有限.文章分别给出有限域F_(2~n)上两类形如(x^(2^i)+ηx+δ)~s+x和两类形x^r+δx^s+δ~tx的置换多项式.

Permutation polynomials over finite fields have been an important subject of study for a long time and have wide applications in coding theory, cryptography and sequence designs. However, only some specific classes of permutation polynomials have been described in the literature so far. In this paper, based on the knowledge of finite fields, such as the properties of the trace function, we propose two classes of permutation polynomials having the form （x2i＋ηx＋δ）s＋x and two classes of permutation trinomials having the form xr＋δxs＋δtx over the finite field F2n. The permutation polynomials of the first form are studied along with the work of Yuan and Ding, and the method studying the permutation polynomials of the second form relies a sufficient condition for a trinomial having no root, which is established in this paper. There are rare known classes of permutation trinomials over F2n in the literature, and most of them are of the simple form, i.e., all nonzero coefficients equal to the identity. The permutation trinomials presented in this paper have the coefficients which can take any nonzero elements in F2n.