摘要
研究了特征为2的有限域上一类正形置换多项式的非存在性.利用乘积多项式中次数的分布规律和整数的m进制表示的有关技巧,证明了在有限域F2n上不存在次数为2d-1的正形置换多项式的充分条件是:n(modd)≡0,1,或者当n(modd)≡r(1<r<d,1<d<log2n)时,这个多项式的2r-1次项的系数为0.进一步,给出了在有限域F2n上次数为2d的多项式是正形置换多项式的必要条件是:当n(modd)≡0,1时,这个多项式的2d-1次项的系数必为0;或者当n(modd)≡r(1<r<d,1<d<log2n),且这个多项式的2r-1次项的系数不为0时,它的2d-1次项的系数必为0.利用这个结果给出了有限域F2n上所有4次正形置换多项式的一个计数.
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 2^d - 1 is either n （rood d ）≡0,1, or the coefficient of the term with degree 2r - 1 of this polynomial is zero whenever n （mod d ）≡r （ 1 〈 r 〈 d ）, where 1 〈 d 〈log2n. 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 （modd ） ≡ r （ 1 〈 r 〈 d, 1 〈 d 〈 log2 n ）, and the coefficient of the term with degree 2^r - 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 F2^n is given.
出处
《陕西师范大学学报（自然科学版）》
CAS
CSCD
北大核心
2008年第2期5-10,共6页
Journal of Shaanxi Normal University：Natural Science Edition
基金
国家自然科学基金资助项目(10571112)
陕西省自然科学基础研究资助项目(2007A06)
关键词
有限域
置换
正形置换多项式
特征
finite field
permutation
orthomorphic permutation polynomial
characteristic
