本原多项式的判别新算法

A New Algorithm for Determining Primitive Binary Polynomials

设0-1域上多项式f(x)=x^m+b_(m-1)x^(m-1)+…+b_1x+1,又设g(x)=x^n+a_(n-1)x^(n-1)+…+a_1x+1是0-1域上不可约多项式,并假定m≥n。基于整除关系式g(x)|f(x)看成由f(x)系数产生的向量经由g(x)系数产生的向量线性表出的基础上,设计了求解最小正整数m的算法,使得g(x)不仅有g(x)|x^m-1,而且还可判别g(x)是否是本原多项式。

Let f(x)=x^m+bm-1x^m-1 +…+b1x+1 be a binary polynomial.Also let g(x)=x^n+an-1x^n-1 +…+a1x+1 be an irreducible binary polynomial,and suppose that m≥n.We view the relation g(x)|f(x) is that the vector produced by the coefficients of polynomial f(x) is a linear representation of those vectors produced by the coefficients of polynomial g(x),and further develop an algorithm so as to find the smallest positive integer m such that g(x) has not only g(x) x^m-1,but also is determined whether or not the primitive binary polynomial.