有限域上的不可约多项式在密码学的多个领域有着重要应用。随着现代计算机运算能力的提升，对于高次不可约多项式的存在性、分布等的探究更越来越具有实用意义。目前主要的研究方向有三个，第一是直接分析高次不可约多项式的分布特性，第二是通过复合方法迭代产生高次不可约多项式，第三是通过计算直接求某些具有特殊性质的高次不可约多项式。 本文主要针对前两个方向做了相应的研究，研究对象是系数在GF(2)上的不可约多项式。首先分析了一类五项不可约多项式的判别式的计算公式，并提出一个简化公式，针对自反五项不可约多项式提出了一个判别定理；其次针对不可约多项式的计算提出一个算法并针对项链问题提出一个新的生成项链的算法；最后是针对复合方法提出一种计数算法并提出了一种新的不可约多项式。具体内容如下： 1.首先，研究了一类五项不可约多项式的判别式的计算，用于判定其是否可约，提出一个简化公式来减小判别式的计算量。其次，针对五项自反不可约多项式，提出了一种新的定理，可以断定某种条件下某些次数的五项自反不可约多项式不存在，并通过计算机搜索，观察到一类新的不存在的情况并作为猜想。 2.利用牛顿公式和对称多项式理论，给出了一种有限域上迹函数的快速计算方法，并把其应用到不可约多项式和本原多项式的计算当中。针对和不可约多项式关系密切的项链问题，提出了一种新的生成项链的算法，该算法可以保证不会产生冗余的非项链。 3.研究了复合生成高次不可约多项式的算法，并证明了在何种情况下会出现重复情况，由此可以计算在该算法下可以复合生成的不可约多项式的数量。最后研究的是不可约多项式所产生的多项式基，提出一种新的多项式类型，使得其多项式基中元素的迹函数值为1的数量最少。 关键词：有限域上的不可约多项式，自反不可约多项式，牛顿公式，对称多项式，项链问题，复合算法，迹函数为1的元素

 Irreducible polynomials over finite fields play an important role in many applications of cryptography today. As the compute power grows, the existence and distribution of irreducible polynomials with higher degree is more of a practical problem. There are mainly three directions, one of which is on the distribution of the irreducible polynomials of a higher degree, another direction is on generating irreducible polynomials of a higher degree using composing method or other methods, the third direction is to search for irreducible polynomials of a higher degree with special attributes. Our research mainly deals with the first two directions, and is on the irreducible polynomials over the finite field GF(2). We first analyse the discrimination of a special class of irreducible polynomials, and prove a simpler formula, and prove a theorem on self reciprocal irreducible pentanomials. Then an algorithm on computing irreducible polynomials using Newton formula and an algorithm on generating necklaces is proposed in the article. Finally we deal with the iteration method of generating irreducible polynomials, and a new class of polynomials is posed as well. 1.In order to determine whether a special kind of pentanomial is reducible or not, the discriminate of it should be checked, we prove a simplified formula that can be used in the computation procedure. We then prove a theorem to confirm that under the condition of the theorem, there is no self reciprocal irreducible pentanomials of the certain degree given in the theorem. By computer search, we give a conjecture on this topic. 2.Using Newton formula and the theory of symmetric polynomials, we pose an algorithm to make the computation of the trace function faster, and the apply it on the computation of irreducible polynomials and primitive polynomials. As the necklaces are related to the problem of irreducible polynomials, we proposed an algorithm to generate necklaces, there are no non-necklaces in our algorithms. 3.We then research on the composed method of generating irreducible polynomials of higher degrees, we prove under what condition can the generated polynomials be the same, we then use this to compute how many different polynomials can be generated. Finally, as the polynomial basis of an irreducible polynomial is of great importance for computing, especially the basis with less elements whose trace function equals 1, we give a new class of polynomial which suits these demands. KEY WORDS: Irreducible polynomials over finite fields, self reciprocal irreducible pentanomials, Newton formula, symmetric polynomial, necklaces, composed method, elements with trace one