关于多项式函数的计数

On the Number Counting of Polynomial Functions

中文关键词:  多项式函数  置换多项式  有限交换环  计数公式.

英文关键词:polynomial functions  permutation polynomials  finite commutative rings  counting formula.

中文摘要:

      多项式函数(特别是置换多项式)在近代密码体制的设计中有着重要的应用. 本文主要讨论了有限交换环上多项式函数的计数问题. 设 $A$ 是一般有限交换局部环. 本文在一定条件下得到了 $A$ 上的多项式函数的计数公式. 在本文前, 若干学者得到了一些特殊的有限交换环局部环上的多项式函数的计数公式.

英文摘要:

      Polynomial functions (in particular, permutation polynomials) play an important role in the design of modern cryptosystem. In this note the problem of counting the number of polynomial functions over finite commutative rings is discussed. Let $A$ be a general finite commutative local ring. Under a certain condition, the counting formula of the number of polynomial functions over $A$ is obtained. Before this paper, some results over special finite commutative rings were obtained by many authors.