中文摘要：

为了提高对称布尔函数算术Walsh变换的实现效率,利用Krawtchouk多项式研究对称布尔函数的算术Walsh变换。根据算术Walsh变换的定义讨论对称布尔函数的算术Walsh变换和Krawtchouk多项式之间的关系,给出并证明基于Krawtchouk多项式描述的对称布尔函数的算术Walsh变换的简约表达式。利用得出的简约表达式和Krawtchouk多项式的性质即可得出一种实现对称布尔函数的算术Walsh变换的快速算法,该算法具有较低的时间复杂度和空间复杂度。

英文摘要：

In order to improve the algorithm efficiency for computing the arithmetic Walsh spectra of symmetric Boolean functions,Krawtchouk polynomial is used to study arithmetic Walsh trans-form of symmetric Boolean functions.Firstly the relationship of Krawtchouk polynomial and arithmetic Walsh transform of symmetric Boolean functions is discussed.Secondly it is proved that the arithmetic Walsh spectra of symmetric Boolean functions has simple expression related to Krawtchouk polynomial.Finally a fast algorithm for computing the arithmetic Walsh spectra of symmetric Boolean functions using properties of Krawtchouk polynomial is presented.This algorithm has low time and space complexity.