域元素分量代数表达式的研究

Study on Algebraic Representations of Coordinates of Finite Fields

本文提出了有限域上的一个新性质 :用变元为域元素的多项式表示域元素的分量 .基于等价类的划分、线性方程组的求解和标准基之对偶基的计算 ,提出了域元素分量代数表达式的三种求法 .以此解释了Rijndael算法S盒代数表达式复杂度低的本质原因 ,给出其分量函数间等价关系的一种直接证明方法 .

The determination of the algebraic representations of coordinates of finite field elements with the elements themselves as the variables, which is a new property of finite fields, is investigated. Based on the partition of equivalent classes, the resolving of a linear system of equations and the calculation of the dual basis of the standard basis, three methodologies are presented. With those results, we have successfully given an essential explanation to the simplicity of the algebraic representation of Rijndael S-box and provided a direct proof to the equivalence between any two coordinate functions of Rijndael S-box.