Abstract: This paper researches the design of matrix A for a class of S box. It gives necessary and sufficient condition whether a circular matrix is inverse, and shows the expression of S(X)=AXeb in GF(2n) has at least three terms if matrix A is chosen to be an inverse circular matrix in GF(2) which is not identity matrix. So it is appropriate to design A to be an inverse circular matrix in designing this class of S box. Choosing a proper inverse circular matrix A and making the polynomial expression of S(X)=AXeb in GF(2n) be the most terms to guarantee a good resistance against the interpolation attacks and higher order differentials cryptanalysis.

Key words: S box, inverse circular matrix, polynomial expression

摘要： 对一类S盒S(X)=AXeb中矩阵A的构造和设计问题进行研究，给出二元域GF(2)上循环矩阵A可逆的一个充要条件，证明了矩阵A只要选取为与单位阵不等的nn可逆循环矩阵，就可使得S盒S(X)=AXeb在有限域GF(2n)中的多项式表达式至少有3项系数不为0，从而在构造该类S盒时，将矩阵A选取为可逆循环矩阵是可行的。适当地选取可逆循环矩阵A，使得S(X)=AXeb在有限域GF(2n)中的多项式表达式的非零系数尽可能多，就能在一定程度上抵抗插值攻击和高阶差分密码分析。

关键词: S盒, 可逆循环矩阵, 多项式表达式

CLC Number: 

• TN 918.1