AES S盒的代数表达式

The algebraic expression for the AES Sbox

插入攻击是一种针对代数式次数和复杂程度低的分组密码的攻击方法.给出了有限域上一个新的代数性质,即域元素分量可用一个以元素自身为变元的简洁的代数式表示,进而给出RijndaelS盒代数式的一种新的求解方法.证明了GF(2)上的矩阵运算是造成S盒代数式低复杂度的主要原因,无论生成多项式、仿射矩阵和仿射常量怎样改变都不会增加S盒代数式的次数和复杂程度.最后指出改进的思想.

The complexity and the degree of the algebraic expression for the Sbox are symbols of resistance of block ciphers against the interpolation attacks. The algebraic expression for the Rijndael Sbox has a high degree of 254 but low complexity that only nine items are there. We have investigated a new property of the finite field that the component of the field element can be expressed by a simple polynomial with the element itself as the variable. Then a new method to determine the algebraic expression for the Rijndael Sbox is proposed. Conclusions are drawn and proved that the low complexity of the algebraic expression for the Rijndael Sbox is mainly caused by the binary matrix in the construction of the Sbox and that any other choice of the generating polynomial, the affine matrix or the affine constant will not make the expression more complex. Finally, the quideline for improvement is pointed out.