Rijndael MixColumns

The MixColumns operation performed by the Rijndael cipher or Advanced Encryption Standard is, along with the ShiftRows step, its primary source of diffusion. Each column of bytes is treated as a four-term polynomial ${\displaystyle b(x)=b_{3}x^{3}+b_{2}x^{2}+b_{1}x+b_{0}}$, each byte representing an element in the Galois field ${\displaystyle \operatorname {GF} (2^{8})}$. The coefficients are elements within the prime sub-field ${\displaystyle \operatorname {GF} (2)}$.

Each column is multiplied with the fixed polynomial ${\displaystyle a(x)=3x^{3}+x^{2}+x+2}$ modulo ${\displaystyle x^{4}+1}$; the inverse function is ${\displaystyle a^{-1}(x)=11x^{3}+13x^{2}+9x+14}$.