The transformation AddRoundKey

The input to the transformation AddRoundKey is a state $(a_{ij})$, $0\le i<4, 0\le j<$Nb, and a round key, which is an array of bytes $($rk$_{ij})$, $0\le i<4, 0\le j<$Nb. The output of AddRoundKey is the state $(b_{ij}),0\le i<4, 0\le j<$Nb, where

$$b_{ij}=a_{ij}\oplus$$   rk$$_{ij}.$$

The round keys are obtained from the cipher key by expanding the cipher key array $(k_{ij})$ into an array $(k_{ij})$, $0\le i< 4, 0\le j\le$   Nr$\cdot$   Nb, called the expanded key. The round key for the initial application of AddRoundKey is given by the first Nb columns of the expanded key. The round key for the application of AddRoundKey in the $m$-th round of AES is given by columns $m$Nb$,\ldots,(m+1)$Nb$-1$ of the expanded key, $1\le m\le$   Nr.