AES

• A bit-wise XOR between the 16-byte state and the appropriate 16-bytes of the expanded key.
• You pass the block data stored in the state array through an XOR function with the first key generated (K0). It passes the resultant state array on as input to the next step.

\begin{align*} & \begin{bmatrix} block_{1,1} & block_{1,2} & block_{1,3} & block_{1,4} \\ block_{2,1} & block_{2,2} & block_{2,3} & block_{2,4} \\ block_{3,1} & block_{3,2} & block_{3,3} & block_{3,4} \\ block_{4,1} & block_{4,2} & block_{4,3} & block_{4,4} \\ \end{bmatrix} \bigoplus \begin{bmatrix} key_{1,1} & key_{1,2} & key_{1,3} & key_{1,4} \\ key_{2,1} & key_{2,2} & key_{2,3} & key_{2,4} \\ key_{3,1} & key_{3,2} & key_{3,3} & key_{3,4} \\ key_{4,1} & key_{4,2} & key_{4,3} & key_{4,4} \\ \end{bmatrix} \\ = & \begin{bmatrix} block_{1,1} \bigoplus key_{1,1} & block_{1,2} \bigoplus key_{1,2} & block_{1,3} \bigoplus key_{1,3} & block_{1,4} \bigoplus key_{1,4} \\ block_{2,1} \bigoplus key_{2,1} & block_{2,2} \bigoplus key_{2,2} & block_{2,3} \bigoplus key_{2,3} & block_{2,4} \bigoplus key_{2,4} \\ block_{3,1} \bigoplus key_{3,1} & block_{3,2} \bigoplus key_{3,2} & block_{3,3} \bigoplus key_{3,3} & block_{3,4} \bigoplus key_{3,4} \\ block_{4,1} \bigoplus key_{4,1} & block_{4,2} \bigoplus key_{4,2} & block_{4,3} \bigoplus key_{4,3} & block_{4,4} \bigoplus key_{4,4} \\ \end{bmatrix} \end{align*}

• replace each bytes in the block with the values in S-Box

\begin{align*} \begin{bmatrix} block_{1,1} & block_{1,2} & block_{1,3} & block_{1,4} \\ block_{2,1} & block_{2,2} & block_{2,3} & block_{2,4} \\ block_{3,1} & block_{3,2} & block_{3,3} & block_{3,4} \\ block_{4,1} & block_{4,2} & block_{4,3} & block_{4,4} \\ \end{bmatrix} \rightarrow \begin{bmatrix} S-Box(block_{1,1}) & S-Box(block_{1,2}) & S-Box(block_{1,3}) & S-Box(block_{1,4}) \\ S-Box(block_{2,1}) & S-Box(block_{2,2}) & S-Box(block_{2,3}) & S-Box(block_{2,4}) \\ S-Box(block_{3,1}) & S-Box(block_{3,2}) & S-Box(block_{3,3}) & S-Box(block_{3,4}) \\ S-Box(block_{4,1}) & S-Box(block_{4,2}) & S-Box(block_{4,3}) & S-Box(block_{4,4}) \\ \end{bmatrix} \end{align*}

• Matrix production in Galois field

\begin{align*} \begin{bmatrix} 2 & 3 & 1 & 1 \\ 1 & 2 & 3 & 1 \\ 1 & 1 & 2 & 3 \\ 3 & 1 & 1 & 2 \end{bmatrix} \times \begin{bmatrix} block_{1,1} & block_{1,2} & block_{1,3} & block_{1,4} \\ block_{2,1} & block_{2,2} & block_{2,3} & block_{2,4} \\ block_{3,1} & block_{3,2} & block_{3,3} & block_{3,4} \\ block_{4,1} & block_{4,2} & block_{4,3} & block_{4,4} \\ \end{bmatrix} \end{align*}