Writing a specific position of AES state in a specific round in terms of Key bytes

Suppose we manipulate the zeroth byte of AES state after the 9th round ShiftRow operation and we want to see its effect on the ciphertext. The propagation of the error effect is demonstrated in the figure. I have written the equations of each step in the order below and obtained the equation of $$V[0]$$ in terms of key bytes (The ciphertext is known). $$\begin{cases} Z[0] = C[0] \oplus K_{10}[0] \\ Z[7] = C[7] \oplus K_{10}[7] \\ Z[10] = C[10] \oplus K_{10}[10] \\ Z[13] = C[13] \oplus K_{10}[13] \end{cases}$$

$$\begin{cases} Y[0] = Z[0] = C[0] \oplus K_{10}[0] \\ Y[1] = Z[13] = C[13] \oplus K_{10}[13] \\ Y[2] = Z[10] = C[10] \oplus K_{10}[10] \\ Y[3] = Z[7] = C[7] \oplus K_{10}[7] \end{cases}$$

$$\begin{cases} X[0] = S^{-1}(Y[0]) = S^{-1}(C[0] \oplus K_{10}[0]) \\ X[1] = S^{-1}(Y[1]) = S^{-1}(C[13] \oplus K_{10}[13]) \\ X[2] = S^{-1}(Y[2]) = S^{-1}(C[10] \oplus K_{10}[10]) \\ X[3] = S^{-1}(Y[3]) = S^{-1}(C[7] \oplus K_{10}[7]) \end{cases}$$

$$\begin{cases} W[0] = X[0] \oplus K_{9}[0] = S^{-1}(C[0] \oplus K_{10}[0]) \oplus K_{9}[0] \\ W[1] = X[1] \oplus K_{9}[1] = S^{-1}(C[13] \oplus K_{10}[13]) \oplus K_{9}[1] \\ W[2] = X[2] \oplus K_{9}[2] = S^{-1}(C[10] \oplus K_{10}[10]) \oplus K_{9}[2] \\ W[3] = X[3] \oplus K_{9}[3] = S^{-1}(C[7] \oplus K_{10}[7]) \oplus K_{9}[3] \end{cases}$$

$$\Rightarrow V[0] = (0E.W[0]) \oplus (0B.W[1]) \oplus (0D.W[2]) \oplus (09.W[3]) \\ = (0E.(S^{-1}(C[0] \oplus K_{10}[0]) \oplus K_{9}[0])) \oplus (0B.(S^{-1}(C[13] \oplus K_{10}[13]) \oplus K_{9}[1])) \oplus (0D.(S^{-1}(C[10] \oplus K_{10}[10]) \oplus K_{9}[2])) \oplus (09.(S^{-1}(C[7] \oplus K_{10}[7]) \oplus K_{9}[3]))$$

My question is, is there an easier way to do this? Is it possible to write a code to do this for us or is there a special software to do this?

• Do you want just symbolic computation? SageMath can do that ( Maple and mathematical, too) Commented Oct 18, 2023 at 22:25
• Thank you for helping me @kelalaka . Yes, symbolic computation is enough. Can you tell me exactly what to look for in the Sage tutorial? I mean, what do they call this action? Commented Oct 19, 2023 at 7:55
• doc.sagemath.org/html/en/reference/calculus/sage/symbolic/… Commented Oct 19, 2023 at 7:59