In this paper, the following is asserted:
The function $$X\to X^2$$ in $\mathrm{GF}(2^{128})$ is linear. This is due to the fact that $\mathrm{GF}(2^{128})$ is a field of characteristic 2, which implies that $\forall A, B, \quad (A+B)^2 = A^2 + B^2$.
Thus, there is a fixed matrix $M_S$ such that $\overline{X^2} = M_S \overline{X}$ for all $X$. Note that $M_S$ does not depend on anything except the chosen field representation, which is public.
(Here $\overline{X}$ is the vector of coefficients in $\mathrm{GF}(2)$ of the polynomial representation of X.)
I don't find this explanation very helpful. My question is thus: what is the reasoning why the fixed matrix $M_S$ exists?
And, what does the matrix $M_S$ look like?