[EDITED] (Changed the question)
Hello, now I'm making tables to review it, but it's slow... Xor with product in Z/(2^n) are a ring?
I have a "linear" system like this (we know $a,b,c,d,e,f,g,h$, we want to know $x_1,x_2,x_3,x_4$):
$ \begin{array}{ccccccccl} a\cdot x_1 & \oplus & b\cdot x_2 & \oplus & 0 & \oplus & 0 & = e \\ 0 & \oplus & 0 & \oplus & a\cdot x_3 & \oplus & b\cdot x_4 & = f \\ c\cdot x_1 & \oplus & d\cdot x_2 & \oplus & 0 & \oplus & 0 & = g \\ 0 & \oplus & 0 & \oplus & c\cdot x_3 & \oplus & d\cdot x_4 & = h \\ \end{array} $
We can transform the system to two independent equations system, but even if we are working only with 2 unknowns I can't find the way to work with XOR and the typical addition together.
Thanks in advance for any idea :) .