# solving matrix over GF(2)

```
A = matrix(GF(2), 8, 8, [])
b = vector(GF(2), [0, 1, 1, 0, 1, 0, 1, 1])
y = vector(GF(2), [0, 0, 0, 0, 1, 0, 1, 1])
x = vector(GF(2), [1, 0, 0, 0, 0, 0, 0, 0])
```

If the matrix $A$ is unkown, we have $Ax+b = y$.

How can we solve the matrix $A$?

We can write simpler $Ax=b'$ with an obvious $b'$. This is a linear system in the $8^2=64$ entries of $A$, considered as unknowns, if i get the message right, but we have only $8$ equations, corresponding to the components of the constant given known vector $b'=y-b$. We need now all solutions?

Yes, we need all possible solutions of $A$, since we have many equations like $Ax+b=y$, we will solve each equation and take the intersection to get the final $A$.