Let $h$ and $x$ be vectors of Boolean variables. Given a system of Boolean equations like
$$ h_0 = x_0 \oplus x_2 \oplus x_4 \\ h_1 = x_0 \oplus x_3 \oplus x_5 \\ h_2 = x_1 \oplus x_2 \oplus x_5 \\ h_3 = x_1 \oplus x_3 \oplus x_4 $$
and a valuation for $h$, can we efficiently find a valuation for $x$ satisfying the system above?
$\oplus$ is $XOR$ defined as $a \oplus b = (a \land \overline{b}) \lor (\overline{a} \land b)$
Motivation
I would like to create a hash method based on such a system which satisfies $H(A) \otimes H(B) = H(A \otimes B)$. I also asked a more general question on the cryptography forum, but I wanted to explore this specific idea in a separate question. Provided that this system can be designed hard enough to solve ($NP$-hard would be good) then it would solve my problem.
Xor corresponds to addition in the field $GF(2)$ (where $false = 0$ and $true = 1$), so you have a system of linear equations, and that can be solved efficiently by the usual methods of linear algebra.