Given the problem of computing the GCD of two given elements over any finite field with characteristic 2.
$$ r_1 = q_1r_2 + r_3 \\ r_2 = q_2r_3 + r_4 \\ r_3 = q_3r_4 + r_5 \\ \vdots \\ r_{k-1} = q_{k-1}r_{k} + r_{k+1}\\ r_k = q_kr_{k+1}$$
Exist any way of find the aproximate value of $r_{k+1}$, for $k > 3$ without know the $q_{k-2}$ value?