Algebraic attacks have established themselves as a powerful method for the cryptanalysis of LFSR-based keystream generators (e.g., E0 used in Bluetooth). The attack is based on solving an overdetermined system of low-degree equations Rt=0, where Rtis an expression in the state of the LFSRs at clock t and one or several successive keystream bits zt,...,zt+δ. In fast algebraic attacks, new equations of a lower degree are constructed in a precomputation step. This is done by computing appropriate linear combinations of T successive initial equations Rt=0. The successive data complexity of the attack is the number T of successive equations. We propose a new variant of fast algebraic attacks where the same approach is employed to eliminate some unknowns, making a divide-and-conquer attack possible. In some cases, our variant is applicable whereas the first one is not. Both variants can have a high successive data complexity (e.g., T≥ 8.822.188 for E0). We describe how to keep it to a minimum and introduce suitable efficient algorithms for the precomputation step.