Algebraic attacks on stream ciphers apply (at least theoretically) to all LFSR-based stream ciphers that are clocked in a simple and/or easily predictable way. One interesting approach to help resist such attacks is to add a component that de-synchronizes the output bits of the cipher from the clock of the LFSR. The Bit-search generator, recently proposed by Gouget and Sibert, is inspired by the so-called Self-Shrinking Generator which is known for its simplicity (conception and implementation-wise) linked with some interesting properties. In this paper, we introduce two modified versions of the BSG, called MBSG and ABSG, and some of their properties are studied. We apply a range of cryptanalytic techniques in order to compare the security of the BSGs.