This paper proposes a novel approach for cryptanalysis of certain cryptographic pseudorandom sequence (keystream) generators consisting of the composition of a linear finite state machine (LFSM) and nonlinear mapping. The proposed approach includes a dedicated decimation of the sample for cryptanalysis based on the following: Suppose certain B bits of the LFSM initial state as known and identify time instances where certain arguments of the nonlinear function depend only on these B bits and are equal to zero. As opposed to previously reported methods, the proposed one also identifies and uses certain characteristics of the LFSM state-transition matrix in order to reduce the nonlinearity of the system of overdefined equations employed in an algebraic attack scenario, or to reduce the noise introduced by the linearization of the nonlinear function which corrupts the linear equations employed in a correlation attack scenario.