Attack based on direct sum decomposition against the nonlinear filter generator

The nonlinear filter generator (NLFG) is a powerful building block commonly used in stream ciphers. In this paper, we present the direct sum decomposition of the NLFG output sequence that leads to a system of linear equations in the initial state of the NLFG and further to an efficient algebraic attack. The coefficients of the equation system rely only on the NLFG structure. The attack is operated in an online/offline manner, doing most of the work (determining the coefficients of the equation system) in the offline phase. Thus the online phase is very fast, requiring only four multiplications and one diagonalization of n x n matrices. Compared with related works, our attack has the advantages in both online computation cost and success probability. On the one hand, far fewer output bits and significantly less matrix computation are required in our attack, although the online computation complexity O(LC) (LC is the linear complexity of the output sequence) is the same as in the known Rønjom-Helleseth attack. On the other hand, the success probability of the attack is analyzed in this paper, different from most prior work. The success probability of this algebraic attack is 1- 2 ^{-φ(2n - 1)} (φ(·) is the Euler function), which is much greater than 1 - 2 ^-n , the success probability of the Rønjom-Helleseth attack.