Predicting Truncated Galois Linear Feedback Shift Registers

Linear feedback shift registers (LFSRs) over integer residue rings are widely used to generate pseudorandom number, such as ZUC algorithm, truncated LCGs, truncated MRGs. Truncated Galois LFSRs are an important way to generate pseudorandom sequences. Methods to predict the whole sequences by the truncated sequences of the truncated Galois LFSRs are not only a crucial aspect of evaluating their security but also important concerns in their design. This paper studies the predictability of truncated Galois LFSRs. When the modulus and the state transition matrix are known, we first propose a lattice-based method to recover the initial state by the high-order truncated sequences, then discuss the condition that recovering the initial state by the low-order truncated sequences is meaningful, and finally solve the low-order case by transforming it into the high-order case. When the modulus and the state transition matrix are unknown, we generalize our recent work, using the resultant, the greatest common factor, and Kannan's embedding technique in turn to recover the modulus, the characteristic polynomial, and the initial state. Moreover, we heuristically show that the state transition matrix can be successfully recovered only when all registers output sufficiently long truncated sequences. Experiments have verified the effectiveness of our methods.