On Analysis and Synthesis of (n,k)-Non-Linear Feedback Shift Registers

Non-linear feedback shift registers (NLFSRs) have been proposed as an alternative to Linear Feedback Shift Registers (LFSRs) for generating pseudo-random sequences for stream ciphers. In this paper, we introduce (n,k)-NLFSRs which can be considered a generalization of the Galois type of LFSR. In an (n,fc)-NLFSR, the feedback can be taken from any of the n bits, and the next state functions can be any Boolean function of up to k variables. Our motivation for considering this type NLFSRs is that their Galois configuration makes it possible to compute each next state function in parallel, thus increasing the speed of output sequence generation. Thus, for stream cipher application where the encryption speed is important, (n,k)-NLFSRs may be a better alternative than the traditional Fibonacci ones. We derive a number of properties of (n,k)- NLFSRs. First, we demonstrate that they are capable of generating output sequences with good statistical properties which cannot be generated by the Fibonacci type of NLFSRs. Second, we show that the period of the output sequence of an (n,k)-NLFSR is not necessarily equal to the length of the largest cycle of its states. Third, we compute the period of an (n,k)-NLFSR constructed from several parallel NLFSRs whose outputs are XOR-ed and show how to maximize this period. We also present an algorithm for estimating the length of cycles of states of (n,k)-NLFSRs which uses binary decision diagrams for representing the set of states and the transition relation on this set.