Generation of full cycles by a composition of NLFSRs

Non-linear feedback shift registers (NLFSRs) are a generalization of linear feedback shift registers in which a current state is a non-linear function of the previous state. The interest in NLFSRs is motivated by their ability to generate pseudo-random sequences which are typically hard to break with existing cryptanalytic methods. However, it is still not known how to construct large \(n\)-stage NLFSRs which generate full cycles of \(2^n\) possible states. This paper presents a method for generating full cycles by a composition of NLFSRs. First, we show that an \(n*k\)-stage register with period \(O(2^{2n})\) can be constructed from \(k\) NLFSRs with \(n\)-stages by adding to their feedback functions a logic block of size \(O(nk)\), for \(k > 1\). This logic block implements Boolean functions representing pairs of states whose successors have to be exchanged in order to join cycles. Then, we show how to join all cycles into one by using one more logic block of size \(O(nk)\).