Deciding Irreducibility/Indecomposability of Feedback Shift Registers Is NP-Hard

Feedback shift registers (FSRs) are used as a fundamental component in electronics and confidential communication. A FSR f is said to be reducible if all the output sequences of another FSR g can also be generated by f and the FSR g costs less memory than f. A FSR is said to be decomposable if it has the same set of output sequences as a cascade connection of two FSRs. Two polynomial-time computable transformations from Boolean circuits to FSRs are proposed such that the output FSR of the first (resp. second) transformation is irreducible (resp. indecomposable) if and only if the input Boolean circuit is satisfiable. Through the two transformations, it is proved that deciding irreducibility (indecomposability) of FSRs is NP-hard. Additionally, FSRs are constructed to show that there exist infinitely many irreducible (resp. indecomposable) FSRs which are decomposable (resp. reducible).