Deciding Irreducibility/Indecomposability of Feedback Shift Registers is NP-hard

TL;DR

This paper proves that determining whether feedback shift registers are irreducible or indecomposable is computationally NP-hard, highlighting the complexity of analyzing their structural properties in electronics and secure communications.

Contribution

The paper establishes the NP-hardness of deciding irreducibility and indecomposability of feedback shift registers, a novel complexity result in the analysis of FSRs.

Findings

Deciding irreducibility of FSRs is NP-hard.

Deciding indecomposability of FSRs is NP-hard.

Complexity results impact analysis of FSRs in electronics and security.

Abstract

Feedback shift registers(FSRs) are a fundamental component in electronics and secure communication. An 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 $g$ has less memory than $f$. An FSR is said to be decomposable if it has the same set of output sequences as a cascade connection of two FSRs. It is proved that deciding whether FSRs are irreducible/indecomposable is NP-hard.