The Minimal Polynomial over F_q of Linear Recurring Sequence over F_{q^m}

Recently, motivated by the study of vectorized stream cipher systems, the joint linear complexity and joint minimal polynomial of multisequences have been investigated. Let S be a linear recurring sequence over finite field F_{q^m} with minimal polynomial h(x) over F_{q^m}. Since F_{q^m} and F_{q}^m are isomorphic vector spaces over the finite field F_q, S is identified with an m-fold multisequence S^{(m)} over the finite field F_q. The joint minimal polynomial and joint linear complexity of the m-fold multisequence S^{(m)} are the minimal polynomial and linear complexity over F_q of S respectively. In this paper, we study the minimal polynomial and linear complexity over F_q of a linear recurring sequence S over F_{q^m} with minimal polynomial h(x) over F_{q^m}. If the canonical factorization of h(x) in F_{q^m}[x] is known, we determine the minimal polynomial and linear complexity over F_q of the linear recurring sequence S over F_{q^m}.

💡 Research Summary

The paper addresses a fundamental problem that arises when a linear recurring sequence (S) defined over an extension field (\mathbb{F}{q^{m}}) is viewed from the perspective of the base field (\mathbb{F}{q}). Specifically, given the minimal polynomial (h(x)\in\mathbb{F}_{q^{m}}