The minimal polynomial over Fq of linear recurring sequence over Fqm

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 Fqm with minimal polynomial h(x) over Fqm. Since Fqm and are isomorphic vector spaces over the finite field Fq, S is identified with an m-fold multisequence S(m) over the finite field Fq. The joint minimal polynomial and joint linear complexity of the m-fold multisequence S(m) are the minimal polynomial and linear complexity over Fq of S, respectively. In this paper, we study the minimal polynomial and linear complexity over Fq of a linear recurring sequence S over Fqm with minimal polynomial h(x) over Fqm. If the canonical factorization of h(x) in Fqm[x] is known, we determine the minimal polynomial and linear complexity over Fq of the linear recurring sequence S over Fqm.