3 research outputs found
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}.Comment: Submitted to the journal Finite Fields and Their Application
The minimal polynomial of sequence obtained from componentwise linear transformation of linear recurring sequence
Let be a linear recurring sequence with terms in
and be a linear transformation of over . Denote
. In this paper, we first present counter
examples to show the main result in [A.M. Youssef and G. Gong, On linear
complexity of sequences over , Theoretical Computer Science,
352(2006), 288-292] is not correct in general since Lemma 3 in that paper is
incorrect. Then, we determine the minimal polynomial of if the canonical
factorization of the minimal polynomial of without multiple roots is known
and thus present the solution to the problem which was mainly considered in the
above paper but incorrectly solved. Additionally, as a special case, we
determine the minimal polynomial of if the minimal polynomial of is
primitive. Finally, we give an upper bound on the linear complexity of
when exhausts all possible linear transformations of over
. This bound is tight in some cases.Comment: This paper was submitted to the journal Theoretical Computer Scienc
Composed Products and Explicit Factors of Cyclotomic Polynomials over Finite Fields
Let be a power of a prime number and let be the
finite field with elements. In this paper we obtain the explicit
factorization of the cyclotomic polynomial over
where both and are odd, , and .
Previously, only the special cases when had been achieved. For
this we make the assumption that the explicit factorization of over
is given to us as a known. Let be the factorization of into powers of distinct
primes . In the case that the orders of modulo all
these prime powers are pairwise coprime we show how to obtain the
explicit factors of from the factors of each . We
also demonstrate how to obtain the factorization of from the
factorization of when is a primitive root modulo and
\gcd(m,n) = \gcd(\phi(m),\ord_n(q)) = 1. Here is the Euler's totient
function, and \ord_n(q) denotes the multiplicative order of modulo .
Moreover, we present the construction of a new class of irreducible polynomials
over and generalize a result due to Varshamov (1984)
\cite{Varshamov}.Comment: 24 page