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A Relation between Self-Reciprocal Transformation and Normal Basis over Odd Characteristic Field

Shigeki KOBAYASHI, Yasuyuki NOGAMI, Tatsuo SUGIMURA

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Summary :

Let q and f(x) be an odd characteristic and an irreducible polynomial of degree m over Fq, respectively. Then, suppose that F(x)=xmf(x+x-1) becomes irreducible over Fq. This paper shows that the conjugate zeros of F(x) with respect to Fq form a normal basis in Fq2m if and only if those of f(x) form a normal basis in Fqm and the compart of conjugates given as follows are linearly independent over Fq, {γ-γ-1,(γ-γ-1)q, …,(γ-γ-1)qm-1} where γ is a zero of F(x) and thus a proper element in Fq2m. In addition, from the viewpoint of q-polynomial, this paper proposes an efficient method for checking whether or not the conjugate zeros of F(x) satisfy the condition.

Publication
IEICE TRANSACTIONS on Fundamentals Vol.E93-A No.11 pp.1923-1931
Publication Date
2010/11/01
Publicized
Online ISSN
1745-1337
DOI
10.1587/transfun.E93.A.1923
Type of Manuscript
Special Section PAPER (Special Section on Information Theory and Its Applications)
Category
Coding Theory

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