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.
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Shigeki KOBAYASHI, Yasuyuki NOGAMI, Tatsuo SUGIMURA, "A Relation between Self-Reciprocal Transformation and Normal Basis over Odd Characteristic Field" in IEICE TRANSACTIONS on Fundamentals,
vol. E93-A, no. 11, pp. 1923-1931, November 2010, doi: 10.1587/transfun.E93.A.1923.
Abstract: 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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E93.A.1923/_p
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@ARTICLE{e93-a_11_1923,
author={Shigeki KOBAYASHI, Yasuyuki NOGAMI, Tatsuo SUGIMURA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Relation between Self-Reciprocal Transformation and Normal Basis over Odd Characteristic Field},
year={2010},
volume={E93-A},
number={11},
pages={1923-1931},
abstract={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.},
keywords={},
doi={10.1587/transfun.E93.A.1923},
ISSN={1745-1337},
month={November},}
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TY - JOUR
TI - A Relation between Self-Reciprocal Transformation and Normal Basis over Odd Characteristic Field
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 1923
EP - 1931
AU - Shigeki KOBAYASHI
AU - Yasuyuki NOGAMI
AU - Tatsuo SUGIMURA
PY - 2010
DO - 10.1587/transfun.E93.A.1923
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E93-A
IS - 11
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - November 2010
AB - 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.
ER -