### Abstract

Let q and f (x) be an odd characteristic and an irreducible polynomial of degree m over F_{q} , respectively. Then, suppose that F(x) = x ^{m} f (x + x-1) becomes irreducible over F_{q} . This paper shows that the conjugate zeros of F(x) with respect to F_{q} form a normal basis in F_{q}2m if and only if those of f (x) form a normal basis in F_{q}^{m} and the part of conjugates given as follows are linearly independent over F_{q} , {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 F_{q2m} . 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.

Original language | English |
---|---|

Pages (from-to) | 1923-1931 |

Number of pages | 9 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E93-A |

Issue number | 11 |

DOIs | |

Publication status | Published - Nov 2010 |

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### Keywords

- Normal basis
- Self-reciprocal irreducible polynomial

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Computer Graphics and Computer-Aided Design
- Applied Mathematics
- Signal Processing

### Cite this

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E93-A*(11), 1923-1931. https://doi.org/10.1587/transfun.E93.A.1923

**A relation between self-re ciprocal transformation and normal basis over odd characteristic field.** / Kobayashi, Shigeki; Nogami, Yasuyuki; Sugimura, Tatsuo.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E93-A, no. 11, pp. 1923-1931. https://doi.org/10.1587/transfun.E93.A.1923

}

TY - JOUR

T1 - A relation between self-re ciprocal transformation and normal basis over odd characteristic field

AU - Kobayashi, Shigeki

AU - Nogami, Yasuyuki

AU - Sugimura, Tatsuo

PY - 2010/11

Y1 - 2010/11

N2 - Let q and f (x) be an odd characteristic and an irreducible polynomial of degree m over Fq , respectively. Then, suppose that F(x) = x m f (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 part 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.

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) = x m f (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 part 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.

KW - Normal basis

KW - Self-reciprocal irreducible polynomial

UR - http://www.scopus.com/inward/record.url?scp=78049501786&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78049501786&partnerID=8YFLogxK

U2 - 10.1587/transfun.E93.A.1923

DO - 10.1587/transfun.E93.A.1923

M3 - Article

AN - SCOPUS:78049501786

VL - E93-A

SP - 1923

EP - 1931

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 11

ER -