### 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}) is 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 partial conjugates given as follows are linearly independent over F_{q}, {γ - γ^{-1}, (γ - γ^{-1})^{q}, · · · , (γ - γ^{-1})qm-1}, (1) where γ is a zero of F(x) and thus a proper element in F_{q}2m. 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 |
---|---|

Title of host publication | ICCIT 2009 - 4th International Conference on Computer Sciences and Convergence Information Technology |

Pages | 999-1004 |

Number of pages | 6 |

DOIs | |

Publication status | Published - 2009 |

Event | 4th International Conference on Computer Sciences and Convergence Information Technology, ICCIT 2009 - Seoul, Korea, Republic of Duration: Nov 24 2009 → Nov 26 2009 |

### Other

Other | 4th International Conference on Computer Sciences and Convergence Information Technology, ICCIT 2009 |
---|---|

Country | Korea, Republic of |

City | Seoul |

Period | 11/24/09 → 11/26/09 |

### Fingerprint

### Keywords

- Normal basis
- Polynomial transformation
- Self-reciprocal irreducible polynomial

### ASJC Scopus subject areas

- Computer Science(all)
- Information Systems and Management

### Cite this

*ICCIT 2009 - 4th International Conference on Computer Sciences and Convergence Information Technology*(pp. 999-1004). [5369570] https://doi.org/10.1109/ICCIT.2009.119

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*ICCIT 2009 - 4th International Conference on Computer Sciences and Convergence Information Technology.*, 5369570, pp. 999-1004, 4th International Conference on Computer Sciences and Convergence Information Technology, ICCIT 2009, Seoul, Korea, Republic of, 11/24/09. https://doi.org/10.1109/ICCIT.2009.119

}

TY - GEN

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

AU - Kobayashi, Shigeki

AU - Nogami, Yasuyuki

AU - Sugimura, Tatsuo

PY - 2009

Y1 - 2009

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) is 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 partial conjugates given as follows are linearly independent over Fq, {γ - γ-1, (γ - γ-1)q, · · · , (γ - γ-1)qm-1}, (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) is 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 partial conjugates given as follows are linearly independent over Fq, {γ - γ-1, (γ - γ-1)q, · · · , (γ - γ-1)qm-1}, (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 - Polynomial transformation

KW - Self-reciprocal irreducible polynomial

UR - http://www.scopus.com/inward/record.url?scp=77749277591&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77749277591&partnerID=8YFLogxK

U2 - 10.1109/ICCIT.2009.119

DO - 10.1109/ICCIT.2009.119

M3 - Conference contribution

AN - SCOPUS:77749277591

SN - 9780769538969

SP - 999

EP - 1004

BT - ICCIT 2009 - 4th International Conference on Computer Sciences and Convergence Information Technology

ER -