### 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 |

### Keywords

- Normal basis
- Self-reciprocal irreducible polynomial

### ASJC Scopus subject areas

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

## Fingerprint Dive into the research topics of 'A relation between self-re ciprocal transformation and normal basis over odd characteristic field'. Together they form a unique fingerprint.

## 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