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

Shigeki Kobayashi, Yasuyuki Nogami, Tatsuo Sugimura

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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) = 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.

Original languageEnglish
Title of host publicationICCIT 2009 - 4th International Conference on Computer Sciences and Convergence Information Technology
Pages999-1004
Number of pages6
DOIs
Publication statusPublished - 2009
Event4th International Conference on Computer Sciences and Convergence Information Technology, ICCIT 2009 - Seoul, Korea, Republic of
Duration: Nov 24 2009Nov 26 2009

Other

Other4th International Conference on Computer Sciences and Convergence Information Technology, ICCIT 2009
CountryKorea, Republic of
CitySeoul
Period11/24/0911/26/09

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Polynomials

Keywords

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

ASJC Scopus subject areas

  • Computer Science(all)
  • Information Systems and Management

Cite this

Kobayashi, S., Nogami, Y., & Sugimura, T. (2009). A relation between self-reciprocal transformation and normal basis over odd characteristic field. In 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.

ICCIT 2009 - 4th International Conference on Computer Sciences and Convergence Information Technology. 2009. p. 999-1004 5369570.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Kobayashi, S, Nogami, Y & Sugimura, T 2009, A relation between self-reciprocal transformation and normal basis over odd characteristic field. in 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
Kobayashi S, Nogami Y, Sugimura T. A relation between self-reciprocal transformation and normal basis over odd characteristic field. In ICCIT 2009 - 4th International Conference on Computer Sciences and Convergence Information Technology. 2009. p. 999-1004. 5369570 https://doi.org/10.1109/ICCIT.2009.119
Kobayashi, Shigeki ; Nogami, Yasuyuki ; Sugimura, Tatsuo. / A relation between self-reciprocal transformation and normal basis over odd characteristic field. ICCIT 2009 - 4th International Conference on Computer Sciences and Convergence Information Technology. 2009. pp. 999-1004
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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.

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