有限域上一类高斯正规基复杂度的准确计算公式

廖群英, 胡晓兰

数学学报 ›› 2014, Vol. 57 ›› Issue (5) : 863-874.

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数学学报 ›› 2014, Vol. 57 ›› Issue (5) : 863-874. DOI: 10.12386/A2014sxxb0080
论文

有限域上一类高斯正规基复杂度的准确计算公式

    廖群英, 胡晓兰
作者信息 +

The Explicit Formula for the Complexity of a Class of Gauss Period Normal Bases over Finite Fields

    Qun Ying LIAO, Xiao Lan HU
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摘要

通过刻画有限域上分圆数的性质，给出了有限域上一类高斯正规基复杂度的准确计算公式. 进而证明了有限域Fqn在Fq上的7型高斯正规基满足所给条件当且仅当n≠4.

Abstract

In this paper, by characterizing some properties of cyclotomic numbers, we obtain the explicit formula for the complexity of a class of Gauss period normal bases over finite fields. Furthermore, we show that the type 7 Gauss period normal basis of Fqn over Fq is just the desired basis except for n= 4.

关键词

有限域 / 正规基 / 复杂度 / 分圆数

Key words

finite field / normal basis / complexity / cyclotomic number

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廖群英, 胡晓兰. 有限域上一类高斯正规基复杂度的准确计算公式. 数学学报, 2014, 57(5): 863-874 https://doi.org/10.12386/A2014sxxb0080
Qun Ying LIAO, Xiao Lan HU. The Explicit Formula for the Complexity of a Class of Gauss Period Normal Bases over Finite Fields. Acta Mathematica Sinica, Chinese Series, 2014, 57(5): 863-874 https://doi.org/10.12386/A2014sxxb0080

参考文献

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[5] Gao S., Lenstra H. W., Optimal normal bases, Des. Codes Cryptogr., 1992, 2: 315-323.

[6] Gathen J. Z., Pappalardi F., Density estimates related to Gauss periods, Prog. Comput. Sci. Appl. Log.,2001, 20: 33-41.

[7] Gauss C. F., Disquisitiones Arithmeticae, English Edition, Springer-Verlag, New York, 1986.

[8] Mullin R. C., Onyszchuk I. M., Vanstone S. A., et al., Optimal normal bases in GFqn, Discrete Appl. Math.,1989, 22: 149-161.

[9] Silva D., Kschischang F. R., Fast encoding and decoding of Gabidulin codes, In: Proceedings of the IEEEInternational Symposium of Information Theory, Seoul, Korea, 2009: 2858-2862.

[10] Wassermann A., Konstruktion von Normal basen, Bayreuther Mathematische Scriften, 1990, 31: 155-164.

基金

四川省杰出青年学术技术带头人培育计划（2011JQ0037）；四川省教育厅科研重点项目（14ZA0034）；四川师范大学科研基金重点培育项目（13ZDL06）

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