The Gaussian normal basis and its trace basis over finite fields

被引:14
作者
Liao, Qunying [1 ]
机构
[1] Sichuan Normal Univ, Inst Math & Software Sci, Chengdu 610066, Peoples R China
基金
美国国家科学基金会;
关键词
Finite fields; Complexity; Trace map; Dual bases; Gaussian normal bases; NORMAL BASES; COMPLEXITY;
D O I
10.1016/j.jnt.2012.01.013
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
It is well known that normal bases are useful for implementations of finite fields in various applications including coding theory, cryptography, signal processing, and so on. In particular, optimal normal bases are desirable. When no optimal normal basis exists, it is useful to have normal bases with low complexity. In this paper, we study the type k (>= 1) Gaussian normal basis N of the finite field extension F-qn/F-q, which is a classical normal basis with low complexity. By studying the multiplication table of N, we obtain the dual basis of N and the trace basis of N via arbitrary medium subfields F-qm/F-q with m vertical bar n and 1 <= m <= n. And then we determine all self-dual Gaussian normal bases. As an application, we obtain the precise multiplication table and the complexity of the type 2 Gaussian normal basis and then determine all optimal type 2 Gaussian normal bases. (C) 2012 Elsevier Inc. All rights reserved.
引用
收藏
页码:1507 / 1518
页数:12
相关论文
共 50 条
[41]   EXISTENCE OF RATIONAL PRIMITIVE NORMAL PAIRS OVER FINITE FIELDS [J].
Sharma, Rajendra Kumar ;
Takshak, Soniya ;
Awasthi, Ambrish ;
Sharma, Hariom .
INTERNATIONAL JOURNAL OF GROUP THEORY, 2024, 13 (01)
[42]   On the Existence of Pairs of Primitive and Normal Elements Over Finite Fields [J].
Carvalho, Cicero ;
Guardieiro, Joao Paulo ;
Neumann, Victor G. L. ;
Tizziotti, Guilherme .
BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY, 2022, 53 (03) :677-699
[43]   Efficient FPGA Implementations of Point Multiplication on Binary Edwards and Generalized Hessian Curves Using Gaussian Normal Basis [J].
Azarderakhsh, Reza ;
Reyhani-Masoleh, Arash .
IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, 2012, 20 (08) :1453-1466
[44]   On the bound of the complexity of the normal basis generated by the trace of the dual element of a Type I optimal normal element [J].
Mishra, Alok ;
Sharma, Rajendra Kumar ;
Shukla, Wagish .
ASIAN-EUROPEAN JOURNAL OF MATHEMATICS, 2014, 7 (03)
[45]   Low-Complexity Digit-Level Systolic Gaussian Normal Basis Multiplier [J].
Shao, Qiliang ;
Hu, Zhenji ;
Chen, Shaobo ;
Chen, Pingxiuqi ;
Xie, Jiafeng .
IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, 2017, 25 (10) :2817-2827
[46]   Common Subexpression Algorithms for Space-Complexity Reduction of Gaussian Normal Basis Multiplication [J].
Azarderakhsh, Reza ;
Jao, David ;
Lee, Hao .
IEEE TRANSACTIONS ON INFORMATION THEORY, 2015, 61 (05) :2357-2369
[47]   On the complexity of the dual basis of a type I optimal normal basis [J].
Wan, Zhe-Xian ;
Zhou, Kai .
FINITE FIELDS AND THEIR APPLICATIONS, 2007, 13 (02) :411-417
[48]   Inverses of r-primitive k-normal elements over finite fields [J].
Rani, Mamta ;
Sharma, Avnish K. ;
Tiwari, Sharwan K. ;
Panigrahi, Anupama .
RAMANUJAN JOURNAL, 2024, 63 (03) :723-747
[49]   Trace of products in finite fields [J].
Swaenepoel, Cathy .
FINITE FIELDS AND THEIR APPLICATIONS, 2018, 51 :93-129
[50]   Analytic normal basis theorem [J].
Alexandru, Victor ;
Popescu, Nicolae ;
Zaharescu, Alexandru .
CENTRAL EUROPEAN JOURNAL OF MATHEMATICS, 2008, 6 (03) :351-356