有限域上线性化多项式与正规基

2022-03-16徐碧云肖义丽

宁波大学学报（理工版） 2022年2期

关键词：宁波大学数论密码学

徐碧云, 肖义丽, 曹炜

有限域上线性化多项式与正规基

徐碧云1, 肖义丽1, 曹炜2*

（1.宁波大学数学与统计学院, 浙江宁波 315211; 2.闽南师范大学数学与统计学院, 福建漳州 363000）

有限域; 正规基; 线性化多项式

1986年, Pei等[4]证明了定理2.

2001年, Chang等[5]进一步证明了定理1和定理2的充分条件也是必要的, 从而得到了定理3.

1 线性化多项式

线性化多项式的许多相关性质与结论可参考文献[1,8-11].

引理2[1]

推论2

引理3[1]

定理5[12]

证明

由容斥原理得:

引理4 由定义4, 得到如下集合:

证明由于

又因为

所以

可得:

由定义4可得式(3), 同理可得式(4)～(6).

引理5 假设

由引理4可以得到:

由欧拉函数性质得:

由定义4得:

即式(8)得证.

所以

其中:

定理6

由容斥原理和引理5可得:

则由式(10), 得到:

[1] Lidl R, Niederreiter H. Finite Fields[M]. Cambridge: Cambridge University Press, 1997.

[2] Mullen G L, Panario D. Handbook of Finite Fields[M]. London: Chapman and Hall, 2013.

[3] Perlis S. Normal bases of cyclic fields of prime-power degree[J]. Duke Mathematical Journal, 1942, 9(3):507- 517.

[4] Pei D, Wang C, Omura J. Normal basis of finite field GF(2)[J]. IEEE Transactions on Information Theory, 1986, 32(2):285-287.

[5] Chang Y, Truong T K, Reed I S. Normal bases over GF(q)[J]. Journal of Algebra, 2001, 241(1):89-101.

[6] Huang H, Han S M, Cao W. Normal bases and irreducible polynomials[J]. Finite Fields and Their Applications, 2018, 50:272-278.

[7] Hachenberger D. Characterizing normal bases via the trace map[J]. Communications in Algebra, 2004, 32(1): 269-277.

[8] Zan H X, Cao W. Powers of polynomials and bounds of value sets[J]. Journal of Number Theory, 2014, 143:286- 292.

[9] Polverino O, Zullo F. On the number of roots of some linearized polynomials[J]. Linear Algebra and Its Applications, 2020, 601:189-218.

[10] Polverino O, Zini G, Zullo F. On certain linearized polynomials with high degree and kernel of small dimension[J]. Journal of Pure and Applied Algebra, 2021, 225(2):106491.

[11] Hu W Y, Zhang Z C, Tu Z H, et al. Efficient computation of the nearest polynomial by linearized alternating direction method[J]. Applied Mathematics and Computation, 2021, 395:125860.

[12] von zur Gathen J, Giesbrecht M. Constructing normal bases in finite fields[J]. Journal of Symbolic Computation, 1990, 10(6):547-570.

Linearized polynomials and normal bases in finite fields

XU Biyun1, XIAO Yili1, CAO Wei2*

( 1.School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China;2.School of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China )