Construction of Negabent Function Based on Trace Function over Finite Field
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摘要: Negabent函数是一种具有最优自相关性、较高非线性度的布尔函数,在密码学、编码理论及组合设计中都有着广泛的应用。该文基于有限域上的迹函数,将其与置换多项式相结合,提出两种构造negabent函数的方法。所构造的两类negabent函数均具备${\text{Tr}}_1^k(\lambda {x^{{2^k} + 1}}) + {\text{Tr}}_1^n(ux){\text{Tr}}_1^n(vx) + {\text{Tr}}_1^n(mx){{\rm{Tr}}} _1^n(dx)$形式:构造方法1通过调整$\lambda ,{\text{ }}u,{\text{ }}v,{\text{ }}m$中的3个参数来获得negabent函数,特别地,当$\lambda $≠1时,能得到$({2^{n - 1}} - 2)({2^n} - 1)({2^n} - 4)$个negabent函数;构造方法2通过调整$\lambda ,{\text{ }}u,{\text{ }}v,{\text{ }}m,{\text{ }}d$中的4个参数来获得negabent函数,特别地,当$\lambda$≠1时,至少能够得到${2^{n - 1}}[({2^{n - 1}} - 2)({2^{n - 1}} - 3) + {2^{n - 1}} - 4]$个negabent函数。
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关键词:
- Negabent函数 /
- 迹函数 /
- 置换多项式
Abstract: Negabent function is a Boolean function with optimal autocorrelation and high nonlinearity, which has been widely used in cryptography, coding theory and combination design. In this paper, by combining trace function on a finite field with permutation polynomials, two methods for constructing negabent functions are proposed. Both the two kinds of constructed negabent functions take on such form: ${\text{Tr}}_1^k(\lambda {x^{{2^k} + 1}}) + $$ {\text{Tr}}_1^n(ux){\text{Tr}}_1^n(vx) + {\text{Tr}}_1^n(mx){{\rm{Tr}}} _1^n(dx)$. In the first construction method, negabent functions can be obtained by adjusting the three parameters in $\lambda ,{\text{ }}u,{\text{ }}v,{\text{ }}m$. In particular, when $\lambda \ne 1$, $({2^{n - 1}} - 2)({2^n} - 1)({2^n} - 4)$ negabent functions can be obtained. In the second construction method, negabent functions can be obtained by adjusting the four parameters in $\lambda ,{\text{ }}u,{\text{ }}v,{\text{ }}m,{\text{ }}d$. In particular, when $\lambda \ne 1$, at least ${2^{n - 1}} [({2^{n - 1}} - 2) $$ ({2^{n - 1}} - 3) + {2^{n - 1}} - 4]$ negabent functions can be obtained.-
Key words:
- Negabent function /
- Trace function /
- Permutation polynomial
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表 1 不同调参方案所构造函数数量
调整的参数 计数 文献[12] $ u,v $ $({2^{n - 1}} - 2)({2^n} - 1)$ 定理2 $ u,v,m $ $({2^{n - 1}} - 2)({2^n} - 1)({2^n} - 4)$ 定理3 $ u,v,m,d $ $\ge {2^{n - 1} }[({2^{n - 1} } - 2)({2^{n - 1} } - 3) + {2^{n - 1} } - 4]$ 定理3中$\lambda \ne 1$时$\left( {A,B,C,D,E,F,G,H,I,J} \right)$不取的10元向量 1100000000 1100000100 1111010001 1100000001 1111110001 1100000101 1111110110 1100000111 1110000000 1110000100 1111001100 1111110000 1111101100 1111100101 1111101111 1111111011 0000000010 1101000001 1111000100 1111100100 1111011101 1111001110 1111101011 1111111010 0000000101 0000000011 1110110101 1111100001 1111011010 1111000110 1111011111 1111110101 0000001101 0000000110 1110101001 1111000010 1111011000 1110101101 1111010101 1111101101 0000010101 0000001110 1110010010 1110111000 1111001111 1110100110 1111001101 1111101001 0000011010 0000010011 1110010001 1110110001 1111000011 1110011011 1111000101 1111100010 0000011101 0000100101 1110001110 1110100111 1110111100 1101011101 1110110011 1111000111 0001000010 0000111010 1110000010 1110100010 1110111001 1101011010 1110101000 1110111101 0001001000 0000111101 1101110111 1110011001 1110110010 1101011001 1110011101 1110110111 0001010000 0001000011 1101000100 1110000001 1110101110 1101010010 1110010110 1110110110 0001100000 0001000110 1100011111 1101110000 1110011100 1101000101 1110001111 1110101111 0001111100 0001001001 1100010010 1101100101 1110001000 1100110111 1110000110 1110000111 0010000010 0001010001 1100000010 1101100010 1110000011 1100101111 1101111111 1101111101 0010000110 0001100100 1011111010 1101000010 1101101011 1100100110 1101101001 1101111011 0010010000 0001101100 1011011100 1100111100 1101011110 1100010100 1101000110 1101111010 0010100001 0001110111 1011010011 1100101011 1101010110 1100001001 1100010110 1101101111 0010101101 0001111101 1011001010 1100100010 1101010000 1011111011 1100000110 1101101101 0010111001 0010000011 1010011110 1100010000 1101000011 1011110101 1011111101 1101101010 0011000000 0010101111 1010011100 1100001000 1100110010 1011100101 1011010111 1101010100 0011000010 0010111101 1010011011 1011111100 1100011101 1011100011 1011001011 1100111101 0011000101 0011000011 1010010010 1011110100 1100011010 1011001110 1010110011 1100111010 0011000111 0011100000 1010001010 1011100111 1100000011 1010111101 1010011111 1100110110 0011011010 0011100100 1001100101 1011100010 1011111110 1010110101 1010010110 1100100101 0011011101 0011100101 1001011111 1011011101 1011110110 1010101111 1010001011 1011111111 0011100110 0011100111 1001010101 1011011010 1011110011 1010101110 1001111010 1011110111 0100000000 0011111010 1001001010 1011000101 1011101010 1010100110 1001101101 1011101111 0100001000 0011111101 1000011110 1010111001 1011010000 1010100011 1001001011 1011101011 0100010000 0100000001 1000010010 1010110100 1011001000 1010010100 1000111000 1011001001 0100011000 0100001001 1000010001 1010100010 1010111110 1001111101 1000010111 1011000110 0100100000 0100010001 1000001100 1010010000 1010111011 1001110111 1000010110 1010111111 0100101000 0100011100 1000001010 1001111100 1010110010 1001100011 1000001011 1010110111 0100110000 0100100100 0111110010 1001110110 1010101100 1001001001 0111111101 1010110110 0100111000 0100101000 0111011010 1001101100 1010101010 1000111101 0111111010 1010101011 0101000000 0100101100 0111000100 1001100010 1010011000 1000111010 0111100101 1001101111 0110000000 0100110100 0110111100 1001010100 1010001110 1000100110 0111100010 1001101011 0110010000 0100111100 0110010010 1001001000 1001101110 1000100101 0111011011 1001011110 0111000000 0101000100 0110001000 1000100100 1001101010 1000100011 0111000101 1000111111 0111011000 0101001001 0110000010 1000100010 1001011101 0111110001 0110101110 1000110110 1000000010 0110000100 0101111001 1000011101 1001011010 0111101110 0110010110 1000101011 1000100000 0110010001 0101110101 1000011100 1000110010 0111101100 0110010011 1000100111 1000101001 0111000001 0101101011 1000011010 1000101010 0111011101 0110001001 0111111111 1000110111 0111011001 0101010000 1000001000 1000001110 0111000011 0110000110 0111111000 1001000010 1000000011 0101001010 1000000101 0111101001 0110100110 0101111111 0111110110 1001000101 1000000110 0101001000 0111110100 0111011110 0110010101 0101101111 0111101011 1001011000 1000100001 0101000010 0111100111 0111000110 0110000101 0101010100 0111011111 1001100001 1000101000 0100111010 0111100000 0110110010 0101100100 0101001011 0111000111 1001101000 1000110011 0100110010 0111011100 0110000011 0101100001 0101000110 0110111001 1001110001 1001000011 0100101010 0111000010 0101110000 0101001101 0100111011 0110110110 1010000010 1001100000 0100100010 0110100010 0101101000 0101000101 0100110011 0110101010 1010000101 1001101001 0100011010 0110010100 0101001110 0100111101 0100101011 0110010111 1010001001 1010000011 0100010010 0110000001 0101000011 0100110101 0100100011 0110000111 1010010001 1010000110 0100001010 0101100000 0100111110 0100101101 0100011011 0101111110 1010100111 1010001111 0100000010 0101001100 0100110110 0100100101 0100010101 0101110100 1010110000 1010010111 0011111100 0101000001 0100101110 0100011101 0100001101 0101101001 1011000010 1010101001 0011111001 0100111001 0100100110 0100010110 0100000101 0101001111 1011010100 1010110001 0011110110 0100110001 0100011110 0100001110 0011110111 0101000111 1011100000 1011000011 0011010001 0100101001 0100010011 0100000011 0011101111 0100111111 1011111000 1011000111 0011010000 0100100001 0100001011 0011111111 0011011011 0100110111 1100011000 1011010101 0011001111 0100011001 0100000110 0011111000 0011010100 0100101111 1100101100 1011111001 0011001101 0100010100 0011101010 0011101110 0011001011 0100100111 1100110001 1110001100 0011001010 0100001100 0011011111 0011100011 0011001001 0100011111 1101000000 1110001101 0011001000 0100000100 0011011000 0010110001 0011000110 0100010111 1110001001 1111000001 0010011001 0011100010 0010110010 0010101100 0010111011 0100001111 1111000000 0001110110 0010010010 0010100010 0010101010 0010100110 0010010110 0100000111 0000100110 0001101011 0010001010 0010010101 0010100111 0010100011 0010010011 0011101011 0000100011 0000111110 0001011000 0001110000 0010001000 0010011000 0010001011 0010111100 1110000101 0000111011 0001010010 0001100010 0001111000 0010001111 0001011001 0010110110 0000110011 0000110110 0001001010 0001011111 0001110010 0001110001 0001010110 0010101011 0000101110 0000101011 0000110101 0001011011 0001101010 0001100110 0001001011 0010011111 0000010110 1101000111 0000101101 0001010111 0000110010 0001100011 0000011110 0001111110 0000001010 0000100010 0000010010 0001001101 0000101010 0001000101 0000011011 0001111001 0000001011 -
[1] ROTHAUS O S. On “bent” functions[J]. Journal of Combinatorial Theory, Series A, 1976, 20(3): 300–305. doi: 10.1016/0097-3165(76)90024-8 [2] RIERA C and PARKER M G. Generalized bent criteria for Boolean functions (I)[J]. IEEE Transactions on Information Theory, 2006, 52(9): 4142–4159. doi: 10.1109/TIT.2006.880069 [3] 任明生. Nega-Hadamard变换和Negabent函数的研究[D]. [硕士论文], 淮北师范大学, 2017.REN Mingsheng. Research on Nega-Hadamard transform and Negabent function[D]. [Master dissertation], Huaibei Normal University, 2017. [4] SCHMIDT K U, PARKER M G, and POTT A. Negabent functions in the Maiorana–McFarland class[C]. 5th International Conference on Sequences and Their Applications-SETA 2008, Lexington, USA, 2008: 390–402. [5] PARKER M G and POTT A. On Boolean functions which are bent and Negabent[C]. International Workshop on Sequences, Subsequences, and Consequences 2007, Los Angeles, USA, 2007: 9–23. [6] STĂNICĂ P, GANGOPADHYAY S, CHATURVEDI A, et al. Nega-Hadamard transform, bent and Negabent functions[C]. 6th International Conference on Sequences and Their Applications-SETA 2010, Paris, France, 2010: 359–372. [7] STANICA P, GANGOPADHYAY S, CHATURVEDI A, et al. Investigations on bent and Negabent functions via the Nega-Hadamard transform[J]. IEEE Transactions on Information Theory, 2012, 58(6): 4064–4072. doi: 10.1109/TIT.2012.2186785 [8] SU Wei, POTT A, and TANG Xiaohu. Characterization of Negabent functions and construction of bent-Negabent functions with maximum algebraic degree[J]. IEEE Transactions on Information Theory, 2013, 59(6): 3387–3395. doi: 10.1109/TIT.2013.2245938 [9] MANDAL B, MAITRA Su, and STĂNICĂ P. On the existence and non-existence of some classes of bent-Negabent functions[J]. Applicable Algebra in Engineering, Communication and Computing, 2022, 33(3): 237–260. doi: 10.1007/s00200-020-00444-w [10] SARKAR S. Characterizing Negabent Boolean functions over finite fields[C]. 7th International Conference on Sequences and Their Applications-SETA 2012, Waterloo, Canada, 2012: 77–88. [11] ZHOU Yue and QU Longjiang. Constructions of Negabent functions over finite fields[J]. Cryptography and Communications, 2017, 9(2): 165–180. doi: 10.1007/s12095-015-0167-0 [12] WU Gaofei, LI Nian, ZHANG Yuqing, et al. Several classes of Negabent functions over finite fields[J]. Science China Information Sciences, 2018, 61(3): 038102. doi: 10.1007/s11432-017-9096-0 [13] JIANG Niu, ZHAO Min, YANG Zhiyao, et al. Characterization and properties of bent-Negabent functions[J]. Chinese Journal of Electronics, 2022, 31(4): 786–792. doi: 10.1049/cje.2021.00.417 [14] GUO Fei, WANG Zilong, and GONG Guang. Several secondary methods for constructing bent-Negabent functions[J]. Designs, Codes and Cryptography, 2023, 91(3): 971–995. doi: 10.1007/s10623-022-01133-0 [15] STĂNICĂ P, MANDAL B, and MAITRA S. The connection between quadratic bent-Negabent functions and the Kerdock code[J]. Applicable Algebra in Engineering, Communication and Computing, 2019, 30(5): 387–401. doi: 10.1007/s00200-019-00380-4 [16] 周宇, 胡予濮, 董新锋. 布尔函数的设计与分析[M]. 北京: 国防工业出版社, 2015: 20–55.ZHOU Yu, HU Yupu, and DONG Xinfeng. Design and Analysis of Boolean Functions[M]. Beijing: National Defense Industry Press, 2015: 20–55. [17] LIDL R and NIEDERREITER H. Finite Fields[M]. 2nd ed. London: Cambridge University Press, 1996: 54–62. [18] WU Danyao and YUAN Pingzhi. Further results on permutation polynomials from trace functions[J]. Applicable Algebra in Engineering, Communication and Computing, 2022, 33(4): 341–351. doi: 10.1007/s00200-020-00456-6 [19] SARKAR S. Some results on bent-Negabent Boolean functions over finite fields[EB/OL]. https://arxiv.org/abs/1406.1036, 2014.
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