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 $ $\ge {2^{n - 1} }[({2^{n - 1} } - 2)({2^{n - 1} } - 3) + {2^{n - 1} } - 4]$ 
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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 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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 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