Construction of Negabent Function Based on Trace Function over Finite Field

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.