# Several classes of bent functions over finite fields

Let ๐ฝ_p^n be the finite field with p^n elements and Tr(ยท) be the trace function from ๐ฝ_p^n to ๐ฝ_p, where p is a prime and n is an integer. Inspired by the works of Mesnager (IEEE Trans. Inf. Theory 60(7): 4397-4407, 2014) and Tang et al. (IEEE Trans. Inf. Theory 63(10): 6149-6157, 2017), we study a class of bent functions of the form f(x)=g(x)+F(Tr(u_1x),Tr(u_2x),โฏ,Tr(u_ฯx)), where g(x) is a function from ๐ฝ_p^n to ๐ฝ_p, ฯโฅ2 is an integer, F(x_1,โฏ,x_n) is a reduced polynomial in ๐ฝ_p[x_1,โฏ,x_n] and u_iโ๐ฝ^*_p^n for 1โค i โคฯ. As a consequence, we obtain a generic result on the Walsh transform of f(x) and characterize the bentness of f(x) when g(x) is bent for p=2 and p>2 respectively. Our results generalize some earlier works. In addition, we study the construction of bent functions f(x) when g(x) is not bent for the first time and present a class of bent functions from non-bent Gold functions.

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