Some semi-bent functions with polynomial trace form

Abstract

This paper is devoted to the study of semi-bent functions with several parameters flexible on the finite field \(\mathbb{F}_{2^n } \). Boolean functions defined on \(\mathbb{F}_{2^n } \) of the form $$f_{a,b}^{(r)} (x) = Tr_1^n (ax^{r(2^m - 1)} ) + Tr_1^4 (bx^{\tfrac{{2^n - 1}} {5}} ) $$ and the form $$g_{a,b,c,d}^{(r,s)} (x) = Tr_1^n (ax^{r(2^m - 1)} ) + Tr_1^4 (bx^{\tfrac{{2^n - 1}} {5}} ) + Tr_1^n (cx^{(2^m - 1)\tfrac{1} {2} + 1} ) + Tr_1^n (dx^{(2^m - 1)s + 1} ) $$ where n = 2m, m ≡ 2 (mod 4), a, c ∈ \(\mathbb{F}_{16} \), and b ∈ \(\mathbb{F}_2 \), d ∈ \(\mathbb{F}_2 \), are investigated in constructing new classes of semi-bent functions. Some characteristic sums such as Kloosterman sums and Weil sums are employed to determine whether the above functions are semi-bent or not.