A Note on Semi-bent and Hyper-bent Boolean Functions

摘要

Semi-bent and hyper-bent funcitons as two classes of Boolean functions with low Walsh transform, are applied in cryptography and commnunications. This paper considers a new class of semi-bent quadratic Boolean function and a generalization of a new class of hyper-bent Boolean functions. The new class of semi-bent quadratic Boolean function of the form f(x) = ∑_(i=1)~(「(m-1)/2」) Tr_1~n(c_ix~(1+4~i)) (c_i ∈ F_4,n = 2m) is simply characterized and enumerated. Then we present the characterization of a generalization of a new class of hyper-bent Boolean functions of the form f_(a,b)~((r)) := Tr_1~n(ax~(r(2~m-1))) + Tr_1~4(bx (2~n-1)/5), where n = 2m,m ≡ 2 (mod 4), a ∈ F_(2~m) and b ∈ F_(16).