International Association
for Cryptologic Research

In this paper, we obtain a characterization of generalized Boolean functions based on spectral analysis. We investigate a relationship between the Walsh-Hadamard spectrum and $\\sigma_f$, the sum-of-squares-modulus indicator (SSMI) of the generalized Boolean function. It is demonstrated that $\\sigma_f = 2^{2n + s}$ for every $s$-plateaued generalized Boolean function in $n$ variables. Two classes of generalized semi-bent Boolean functions are constructed.% and it is demonstrated that their SSMI is over generalized $s$-plateaued Boolean functions is $2^{2n + s}$. We have constructed a class of generalized semi-bent functions in $(n+1)$ variables from generalized semi-bent functions in $n$ variables and identify a subclass of it for which $\\sigma_f$ and $\\triangle_{f}$ both have optimal value. Finally, some construction on generalized partially bent Boolean functions are given.