In this paper, some properties of Boolean functions via the unitary transform and c-correlation functions are presented. Based on the unitary transform, we present two classes of secondary constructions for c-bent4 functions. Also, by using the c-correlation functions, a direct link between c-autocorrelation function and the unitary transform of Boolean functions is provided, and the relationship among c-crosscorrelation functions of arbitrary four Boolean functions can be obtained.

In this paper, we study the properties of the sum-of-squares indicator of vectorial Boolean functions. Firstly, we give the upper bound of $sum_{uin mathbb{F}_2^n,vin mathbb{F}_2^m}mathcal{W}_F^3(u,v)$. Secondly, based on the Walsh-Hadamard transform, we give a secondary construction of vectorial bent functions. Further, three kinds of sum-of-squares indicators of vectorial Boolean functions are defined by autocorrelation function and the lower and upper bounds of the sum-of-squares indicators are derived. Finally, we study the sum-of-squares indicators with respect to several equivalence relations, and get the sum-of-squares indicator which have the best cryptographic properties.

In this paper, we consider the spectra of Boolean functions with respect to the nega-Hadamard transform. Based on the properties of the nega-Hadamard transform and the solutions of the Diophantine equations, we investigate all possibilities of the nega-Hadamard transform of Boolean functions with exactly two distinct nega-Hadamard coefficients.