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In this paper, we study the properties of the sum-of-squares indicator of vectorial Boolean functions. Firstly, we give the upper bound ofSecondly, 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-ofsquares indicators with respect to several equivalence relations, and get the sum-of-squares indicator which have the best cryptographic properties.

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-bent 4 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.

The Walsh transform is an important tool to investigate cryptographic properties of Boolean functions. This paper is devoted to study the Walsh transform of a class of Boolean functions defined as [see formula in PDF], by making use of the known conclusions of Walsh transform and the properties of trace function, and the conclusion is obtained by generalizing an existing result.