In this letter, we give a characterization for a generic construction of bent functions. This characterization enables us to obtain another efficient construction of bent functions and to give a positive answer on a problem of bent functions.

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 2017, Tang et al. provided a complete characterization of generalized bent functions from ℤ2n to ℤq(q = 2m) in terms of their component functions (IEEE Trans. Inf. Theory. vol.63, no.7, pp.4668-4674). In this letter, for a general even q, we aim to provide some characterizations and more constructions of generalized bent functions with flexible coefficients. Firstly, we present some sufficient conditions for a generalized Boolean function with at most three terms to be gbent. Based on these results, we give a positive answer to a remaining question proposed by Hodžić in 2015. We also prove that the sufficient conditions are also necessary in some special cases. However, these sufficient conditions whether they are also necessary, in general, is left as an open problem. Secondly, from a uniform point of view, we provide a secondary construction of gbent function, which includes several known constructions as special cases.

As a with-carry analog (based on modular arithmetic) of the usual Walsh-Hadamard transform (WHT), arithmetic Walsh transform (AWT) has been used to obtain analogs of some properties of Boolean functions which are important in the design and analysis of cryptosystems. The existence of nonzero linear structure of Boolean functions is an important criterion to measure the weakness of these functions in their cryptographic applications. In this paper, we find more analogs of linear structures of Boolean functions from AWT. For some classes of n-variable Boolean functions f, we find necessary and sufficient conditions for the existence of an invariant linear structure and a complementary linear structure 1n of f. We abstract out a sectionally linear relationship between AWT and WHT of n-variable balanced Boolean functions f with linear structure 1n. This result show that AWT can characterize cryptographic properties of these functions as long as WHT can. In addition, for a diagonal Boolean function f, a recent result by Carlet and Klapper says that the AWT of f can be expressed in terms of the AWT of a diagonal Boolean function of algebraic degree at most 3 in a larger number of variables. We provide for the result a complete and more modular proof which works for both even and odd weights (of the parameter c in the Corollary 19 by Carlet and Klapper (DCC 73(2): 299-318, 2014).

This paper presents an integer discrete cosine transform (IntDCT) with only dyadic values such as k/2n (k, n∈ in N). Although some conventional IntDCTs have been proposed, they are not suitable for lossless-to-lossy image coding in low-bit-word-length (coefficients) due to the degradation of the frequency decomposition performance in the system. First, the proposed M-channel lossless Walsh-Hadamard transform (LWHT) can be constructed by only (log2M)-bit-word-length and has structural regularity. Then, our 8-channel IntDCT via LWHT keeps good coding performance even if low-bit-word-length is used because LWHT, which is main part of IntDCT, can be implemented by only 3-bit-word-length. Finally, the validity of our method is proved by showing the results of lossless-to-lossy image coding in low-bit-word-length.