The main contribution of this paper is to characterize the hyperbentness of two infinite classes of Boolean functions via Dillon-like exponents, and give new classes of semibent functions with Dillon-like exponents and Niho exponents. In this paper, the approaches of Mesnager and Wang et al. are generalized to Charpin-Gong like functions with two additional trace terms. By using the partial exponential sums and Dickson polynomials, it also gives the necessary and sufficient conditions of the hyperbent properties for their subclasses of Boolean functions, and gives two corresponding examples on F230. Thanks to the result of Carlet et al., new classes of semibent functions are obtained by using new hyperbent functions and the known Niho bent functions. Finally, this paper extends the Works of Lisonek and Flori and Mesnager, and gives different characterizations of new hyperbent functions and new semibent functions with some restrictions in terms of the number of points on hyperelliptic curves. These results provide more nonlinear functions for designing the filter generators of stream ciphers.

A new construction method for polyphase sequences with two-valued periodic auto- and crosscorrelation functions is proposed. This method gives L families of polyphase sequences for each prime length L which is bigger than three. For each family of sequences, the out-of-phase auto- and crosscorrelation functions are proved to be constant and asymptotically reach the Sarwate bound. Furthermore, it is shown that sequences of each family are mutually orthogonal.

A class of balanced semi-bent functions with an even number of variables is proposed. It is shown that they include one subclass of semi-bent functions with maximum algebraic degrees. Furthermore, an example of semi-bent functions in a small field is given by using the zeros of some Kloosterman sums. Based on the result given by S.Kim et al., an example of infinite families of semi-bent functions is also obtained.