  Let $m$ be an even positive integer. A Boolean bent function $f$ on $\GF{m-1}
\times \GF {}$ is called a \emph{cyclic bent function} if for any $a\neq b\in
\GF {m-1}$ and $\epsilon \in \GF{}$, $f(ax_1,x_2)+f(bx_1,x_2+\epsilon)$ is
always bent, where $x_1\in \GF {m-1}, x_2 \in \GF {}$. Cyclic bent functions
look extremely rare. This paper focuses on cyclic bent functions on $\GF {m-1}
\times \GF {}$ and their applications. The first objective of this paper is to
construct a new class of cyclic bent functions, which includes all known
constructions of cyclic bent functions as special cases. The second objective
is to use cyclic bent functions to construct good mutually unbiased bases
(MUBs), codebooks and sequence families. The third objective is to study cyclic
semi-bent functions and their applications. The fourth objective is to present
a family of binary codes containing the Kerdock code as a special case, and
describe their support designs. The results of this paper show that cyclic bent
functions and cyclic semi-bent functions have nice applications in several
fields such as symmetric cryptography, quantum physics, compressed sensing and
CDMA communication.
