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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 $\epsilon \in \GF{}$, $f(ax_1,x_2)+f(bx_1,x_2+\epsilon)$ isalways bent, where $x_1\in \GF {m-1}, x_2 \in \GF {}$. Cyclic bent functionslook 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 toconstruct a new class of cyclic bent functions, which includes all knownconstructions of cyclic bent functions as special cases. The second objectiveis to use cyclic bent functions to construct good mutually unbiased bases(MUBs), codebooks and sequence families. The third objective is to study cyclicsemi-bent functions and their applications. The fourth objective is to presenta family of binary codes containing the Kerdock code as a special case, anddescribe their support designs. The results of this paper show that cyclic bentfunctions and cyclic semi-bent functions have nice applications in severalfields such as symmetric cryptography, quantum physics, compressed sensing andCDMA communication.

Cyclic bent functions and their applications in codes, codebooks, designs, MUBs and sequences