The Notes on the Linear Structures of Rotation Symmetric Boolean Functions

In this paper, the linear structure of Rotation Symmetric Boolean Functions (RSBF) is studied. The relationship between the degree and the existence of linear structures in RSBFs is investigated. The open problem that an-variable RSBF being balanced and of degree n-1 has no linear structure except the all-zero vector is proved. A formula for enumerating the self-conjugate orbits is presented. By this formula, the number of RSBFs, which have no linear structure except all-one vectors, is obtained.