On the Conjecture About the Linear Structures of Rotation Symmetric Boolean Functions

In this paper, we study the conjecture that [Formula: see text]-variable ([Formula: see text] odd) rotation symmetric Boolean functions with degree [Formula: see text] have no non-zero linear structures. We show that if this class of RSBFs have non-zero linear structures, then the linear structures are invariant linear structures and the homogeneous component of degree [Formula: see text] in the function’s algebraic normal form has only two possibilities. Moreover, it is checked that the conjecture is true for [Formula: see text], and then a more explicit conjecture is proposed.