Permutation rotation-symmetric S-boxes, liftings and affine equivalence

In this paper, we investigate permutation rotation-symmetric (shift-invariant) vectorial Boolean functions on $n$ bits that are liftings from Boolean functions on $k$ bits, for $k\leq n$. These functions generalize the well-known map used in the current Keccak hash function, which is generated via the Boolean function on $3$ variables, $x1+(x2+1)x_3$. We provide some general constructions, and also study the affine equivalence between rotation-symmetric S-boxes and describe the corresponding relationship between the Boolean function they are associated with.