  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, $x_1+(x_2+1)x_3$. 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.
