Cellular automata find applications in cryptography, error correction, theoretical computer science, and the simulation of physical and biological processes. We say that a Boolean function is a proper lifting if its induced cellular automaton is reversible. This paper introduces new classes of proper liftings, constructed through compositions of landscape functions, and explores their cryptographic potential by computing indicators relevant to differential and linear cryptanalysis. We combine theoretical and computational methods to obtain a complete classification of proper liftings for diameters up to $6$. Several open problems and conjectures are posed along the way.
A rotation-symmetric S-box on $n$ bits is a vectorial Boolean function that commutes with the right cyclic shift and is therefore determined by a single Boolean function. The map $χ$ used in Keccak is a well-known example. Ordinary affine equivalence allows arbitrary invertible affine transformations, which need not preserve rotation symmetry. We introduce \emph{cyclic affine equivalence}, in which the implementing transformations each have an invertible $k$-circulant linear part with $\gcd(k,n)=1$ and a translation vector fixed by the shift. For $n\ge3$, the two relations give the same classes of rotation-symmetric permutations exactly when $n$ is an odd prime for which $2$ is a primitive root. In every other dimension $n\ge3$, explicit constructions exhibit affine classes containing several cyclic affine classes. We derive exact class-size formulas, count cyclic affine classes for $3\le n\le9$, and classify a four-bit family of $χ$-siblings under both relations. Finally, we characterize when a permutation is affinely equivalent to a rotation-symmetric one and apply the criterion to the Ascon S-box.