Reversible computing refers to the type of computing where one minimizes the the amount of data to be deleted usually to save energy (Reversible computing could also thwart side-channel attacks). Energy efficient reversible computers do not exist yet in the free market although some have constructed prototypes and they should become the computers of the future. In reversible computing, the AND and OR gates are forbidden since they delete data (the AND gate has 2 inputs but 1 output so one bit is necessarily deleted).
With purely reversible computing, only the bijective logic gates are allowed for computation since non-bijective gates delete information. For example, the CNOT $(x,y)\mapsto(x,x\oplus y)$, Toffoli $(x,y,z)\mapsto(x,y,(x\wedge y)\oplus z)$, and Fredkin $(0,y,z)\mapsto(0,y,z),(1,y,z)\mapsto(1,z,y)$ gates are all reversible gates.
While reversible computing theoretically should save energy, it typically takes more steps to compute something reversibly than it does to perform the same computation irreversibly. However, in symmetric cryptography, one could potentially design cryptosystems such as encryption systems, hash functions, (cryptographic and non-cryptographic) pseudorandom number generators which do not contain any such algorithmic overhead (the only irreversibility that is required in a cryptographic hash function is that unless one wants to keep the hash around forever, one must eventually delete the hash).
1) Are there any symmetric cryptosystems which are specifically designed to be used by reversible computers?
2) Are there any symmetric cryptosystems which were not necessarily designed to be used by reversible computers but which just-so-happen to be reversible?
3) Are there any symmetric cryptosystems which are nearly reversible and which can potentially be modified slightly to become reversible?
Here are some characteristics that one would like in reversible cryptosystem.
-The cryptosystem should ideally be computed using solely reversible gates without any garbage bits, ancilla bits, uncomputation, or irreversible gates.
-The inverses of all parts included in the cryptosystem should be computed in exactly the same way as the forward direction (for example, the inverse $\chi$ in Keccak is computed differently than the forward direction; $\chi$ has algebraic degree 2 while the inverse of $\chi$ has algebraic degree 3 (see page 16 of these slides)).
-The cryptosystem must not require any uncomputing. Uncomputing is an overhead that arises in reversible computing even though irreversible computing does not require any uncomputing.
-The cryptosystem should not use any lookup tables.
-The cryptosystem should not use modular multiplication, finite field multiplication, or finite field inversion at all these operations require uncomputing.