Efficient implementation of standard cryptographic primitives without auxiliary space

Determine whether standard cryptographic primitives, in particular keyed pseudorandom permutations, can be implemented efficiently without auxiliary qubits beyond the key and input registers.

Background

The paper motivates catalytic computation by considering the task of implementing a keyed pseudorandom permutation in place, namely mapping the key and input registers to the key and permuted input while using only those registers. Standard reversible compilations generally use auxiliary space to store intermediate computations, and the paper introduces catalysts as auxiliary qubits that are returned exactly to their initial state. The authors then show that catalysts suffice for implementing the permutation, but do not resolve whether the same task can be achieved efficiently without auxiliary space.