Coherent-label lifting from PRUs to PRFUs

Determine whether a general lifting from pseudorandom unitaries to pseudorandom function-like unitaries is possible when queries may involve superpositions of labels, or establish a separation showing that no comparable lifting exists.

Background

The paper establishes a lifting from a pseudorandom unitary and a post-quantum pseudorandom function to a pseudorandom function-like unitary for classical-label access. For coherent-label access, however, the construction directly incorporates the label into quantum-secure phase and permutation primitives rather than deriving a general transformation from a single-instance pseudorandom unitary.

The unresolved issue is whether coherent superposition queries impose additional cryptographic requirements that prevent a general PRU-to-PRFU transformation. Resolving this by constructing such a lifting or proving a separation would clarify the relationship between single-instance and function-like quantum pseudorandomness.