Relaxing the one-way permutation assumption

Construct a hard PAC learning and PAC generation instance for which the separation established for the one-way-permutation construction holds under weaker cryptographic assumptions, such as the existence of one-way functions, or without relying on one-way functions, for example by using trapdoor functions.

Background

The first separation in the paper assumes an oracle encoding a one-way permutation. The permutation property is used so that a uniformly sampled output has a uniformly sampled preimage, allowing the generator to reproduce the target distribution while avoiding inversion of the permutation.

The authors explicitly identify relaxing this assumption as unresolved. In particular, they ask whether one-way functions suffice, or whether a different cryptographic primitive such as trapdoor functions can yield a comparable hard instance.

References

We leave it as an open question to relax this requirement to only needing one-way functions, or indeed a separate construction that does not require this at all.

— A Quantum/Classical Example Oracle Separation for Making Things Up  (2608.11648 - Chen, 12 Aug 2026) in Concluding remark after the proof of Theorem 3, Section 3, subsection “A Counterexample to Function Hardness Implies Distribution Hardness”

However, it is not clear if proving the converse is possible with this approach, as this variant of OWSGs requires an orthogonality property (statistical invertibility) which an arbitrary AHL instance does not satisfy in general.

— Equivalence Between Average-Case Hardness of Learning and Cryptography for Mixed Quantum States  (2608.14331 - Cojocaru et al., 14 Aug 2026) in Section 1, Introduction