No fully-black-box reduction from Hash function to PKE

A reduction of class fully-black-box from Hash function to PKE would imply a contradiction.

Statement

Migrated verbatim from impagliazzos-five-worlds § Breaking up Cryptomania:

Early work of GKM+00 lays out how these different primitives relate to each other. Importantly, there is a large gap between OWF and TDP in Cryptomania: OWFs imply PRGs, PRFs, SKE, MACs, and digital signatures, but not public-key encryption. TDPs (equivalently, the existence of PKE or OT) unlock the full power of asymmetric cryptography. OT is complete for all of MPC, so Cryptomania is also the world where general secure computation is possible.

Notes

Recorded during migration and not fixed — these are claims about the source text, not changes to it:

• SUSPECTED OVERSTATEMENT: “OWFs imply … but not public-key encryption” is written as an unconditional non-implication. The known result (Impagliazzo–Rudich) is a BLACK-BOX separation only; whether OWFs imply PKE is open. Recorded, not fixed.
• No citation — IR89 is neither cited nor linked anywhere on the page.
• class must be “fully-black-box” for the claim to be true, but the page states no class; left “unstated”.
• SUSPECTED MATH ERROR: ‘OWFs imply PRGs, PRFs, SKE, MACs, and digital signatures, but not public-key encryption’ states the barrier as an unconditional NON-IMPLICATION. It is not known that OWFs do not imply PKE; only that no black-box construction exists (IR89). The bolded ‘not’ asserts something strictly stronger than anything proven, and the page’s own black-box-separations glossary entry (line 89) explicitly warns against this reading.
• Cited to GKM+00; the separation is IR89’s. IR89 is not linked anywhere on this page.
• Same sentence contains the already-flagged error ‘TDPs (equivalently, the existence of PKE or OT)’ — TODO_SUMMARY.md:33 records this as an open high-severity finding.
• The bullet is a composite chain (OWF → PRG → PRF → SKE/MAC/DS) plus a barrier, in one sentence, with one citation. Under the target model the positive chain must be split into separate reductions with separate citations (HILL99, GGM86, Lam79) and the barrier separated out entirely.