Oracles for structural properties: the isomorphism problem and public-key cryptography

There exists an oracle, relative to which P ≠ NP and each of the following properties hold: (i) all Σ^p₂-complete sets are p-isomorphic; (ii) P-inseparable pairs of sets in NP do not exist; (iii) intractable public-key cryptosystems do not exist; and (iv) NP-complete sets are closed under union of disjoint sets. Remarkably, these properties all follow from one oracle construction, namely, it is proved that there is an oracle A such that every two disjoint sets in NP^A are P-separable, and Σ^P₂=∪{DTIME(2^p )| p is a polynomial}. Additional related relativization results are presented