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

There exists an oracle, relative to which P not= NP and each of the following properties hold: (i) all Sigma /sup p//sub 2/-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/sup A/ are P-separable, and Sigma /sup P//sub 2/= union (DTIME(2/sup p/) mod p is a polynomial). Additional related relativization results are presented.<
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