On Basing One-way Permutations on NP-hard Problems under Quantum Reductions
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A fundamental pursuit in complexity theory concerns reducing worst-case problems to average-case problems. There exist complexity classes such as PSPACE that admit worst-case to average-case reductions. Many other classes such as NP, however, the evidence so far is typically negative, in the sense that the existence of such reductions would cause collapses of the polynomial hierarchy. Basing cryptography, e.g., the average-case hardness of inverting one-way permutations, on NP-completeness is a particularly intriguing instance. We initiate a study of the quantum analogues of these questions and show that if NP-complete problems reduce to inverting one-way permutations using certain types of quantum reductions, then coNP ⊆ QIP(2).
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