On Sets with Easy Certificates and the Existence of One-Way Permutations

Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. We give structural conditions that control the size of these classes. We also provide negative results showing that some of our positive claims are optimal. Our negative results are proven using a novel observation: The classic “wide spacing” oracle construction technique instantly yields non-bi-immunity results.

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