Average-case complexity has two standard formulations, i.e., errorless complexity and error-prone complexity. In average-case complexity, a critical topic of research is to show the equivalence between these formulations, especially on the average-case complexity of NP. In this study, we present a relativization barrier for such an equivalence. Specifically, we construct an oracle relative to which NP is easy on average in the error-prone setting (i.e., DistNP ⊆ HeurP) but hard on average in the errorless setting even by 2^o(n/log n)-size circuits (i.e., DistNP ⊈ AvgSIZE[2^o(n/log n)]), which provides an answer to the open question posed by Impagliazzo (CCC 2011). Additionally, we show the following in the same relativized world: - Lower bound of meta-complexity: GapMINKT^𝒪 ∉ prSIZE^𝒪[2^o(n/log n)] and GapMCSP^𝒪 ∉ prSIZE^𝒪[2^(n^ε)] for some ε > 0. - Worst-case hardness of learning on uniform distributions: P/poly is not weakly PAC learnable with membership queries on the uniform distribution by nonuniform 2ⁿ/n^ω(1)-time algorithms. - Average-case hardness of distribution-free learning: P/poly is not weakly PAC learnable on average by nonuniform 2^o(n/log n)-time algorithms. - Weak cryptographic primitives: There exist a hitting set generator, an auxiliary-input one-way function, an auxiliary-input pseudorandom generator, and an auxiliary-input pseudorandom function against SIZE^𝒪[2^o(n/log n)]. This provides considerable insights into Pessiland (i.e., the world in which no one-way function exists, and NP is hard on average), such as the relativized separation of the error-prone average-case hardness of NP and auxiliary-input cryptography. At the core of our oracle construction is a new notion of random restriction with masks.
@InProceedings{hirahara_et_al:LIPIcs.CCC.2022.25, author = {Hirahara, Shuichi and Nanashima, Mikito}, title = {{Finding Errorless Pessiland in Error-Prone Heuristica}}, booktitle = {37th Computational Complexity Conference (CCC 2022)}, pages = {25:1--25:28}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-241-9}, ISSN = {1868-8969}, year = {2022}, volume = {234}, editor = {Lovett, Shachar}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.25}, URN = {urn:nbn:de:0030-drops-165875}, doi = {10.4230/LIPIcs.CCC.2022.25}, annote = {Keywords: average-case complexity, oracle separation, relativization barrier, meta-complexity, learning, auxiliary-input cryptography} }
Feedback for Dagstuhl Publishing