← 研踪 PaperTrailarXiv:2611.05308 · v1
TFNP in the Random Oracle Model
首次提交 2026-11-04 02:17
版本更新 2026-11-04 02:17
雷达观测 2026-11-04 23:39
摘要
We determine which major $\TFNP$ classes are hard in the random oracle model (ROM). Prior work establishes average-case hardness of highly structured subclasses of $\TFNP$ from strong cryptographic assumptions such as LWE and indistinguishability obfuscation. It is not known whether weaker assumptions such as one-way functions imply hardness of any of the standard $\TFNP$ subclasses, even in the worst case. The ROM, where all parties have query access to a random function, lies, morally, between these two regimes: a random function is one-way, but public-key cryptography cannot be built from it in a black-box way. We refer to this setting as \emph{Romantica}. Our first result settles a long-standing open problem: $\PPAD$, the class of total search problems reducible to finding a Nash equilibrium, is hard on average in Romantica. Combined with the known hardness of PLS (local search), this gives average-case hardness of $\EoPL = \PPAD \cap \PLS$. In contrast, we show that $\UEoPL$ can be solved in Romantica with polynomially many oracle queries by a computationally unbounded solver, a sharp distinction between $\EoPL$ and $\UEoPL$. This resolves the black-box hardness landscape of all major $\TFNP$ classes in Romantica. Finally, we initiate the study of separations between $\TFNP$ subclasses in Romantica: $\PWPP$ and $\PPA$ do not reduce to $\PLS$ under \emph{public-coin} reductions.
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v1 · 2026-11-04 02:17 · 最新观测版本
TFNP in the Random Oracle Model
We determine which major $\TFNP$ classes are hard in the random oracle model (ROM). Prior work establishes average-case hardness of highly structured subclasses of $\TFNP$ from strong cryptographic assumptions such as LWE and indistinguishability obfuscation. It is not known whether weaker assumptions such as one-way functions imply hardness of any of the standard $\TFNP$ subclasses, even in the worst case. The ROM, where all parties have query access to a random function, lies, morally, between these two regimes: a random function is one-way, but public-key cryptography cannot be built from it in a black-box way. We refer to this setting as \emph{Romantica}. Our first result settles a long-standing open problem: $\PPAD$, the class of total search problems reducible to finding a Nash equilibrium, is hard on average in Romantica. Combined with the known hardness of PLS (local search), this gives average-case hardness of $\EoPL = \PPAD \cap \PLS$. In contrast, we show that $\UEoPL$ can be solved in Romantica with polynomially many oracle queries by a computationally unbounded solver, a sharp distinction between $\EoPL$ and $\UEoPL$. This resolves the black-box hardness landscape of all major $\TFNP$ classes in Romantica. Finally, we initiate the study of separations between $\TFNP$ subclasses in Romantica: $\PWPP$ and $\PPA$ do not reduce to $\PLS$ under \emph{public-coin} reductions.
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