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Robustness of average-case meta-complexity via pseudorandomness

Rahul Ilango,Hanlin Ren,R. Santhanam

Published 2022 in Symposium on the Theory of Computing

ABSTRACT

We show broad equivalences in the average-case complexity of many different meta-complexity problems, including Kolmogorov complexity, time-bounded Kolmogorov complexity, and the Minimum Circuit Size Problem. These results hold for a wide range of parameters (various thresholds, approximation gaps, weak or strong average-case hardness, etc.) and complexity notions, showing the theory of meta-complexity is very *robust* in the average-case setting. Our results are shown by establishing new and generic connections between meta-complexity and the theory of pseudorandomness and one-way functions. Using these connections, we give the first unconditional characterization of one-way functions based on the average-case hardness of the Minimum Circuit Size Problem. We also give a surprising and clean characterization of one-way functions based on the average-case hardness of (the worst-case uncomputable) Kolmogorov complexity. Moreover, the latter is the first characterization of one-way functions based on the average-case hardness of a fixed problem on *any* samplable distribution. We give various applications of these results to the foundations of cryptography and the theory of meta-complexity. For example, we show that the average-case hardness of deciding k-SAT or Clique on any samplable distribution of high enough entropy implies the existence of one-way functions. We also use our results to unconditionally solve various meta-complexity problems in CZK (computational zero-knowledge) on average, and give implications of our results for the classic question of proving NP-hardness for the Minimum Circuit Size Problem.

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