Average-case fine-grained hardness

Marshall Ball, Alon Rosen, Manuel Sabin, Prashant Nalini Vasudevan

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We present functions that can be computed in some fixed polynomial time but are hard on average for any algorithm that runs in slightly smaller time, assuming widely-conjectured worst-case hardness for problems from the study of fine-grained complexity. Unconditional constructions of such functions are known from before (Goldmann et al., IPL '94), but these have been canonical functions that have not found further use, while our functions are closely related to well-studied problems and have considerable algebraic structure. Based on the average-case hardness and structural properties of our functions, we outline the construction of a Proof of Work scheme and discuss possible approaches to constructing fine-grained One-Way Functions. We also show how our reductions make conjectures regarding the worst-case hardness of the problems we reduce from (and consequently the Strong Exponential Time Hypothesis) heuris-tically falsifiable in a sense similar to that of (Naor, CRYPTO '03). We prove our hardness results in each case by showing finegrained reductions from solving one of three problems - namely, Orthogonal Vectors (OV), 3SUM, and All-Pairs Shortest Paths (APSP) -in the worst case to computing our function correctly on a uniformly random input. The conjectured hardness of OV and 3SUM then gives us functions that require n2-o(1) time to compute on average, and that of APSP gives us a function that requires n3-o(1) time. Using the same techniques we also obtain a conditional average-case time hierarchy of functions.

Original languageEnglish (US)
Title of host publicationSTOC 2017 - Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing
EditorsPierre McKenzie, Valerie King, Hamed Hatami
PublisherAssociation for Computing Machinery
Number of pages14
ISBN (Electronic)9781450345286
StatePublished - Jun 19 2017
Event49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017 - Montreal, Canada
Duration: Jun 19 2017Jun 23 2017

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
VolumePart F128415
ISSN (Print)0737-8017


Other49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017


  • Average-Case complexity
  • Cryptography
  • Fine-grained complexity
  • Heuristic falsifiability
  • Proofs of work
  • Worst-case to average-case reduction

ASJC Scopus subject areas

  • Software


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