### Average-Case Fine-Grained Hardness

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

##### Abstract

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. We prove our hardness results in each case by showing fine-grained 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 $n^{2-o(1)}$ time to compute on average, and that of APSP gives us a function that requires $n^{3-o(1)}$ time. Using the same techniques we also obtain a conditional average-case time hierarchy of functions. 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) heuristically falsifiable in a sense similar to that of (Naor, CRYPTO '03).

Available format(s)
Category
Foundations
Publication info
Published elsewhere. Minor revision.Symposium on the Theory of Computing 2017
Keywords
Proofs of workFine-GrainedAverage-CaseHeuristic Falsifiability
Contact author(s)
msabin @ berkeley edu
marshallball @ gmail com
alon rosen @ idc ac il
prashantv91 @ gmail com
History
Short URL
https://ia.cr/2017/202

CC BY

BibTeX

@misc{cryptoeprint:2017/202,
author = {Marshall Ball and Alon Rosen and Manuel Sabin and Prashant Nalini Vasudevan},
title = {Average-Case Fine-Grained Hardness},
howpublished = {Cryptology ePrint Archive, Paper 2017/202},
year = {2017},
note = {\url{https://eprint.iacr.org/2017/202}},
url = {https://eprint.iacr.org/2017/202}
}

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