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Average-case fine-grained hardness

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Prashant Nalini VasudevanAuthors Info & Claims

STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

Pages 483 - 496

https://doi.org/10.1145/3055399.3055466

Published: 19 June 2017 Publication History

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.

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).

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.

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Index Terms

Average-case fine-grained hardness
1. Theory of computation
  1. Computational complexity and cryptography

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cover image ACM Conferences

STOC 2017: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

June 2017

1268 pages

ISBN:9781450345286

DOI:10.1145/3055399

General Chairs:
Hamed Hatami
McGill University
,
Pierre McKenzie
Université de Montréal
,
Program Chair:
Valerie King
University of Victoria

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Published: 19 June 2017

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https://doi.org/10.1007/978-3-031-68388-6_14
Bauer BCouteau GSadeghi E(2024)Fine-Grained Non-interactive Key Exchange, RevisitedAdvances in Cryptology – CRYPTO 202410.1007/978-3-031-68379-4_9(286-312)Online publication date: 16-Aug-2024
https://doi.org/10.1007/978-3-031-68379-4_9
Ball MGaray JHall PKiayias APanagiotakos G(2024)Towards Permissionless Consensus in the Standard Model via Fine-Grained ComplexityAdvances in Cryptology – CRYPTO 202410.1007/978-3-031-68379-4_4(113-146)Online publication date: 16-Aug-2024
https://doi.org/10.1007/978-3-031-68379-4_4
Agrawal SSaha SSchwartzbach NVanukuri AVasudevan P(2024)k-SUM in the Sparse Regime: Complexity and ApplicationsAdvances in Cryptology – CRYPTO 202410.1007/978-3-031-68379-4_10(315-351)Online publication date: 16-Aug-2024
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