Fine-Grained Complexity in a World without Cryptography
Josh Alman, Columbia University
Yizhi Huang, Columbia University
Kevin Yeo, Columbia University, Google (United States)
Abstract
The study of fine-grained cryptography has proliferated in recent years due to its allure of potentially relying on weaker assumptions compared to standard cryptography. As fine-grained cryptography only requires polynomial gaps between the adversary and honest parties, it seems plausible to build primitives relying upon popular hardness assumptions about problems in such as - or --. The ultimate hope is that fine-grained cryptography could still be viable even if all current cryptographic assumptions are false, such as if or if we live in Pessiland where one-way functions do not exist.
In our work, we consider whether this approach is viable by studying fine-grained complexity when all standard cryptographic assumptions are false. As our main result, we show that many popular fine-grained complexity problems are easy to solve in the average-case when one-way functions do not exist. In other words, many candidate hardness assumptions for building fine-grained cryptography are no longer options in Pessiland. As an example, we prove that the average-case - and -- conjectures are false for sufficiently large constant when no one-way functions exist. The average-case -- assumption was used to build fine-grained key-exchange by Lavigne et al. [CRYPTO'19]. One can also view the contrapositive of our result as providing an explicit construction of one-way functions assuming average-case hardness of - or -- for all constant .
We also show that barriers for reductions in fine-grained complexity may be explained by problems in cryptography. First, we show that finding faster algorithms for computing discrete logarithms is equivalent to designing average-case equivalence between - and - (an extension of - to cyclic groups). In particular, finding such a reduction from - to - could potentially lead to breakthrough algorithms for the discrete logarithm, factoring, RSA and quadratic residuosity problems. Finally, we show that discrete logarithms with preprocessing may be reduced to the -- problem, and we present faster algorithms for average-case -- and --.
@misc{cryptoeprint:2025/324,
author = {Josh Alman and Yizhi Huang and Kevin Yeo},
title = {Fine-Grained Complexity in a World without Cryptography},
howpublished = {Cryptology {ePrint} Archive, Paper 2025/324},
year = {2025},
url = {https://eprint.iacr.org/2025/324}
}
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