On one-way functions and the average time complexity of almost-optimal compression
Marius Zimand
TL;DR
This work proves an equivalence between the existence of one-way functions and the average-case hardness of almost-optimal compression relative to a polynomial-time samplable distribution. It combines the Impagliazzo–Levin–Luby connection between OWFs and hard distributions with the Bauwens–Zimand compression result, and uses a PRG-based argument to bridge the two sides. The main contribution is a formal theorem: OWFs exist if and only if there is a distribution for which almost-optimal compression is hard on average, with robust variants (infinitely often, different slack) and a precise reduction framework. The result has implications for meta-complexity and cryptography, linking cryptographic primitives to compression-hardness and providing a unified picture of their interdependence.
Abstract
We show that one-way functions exist if and only if there exists an efficient distribution relative to which almost-optimal compression is hard on average. The result is obtained by combining a theorem of Ilango, Ren, and Santhanam and one by Bauwens and Zimand.