One-Way Functions and Polynomial Time Dimension

Abstract

This paper demonstrates a duality between the non-robustness of polynomial time dimension and the existence of one-way functions. Polynomial-time dimension (denoted $\mathrm{cdim}_\mathrm{P}$) quantifies the density of information of infinite sequences using polynomial time betting algorithms called $s$-gales. An alternate quantification of the notion of polynomial time density of information is using polynomial-time Kolmogorov complexity rate (denoted $\mathcal{K}_\text{poly}$). Hitchcock and Vinodchandran (CCC 2004) showed that $\mathrm{cdim}_\mathrm{P}$ is always greater than or equal to $\mathcal{K}_\text{poly}$. We first show that if one-way functions exist then there exists a polynomial-time samplable distribution with respect to which $\mathrm{cdim}_\mathrm{P}$ and $\mathcal{K}_\text{poly}$ are separated by a uniform gap with probability $1$. Conversely, we show that if there exists such a polynomial-time samplable distribution, then (infinitely-often) one-way functions exist. Using our main results, we solve a long standing open problem posed by Hitchcock and Vinodchandran (CCC 2004) and Stull under the assumption that one-way functions exist. We demonstrate that if one-way functions exist, then there are individual sequences $X$ whose poly-time dimension strictly exceeds $\mathcal{K}_\text{poly}(X)$, that is $\mathrm{cdim}_\mathrm{P}(X) > \mathcal{K}_\text{poly}(X)$. Further, we show that the gap between these quantities can be made as large as possible (i.e. close to 1). We also establish similar bounds for strong poly-time dimension versus asymptotic upper Kolmogorov complexity rates.