One-Way Functions and Polynomial Time Dimension

Polynomial-time dimension (denoted $\mathrm{cdim}_{P}$) 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 $\mathcal{K}_\text{poly}$). The corresponding unbounded notions, namely, the constructive dimension and unbounded Kolmogorov complexity rates are equal for every sequence. Analogous notions are equal even at finite-state level. In view of this, it is reasonable to conjecture that $\mathrm{cdim}_{P}$ and $\mathcal{K}_\text{poly}$ are identical notions. In this paper we demonstrate that surprisingly, $\mathrm{cdim}_{P}$ and $\mathcal{K}_\text{poly}$ are distinct measures of information density if and only if one-way functions exist. We consider polynomial time samplable distributions over $\Sigma^\infty$ that uses short seeds to sample a finite string $w \in \Sigma^n$. We establish the following results. We first show that if one-way functions exist then there exist a polynomial time samplable distribution with respect to which $\mathrm{cdim}_{P}$ 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. Hence, we provide a new information theoretic characterisation of the existence of one-way functions. Using this new characterization, we solve an open problem posed by Hitchcock and Vinodchandran (CCC 2004) and Stull \cite{stullsurvey}. 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 $\mathrm{cdim}_{P}(X) > \mathcal{K}_\text{poly}(X)$.