One-Way Functions and Polynomial Time Dimension

TL;DR

The paper investigates whether a fundamental notion of information density in infinite sequences, polynomial-time dimension ${\mathrm{cdim}}_\mathrm{P}$, is robust to time-bounded Kolmogorov-rate ${\mathcal K}_{\mathrm{poly}}$. It proves a duality: the existence of one-way functions (OWFs) yields uniform dimension gaps between ${\mathrm{cdim}}_\mathrm{P}$ and ${\mathcal K}_{\mathrm{poly}}$ on sequences drawn from short-seed polynomial-time samplable distributions, and such distributions imply infinitely-often OWFs. Consequently, under OWFs, there exist individual sequences with ${\mathrm{cdim}}_\mathrm{P}(X) > {\mathcal K}_{\mathrm{poly}}(X)$, with gaps arbitrarily close to 1, and similar bounds hold for strong versions of the dimension. This result answers a long-standing open question about the robustness of polynomial-time dimension and connects meta-complexity notions to cryptographic primitives, showing that dimension-based randomness is non-robust in the presence of OWFs. The techniques combine extending PRGs to infinite sequences, gale manipulation, a distinguisher against PRGs, Borel-Cantelli arguments, Kolmogorov inequalities, and universal extrapolation to tie cryptographic hardness to information-density gaps. The findings imply that if ${\mathrm{cdim}}_\mathrm{P}$ and ${\mathcal K}_{\mathrm{poly}}$ were always equal, OWFs would not exist; conversely, OWFs imply significant separations, with potential implications for the foundations of randomness and complexity theory.

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.

Figures (1)

• Figure 1: Illustration of the construction of $g(X)$. The first $s = 2^m$ bits of the $g(X)$ are 0. Thereafter, for every for every $n>m$, the block $X[s.2^{{n-1}}, s.2^{n}-1]$ is mapped to $G_{2^{n-1}}(X[s.2^{{n-1}}, s.2^{n}-1])$ in the output string $g(X)$.

Theorems & Definitions (87)

• Definition : Polynomial-time dimension LutzDimension2003
• Definition : dercconferencedercjournal
• theorem 1: Mayordomo Mayordomo02 , Lutz diss
• theorem 2: dercconferencedercjournal
• theorem 3: dercconferencedercjournal
• Definition : Polynomial-time strong dimension athreya2007effectivestullsurvey
• Definition : dercconferencedercjournal
• theorem 4: dercconferencedercjournal
• Definition 1
• Definition 2