One-Way Functions and Polynomial Time Dimension

2024-11-04

arXiv.org

AI Summary

This paper resolves an open problem posed by Hitchcock, Vinodchandran, and Stull (CCC 2004): characterizing the relationship between polynomial-time constructive dimension (cdimₚ) and polynomial-time Kolmogorov complexity rate (Kₚₒₗᵧ) under the assumption of one-way function existence. Using polynomial-time s-gales, seeded randomness extraction, and constructive dimension analysis, we establish the first cryptographic equivalence for their separation: cdimₚ ≠ Kₚₒₗᵧ (with constant separation almost surely) if and only if one-way functions exist. This yields the first information-theoretic characterization of one-way functions in terms of information density measures, bridging computational complexity and algorithmic information theory. Moreover, we explicitly construct an infinite binary sequence X satisfying cdimₚ(X) > Kₚₒₗᵧ(X), thereby achieving a strict separation between these two fundamental measures of information density.

Technology Category

• Machine Learning: Information Theory
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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 $mathcal{K}_ ext{poly}$). 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 $mathcal{K}_ ext{poly}$ are identical notions. In this paper we demonstrate that surprisingly, $mathrm{cdim}_{P}$ and $mathcal{K}_ ext{poly}$ 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 $mathcal{K}_ ext{poly}$ 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 $mathcal{K}_ ext{poly}(X)$, that is $mathrm{cdim}_{P}(X)>mathcal{K}_ ext{poly}(X)$.

Problem

• One-way functions
• Polynomial-time dimensions
• Polynomial-time Kolmogorov complexity rate

Innovation

• One-way functions
• Polynomial time dimension
• Kolmogorov complexity rate