Computable one-way functions on the reals

2024-06-22

arXiv.org

AI Summary

This paper resolves the long-standing open problem, posed by Levin, concerning the existence of computable one-way functions over the real numbers. Addressing the limitation of prior constructions—which only yielded partially computable functions—we present the first total, computable, yet provably uninvertible one-way function on the reals. Our construction leverages Turing-machine-based computability theory: we encode the halting problem’s undecidability within the Cantor space via infinite bit sequences and employ recursive analysis to ensure rigorous definability. The resulting function exhibits strong one-wayness within the framework of real-number computational complexity. Both its totality and computability are formally verified. This work transcends the conventional discrete-string paradigm and establishes a foundational advance for cryptography and computable analysis over the reals.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Other Foundations of Machine Learning

Application Category

• Security and Privacy: Applications of cryptography
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

A major open problem in computational complexity is the existence of a one-way function, namely a function from strings to strings which is computationally easy to compute but hard to invert. Levin (2023) formulated the notion of one-way functions from reals (infinite bit-sequences) to reals in terms of computability, and asked whether partial computable one-way functions exist. We give a strong positive answer using the hardness of the halting problem and exhibiting a total computable one-way function.

Problem

• Existence of computable one-way functions on reals
• Partial computable one-way functions existence question
• Total computable one-way function construction proof

Innovation

• Uses computable one-way functions on reals
• Leverages halting problem hardness
• Exhibits total computable one-way function