Technology Category
Knowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyMachine Learning: Other Foundations of Machine Learning
Computable one-way functions on the reals
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Existence of computable one-way functions on reals
Partial computable one-way functions existence question
Total computable one-way function construction proof
Innovation
Methods, ideas, or system contributions that make the work stand out.
Uses computable one-way functions on reals
Leverages halting problem hardness
Exhibits total computable one-way function
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