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One-Way Functions and Polynomial-Time Dimension

Authors Satyadev Nandakumar , Subin Pulari , Akhil S , Suronjona Sarma

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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Complexity classes
  • Theory of computation → Cryptographic primitives
Keywords
  • Polynomial-time dimension
  • One-way functions
  • Resource bounded randomness
  • Kolmogorov complexity
  • Polynomial-time martingales

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Abstract

The theory of randomness in computation has developed around several viewpoints on randomness and information, including statistical tests, compression schemes, betting strategies, and efficiently samplable sources. At the level of computability, several of these viewpoints turn out to be equivalent. A fundamental question is how robust these viewpoints remain when the underlying algorithms are subject to computational resource bounds. This paper studies this question for two polynomial-time notions of information density for infinite binary sequences. Polynomial-time dimension, denoted dim_P, quantifies information density using polynomial-time betting strategies called s-gales. Polynomial-time Kolmogorov complexity rate, denoted 𝒦_poly, gives a compression-based notion of polynomial-time information density using polynomial-time descriptions. Hitchcock and Vinodchandran (CCC 2004) showed that dim_P(X) ≥ 𝒦_poly(X) for every sequence X, and asked whether equality always holds. This question was later also posed by Stull. 
Our main result proves a duality between the non-robustness of these polynomial-time information-density notions and the existence of one-way functions. Assuming one-way functions exist, we construct a polynomial-time samplable distribution over infinite sequences such that, with probability 1, the sampled sequence X satisfies dim_P(X) > 𝒦_poly(X) by a uniform gap. Conversely, we show that if some polynomial-time samplable distribution yields such an almost-sure uniform separation, then infinitely-often one-way functions exist. Thus, separations between these two polynomial-time information-density notions over efficiently samplable sources lie at the same frontier as one-way functions, with the reverse direction yielding infinitely-often one-way functions. This duality gives a negative answer, assuming one-way functions, to the open question posed by Hitchcock, Vinodchandran, and Stull. Furthermore, we show that there are individual sequences witnessing separations between dim_P and 𝒦_poly, and that the gap can be made arbitrarily close to 1. We also establish analogous bounds for strong polynomial-time dimension and asymptotic upper polynomial-time Kolmogorov complexity rates. The main technical challenge is to connect finite-length pseudorandomness assumptions with asymptotic information-density measures on infinite sequences. Our proof addresses this by developing several new constructions and arguments involving probabilistic tools such as the Borel-Cantelli lemma, Kolmogorov’s inequality for martingales, and techniques for approximating probabilities of efficiently samplable distributions. The work shows that the question of non-robustness for polynomial-time information-density notions, which is prima facie different from standard questions about randomness, pseudorandomness, and cryptography, is in fact intimately related to the existence of one-way functions.

Cite As Get BibTex

Satyadev Nandakumar, Subin Pulari, Akhil S, and Suronjona Sarma. One-Way Functions and Polynomial-Time Dimension. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 44:1-44:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.44

Author Details

Satyadev Nandakumar
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India
Subin Pulari
  • National Research University Higher School of Economics Moscow, Russia
Akhil S
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India
Suronjona Sarma
  • Department of Computer Science and Engineering, Indian Institute of Technology Kanpur, India

Funding

  • Pulari, Subin: Supported by the Basic Research Program at HSE University and this paper was prepared in the framework of this program.
  • Sarma, Suronjona: Supported by the Visvesvaraya PhD Scheme Scholarship. This research work was partially supported by the Research-I Foundation of the Department of Computer Science and Engineering, IIT Kanpur for conference travel.

Acknowledgements

The authors would like to thank CS Bhargav, Laurent Bienvenu, John Hitchcock, Elvira Mayordomo, Jaikumar Radhakrishnan, Mahesh Sreekumar Rajasree, Bruno Bauwens and Marius Zimand for helpful comments and discussions.

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