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Algebraic Barriers to Halving Algorithmic Information Quantities in Correlated Strings

Author Andrei Romashchenko

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Information theory
  • Theory of computation → Communication complexity
  • Security and privacy → Information-theoretic techniques
  • Mathematics of computing → Combinatoric problems
Keywords
  • Kolmogorov complexity
  • algorithmic information theory
  • communication complexity
  • discrete geometry

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Abstract

We study the possibility of scaling down algorithmic information quantities in tuples of correlated strings. In particular, we address a question raised by Alexander Shen: whether, for any triple of strings (a, b, c), there exists a string z such that each conditional Kolmogorov complexity C(a|z), C(b|z), C(c|z) is approximately half of the corresponding unconditional Kolmogorov complexity. We provide a negative answer to this question by constructing a triple (a, b, c) for which no such string z exists. Our construction is based on combinatorial properties of incidences in finite projective planes and relies on recent bounds for point-line incidences over prime fields, obtained using tools from additive combinatorics and algebraic methods, notably results by Bourgain-Katz-Tao and Stevens-De Zeeuw. As an application, we show that this impossibility yields lower bounds on the communication complexity of secret key agreement protocols in certain settings. These results reveal algebraic obstructions to efficient information exchange and highlight a separation in information-theoretic behavior between fields with and without proper subfields.

Cite As Get BibTex

Andrei Romashchenko. Algebraic Barriers to Halving Algorithmic Information Quantities in Correlated Strings. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 84:1-84:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.84

Author Details

Andrei Romashchenko
  • LIRMM Univ Montpellier & CNRS, France

Funding

The work was supported in part by the ANR project FLITTLA ANR-21-CE48-0023.

Acknowledgements

The author sincerely thanks the anonymous reviewers for their careful reading and insightful suggestions, which helped improve the clarity and presentation of the paper.

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