Spectral Approach to the Communication Complexity of Multi-Party Key Agreement

Authors Geoffroy Caillat-Grenier , Andrei Romashchenko

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Geoffroy Caillat-Grenier
  • LIRMM, University of Montpellier, CNRS, Montpellier, France
Andrei Romashchenko
  • LIRMM, University of Montpellier, CNRS, Montpellier, France


We thank the anonymous referees for valuable and helpful comments.

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Geoffroy Caillat-Grenier and Andrei Romashchenko. Spectral Approach to the Communication Complexity of Multi-Party Key Agreement. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We propose a linear algebraic method, rooted in the spectral properties of graphs, that can be used to prove lower bounds in communication complexity. Our proof technique effectively marries spectral bounds with information-theoretic inequalities. The key insight is the observation that, in specific settings, even when data sets X and Y are closely correlated and have high mutual information, the owner of X cannot convey a reasonably short message that maintains substantial mutual information with Y. In essence, from the perspective of the owner of Y, any sufficiently brief message m = m(X) would appear nearly indistinguishable from a random bit sequence. We employ this argument in several problems of communication complexity. Our main result concerns cryptographic protocols. We establish a lower bound for communication complexity of multi-party secret key agreement with unconditional, i.e., information-theoretic security. Specifically, for one-round protocols (simultaneous messages model) of secret key agreement with three participants we obtain an asymptotically tight lower bound. This bound implies optimality of the previously known omniscience communication protocol (this result applies to a non-interactive secret key agreement with three parties and input data sets with an arbitrary symmetric information profile). We consider communication problems in one-shot scenarios when the parties inputs are not produced by any i.i.d. sources, and there are no ergodicity assumptions on the input data. In this setting, we found it natural to present our results using the framework of Kolmogorov complexity.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Information theory
  • Theory of computation → Communication complexity
  • Security and privacy → Information-theoretic techniques
  • Theory of computation → Expander graphs and randomness extractors
  • communication complexity
  • Kolmogorov complexity
  • information-theoretic cryptography
  • multiparty secret key agreement
  • expander mixing lemma
  • information inequalities


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