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Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory

Authors Emirhan Gürpınar, Andrei Romashchenko

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Emirhan Gürpınar
  • LIRMM, Université de Montpellier, CNRS, France
Andrei Romashchenko
  • LIRMM, Université de Montpellier, CNRS, France


We thank the anonymous reviewers for instructive suggestions and corrections concerning this conference publication as well as the full version of the paper.

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Emirhan Gürpınar and Andrei Romashchenko. Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 44:1-44:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


It is known that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal to the length of the longest shared secret key that two parties can establish via a probabilistic protocol with interaction on a public channel, assuming that the parties hold as their inputs x and y respectively. We determine the worst-case communication complexity of this problem for the setting where the parties can use private sources of random bits. We show that for some x, y the communication complexity of the secret key agreement does not decrease even if the parties have to agree on a secret key the size of which is much smaller than the mutual information between x and y. On the other hand, we provide examples of x, y such that the communication complexity of the protocol declines gradually with the size of the derived secret key. The proof of the main result uses spectral properties of appropriate graphs and the expander mixing lemma as well as various information theoretic techniques.

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
  • Kolmogorov complexity
  • mutual information
  • communication complexity
  • expander mixing lemma
  • finite geometry


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  1. Rudolf Ahlswede and Imre Csiszár. Common randomness in information theory and cryptography. I. Secret sharing. IEEE Transactions on Information Theory, 39(4):1121-1132, 1993. Google Scholar
  2. Noga Alon and Joel H. Spencer. The probabilistic method. John Wiley & Sons, 2004. Google Scholar
  3. Andries E. Brouwer, Sebastian M. Cioabă, Ferdinand Ihringer, and Matt McGinnis. The smallest eigenvalues of hamming graphs, johnson graphs and other distance-regular graphs with classical parameters. Journal of Combinatorial Theory, Series B, 133:88-121, 2018. Google Scholar
  4. Alexei Chernov, Andrej Muchnik, Andrei Romashchenko, Alexander Shen, and Nikolai Vereshchagin. Upper semi-lattice of binary strings with the relation “x is simple conditional to y”. Theoretical Computer Science, 271(1-2):69-95, 2002. Google Scholar
  5. Shai Evra, Konstantin Golubev, and Alexander Lubotzky. Mixing properties and the chromatic number of ramanujan complexes. International Mathematics Research Notices, 2015(22):11520-11548, 2015. Google Scholar
  6. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society, 43(4):439-561, 2006. Google Scholar
  7. Tarik Kaced, Andrei Romashchenko, and Nikolai Vereshchagin. A conditional information inequality and its combinatorial applications. IEEE Transactions on Information Theory, 64(5):3610-3615, 2018. Google Scholar
  8. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 2006. Google Scholar
  9. Ming Li and Paul Vitányi. An introduction to Kolmogorov complexity and its applications. Springer, 4 edition, 2019. Google Scholar
  10. Jingbo Liu, Paul Cuff, and Sergio Verdú. Secret key generation with limited interaction. IEEE Transactions on Information Theory, 63(11):7358-7381, 2017. Google Scholar
  11. Xiaogang Liu and Sanming Zhou. Eigenvalues of cayley graphs. arXiv preprint arXiv:1809.09829, 2018. Google Scholar
  12. Ueli M. Maurer. Secret key agreement by public discussion from common information. IEEE transactions on information theory, 39(3):733-742, 1993. Google Scholar
  13. Archie Medrano, Perla Myers, Harold M. Stark, and Audrey Terras. Finite analogues of euclidean space. Journal of Computational and Applied Mathematics, 68(1-2):221-238, 1996. Google Scholar
  14. Andrej A. Muchnik. Conditional complexity and codes. Theoretical Computer Science, 271(1-2):97-109, 2002. Google Scholar
  15. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 3-13. IEEE, 2000. Google Scholar
  16. Andrei Romashchenko and Marius Zimand. An operational characterization of mutual information in algorithmic information theory. Journal of the ACM (JACM), 66(5):1-42, 2019. Google Scholar
  17. Andrei E. Romashchenko. Pairs of words with nonmaterializable mutual information. Problems of Information Transmission, 36(1), 2000. Google Scholar
  18. Alexander Shen, Vladimir Uspensky, and Nikolay Vereshchagin. Kolmogorov complexity and algorithmic randomness, volume 220. American Mathematical Soc., 2017. Google Scholar
  19. Madhu Sudan, Himanshu Tyagi, and Shun Watanabe. Communication for generating correlation: A unifying survey. IEEE Transactions on Information Theory, 66(1):5-37, 2019. Google Scholar
  20. Himanshu Tyagi. Common information and secret key capacity. IEEE Transactions on Information Theory, 59(9):5627-5640, 2013. Google Scholar
  21. Le Anh Vinh. The Szemerédi-Trotter type theorem and the sum-product estimate in finite fields. European Journal of Combinatorics, 32(8):1177-1181, 2011. Google Scholar
  22. Alexander K. Zvonkin and Leonid A. Levin. The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms. Russian Mathematical Surveys, 25(6):83, 1970. Google Scholar
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