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**Published in:** LIPIcs, Volume 289, 41st International Symposium on Theoretical Aspects of Computer Science (STACS 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.

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)

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@InProceedings{caillatgrenier_et_al:LIPIcs.STACS.2024.22, author = {Caillat-Grenier, Geoffroy and Romashchenko, Andrei}, title = {{Spectral Approach to the Communication Complexity of Multi-Party Key Agreement}}, booktitle = {41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024)}, pages = {22:1--22:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-311-9}, ISSN = {1868-8969}, year = {2024}, volume = {289}, editor = {Beyersdorff, Olaf and Kant\'{e}, Mamadou Moustapha and Kupferman, Orna and Lokshtanov, Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2024.22}, URN = {urn:nbn:de:0030-drops-197323}, doi = {10.4230/LIPIcs.STACS.2024.22}, annote = {Keywords: communication complexity, Kolmogorov complexity, information-theoretic cryptography, multiparty secret key agreement, expander mixing lemma, information inequalities} }

Document

**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 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.

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)

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@InProceedings{gurpinar_et_al:LIPIcs.MFCS.2020.44, author = {G\"{u}rp{\i}nar, Emirhan and Romashchenko, Andrei}, title = {{Communication Complexity of the Secret Key Agreement in Algorithmic Information Theory}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {44:1--44:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.44}, URN = {urn:nbn:de:0030-drops-127102}, doi = {10.4230/LIPIcs.MFCS.2020.44}, annote = {Keywords: Kolmogorov complexity, mutual information, communication complexity, expander mixing lemma, finite geometry} }

Document

**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

We suggest necessary conditions of soficness of multidimensional shifts formulated in terms of resource-bounded Kolmogorov complexity. Using this technique we provide examples of effective and non-sofic shifts on Z^2 with very low block complexity: the number of globally admissible patterns of size n x n grows only as a polynomial in n.

Julien Destombes and Andrei Romashchenko. Resource-Bounded Kolmogorov Complexity Provides an Obstacle to Soficness of Multidimensional Shifts. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 23:1-23:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{destombes_et_al:LIPIcs.STACS.2019.23, author = {Destombes, Julien and Romashchenko, Andrei}, title = {{Resource-Bounded Kolmogorov Complexity Provides an Obstacle to Soficness of Multidimensional Shifts}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {23:1--23:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.23}, URN = {urn:nbn:de:0030-drops-102624}, doi = {10.4230/LIPIcs.STACS.2019.23}, annote = {Keywords: Sofic shifts, Block complexity, Kolmogorov complexity} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We show that the mutual information, in the sense of Kolmogorov complexity, of any pair of strings x and y is equal, up to logarithmic precision, to the length of the longest shared secret key that two parties, one having x and the complexity profile of the pair and the other one having y and the complexity profile of the pair, can establish via a probabilistic protocol with interaction on a public channel. For l > 2, the longest shared secret that can be established from a tuple of strings (x_1, . . . , x_l) by l parties, each one having one component of the tuple and the complexity profile of the tuple, is equal, up to logarithmic precision, to the complexity of the tuple minus the minimum communication necessary for distributing the tuple to all parties. We establish the communication complexity of secret key agreement protocols that produce a secret key of maximal length, for protocols with public randomness. We also show that if the communication complexity drops below the established threshold then only very short secret keys can be obtained.

Andrei Romashchenko and Marius Zimand. An Operational Characterization of Mutual Information in Algorithmic Information Theory. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 95:1-95:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{romashchenko_et_al:LIPIcs.ICALP.2018.95, author = {Romashchenko, Andrei and Zimand, Marius}, title = {{An Operational Characterization of Mutual Information in Algorithmic Information Theory}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {95:1--95:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.95}, URN = {urn:nbn:de:0030-drops-90998}, doi = {10.4230/LIPIcs.ICALP.2018.95}, annote = {Keywords: Kolmogorov complexity, mutual information, communication complexity, secret key agreement} }

Document

**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in Z^d. The minimal shifts are those shifts in which all configurations contain exactly the same patterns. Two classes of shifts play a prominent role in symbolic dynamics, in language theory and in the theory of computability: the shifts of finite type (obtained by forbidding a finite number of finite patterns) and the effective shifts (obtained by forbidding a computably enumerable set of finite patterns).
We prove that every effective minimal shift can be represented as a factor of a projective subdynamics on a minimal shift of finite type in a bigger (by 1) dimension. This result transfers to the class of minimal shifts a theorem by M.Hochman known for the class of all effective shifts and thus answers an open question by E. Jeandel. We prove a similar result for quasiperiodic shifts and also show that there exists a quasiperiodic shift of finite type for which Kolmogorov complexity of all patterns of size n\times n is \Omega(n).

Bruno Durand and Andrei Romashchenko. On the Expressive Power of Quasiperiodic SFT. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 5:1-5:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{durand_et_al:LIPIcs.MFCS.2017.5, author = {Durand, Bruno and Romashchenko, Andrei}, title = {{On the Expressive Power of Quasiperiodic SFT}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {5:1--5:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.5}, URN = {urn:nbn:de:0030-drops-80985}, doi = {10.4230/LIPIcs.MFCS.2017.5}, annote = {Keywords: minimal SFT, tilings, quasiperiodicityIn this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations} }

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