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Computation over the Noisy Broadcast Channel with Malicious Parties

Authors Klim Efremenko, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena



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Author Details

Klim Efremenko
  • Ben Gurion University of the Negev, Beer Sheva, Israel
Gillat Kol
  • Princeton University, NJ, USA
Dmitry Paramonov
  • Princeton University, NJ, USA
Raghuvansh R. Saxena
  • Princeton University, NJ, USA

Acknowledgements

We thank Huacheng Yu for helpful discussions.

Cite AsGet BibTex

Klim Efremenko, Gillat Kol, Dmitry Paramonov, and Raghuvansh R. Saxena. Computation over the Noisy Broadcast Channel with Malicious Parties. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 82:1-82:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ITCS.2021.82

Abstract

We study the n-party noisy broadcast channel with a constant fraction of malicious parties. Specifically, we assume that each non-malicious party holds an input bit, and communicates with the others in order to learn the input bits of all non-malicious parties. In each communication round, one of the parties broadcasts a bit to all other parties, and the bit received by each party is flipped with a fixed constant probability (independently for each recipient). How many rounds are needed? Assuming there are no malicious parties, Gallager gave an 𝒪(n log log n)-round protocol for the above problem, which was later shown to be optimal. This protocol, however, inherently breaks down in the presence of malicious parties. We present a novel n ⋅ 𝒪̃(√{log n})-round protocol, that solves this problem even when almost half of the parties are malicious. Our protocol uses a new type of error correcting code, which we call a locality sensitive code and which may be of independent interest. Roughly speaking, these codes map "close" messages to "close" codewords, while messages that are not close are mapped to codewords that are very far apart. We view our result as a first step towards a theory of property preserving interactive coding, i.e., interactive codes that preserve useful properties of the protocol being encoded. In our case, the naive protocol over the noiseless broadcast channel, where all the parties broadcast their input bit and output all the bits received, works even in the presence of malicious parties. Our simulation of this protocol, unlike Gallager’s, preserves this property of the original protocol.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • Broadcast Network
  • Malicious Parties
  • Communication Complexity

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References

  1. Noga Alon, Mark Braverman, Klim Efremenko, Ran Gelles, and Bernhard Haeupler. Reliable communication over highly connected noisy networks. In Symposium on Principles of Distributed Computing (DISC), pages 165-173. ACM, 2016. Google Scholar
  2. Alexandr Andoni and Piotr Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. Commun. ACM, 51(1):117-122, 2008. URL: https://doi.org/10.1145/1327452.1327494.
  3. Alexandr Andoni, Piotr Indyk, and Ilya P. Razenshteyn. Approximate nearest neighbor search in high dimensions. CoRR, abs/1806.09823, 2018. URL: http://arxiv.org/abs/1806.09823.
  4. Mark Braverman, Klim Efremenko, Ran Gelles, and Bernhard Haeupler. Constant-rate coding for multiparty interactive communication is impossible. In Symposium on Theory of Computing (STOC), pages 999-1010. ACM, 2016. Google Scholar
  5. Klim Efremenko, Gillat Kol, and Raghuvansh Saxena. Interactive coding over the noisy broadcast channel. In Symposium on Theory of Computing (STOC), pages 507-520. ACM, 2018. Google Scholar
  6. Uriel Feige and Joe Kilian. Finding OR in a noisy broadcast network. Information Processing Letters, 73(1-2):69-75, 2000. Google Scholar
  7. Robert G. Gallager. Finding parity in a simple broadcast network. IEEE Transactions on Information Theory, 34(2):176-180, 1988. Google Scholar
  8. Abbas El Gamal. Open problems presented at the 1984 workshop on specific problems in communication and computation sponsored by bell communication research. "Open Problems in Communication and Computation", by Thomas M. Cover and B. Gopinath (editors). Springer-Verlag, 1987. Google Scholar
  9. Ran Gelles, Amit Sahai, and Akshay Wadia. Private interactive communication across an adversarial channel. IEEE Trans. Inf. Theory, 61(12):6860-6875, 2015. URL: https://doi.org/10.1109/TIT.2015.2483323.
  10. Navin Goyal, Guy Kindler, and Michael Saks. Lower bounds for the noisy broadcast problem. SIAM Journal on Computing, 37(6):1806-1841, 2008. Google Scholar
  11. Kokouvi Hounkanli, Avery Miller, and Andrzej Pelc. Global synchronization and consensus using beeps in a fault-prone multiple access channel. Theoretical Computer Science, 806:567-576, 2020. Google Scholar
  12. Eyal Kushilevitz and Yishay Mansour. An ω(dlog (n/d)) lower bound for broadcast in radio networks. SIAM J. Comput., 27(3):702-712, 1998. Google Scholar
  13. Ilan Newman. Computing in fault tolerance broadcast networks. In Computational Complexity Conference (CCC), pages 113-122, 2004. Google Scholar
  14. Alessandro Panconesi and Aravind Srinivasan. Randomized distributed edge coloring via an extension of the chernoff-hoeffding bounds. SIAM J. Comput., 26(2):350-368, 1997. URL: https://doi.org/10.1137/S0097539793250767.
  15. Sridhar Rajagopalan and Leonard J. Schulman. A coding theorem for distributed computation. In Symposium on the Theory of Computing (STOC), pages 790-799, 1994. Google Scholar
  16. Leonard J Schulman. Communication on noisy channels: A coding theorem for computation. In Foundations of Computer Science (FOCS), pages 724-733. IEEE, 1992. Google Scholar
  17. Leonard J Schulman. Deterministic coding for interactive communication. In Symposium on Theory of computing (STOC), pages 747-756. ACM, 1993. Google Scholar
  18. Leonard J. Schulman. Coding for interactive communication. IEEE Transactions on Information Theory, 42(6):1745-1756, 1996. Google Scholar
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