Mohsen Ghaffari
Bernhard Haeupler
Madhu Sudan
ABSTRACT
We consider the task of interactive communication in the presence of adversarial errors and present tight bounds on the tolerable error-rates in a number of different settings.
Most significantly, we explore adaptive interactive communication where the communicating parties decide who should speak next based on the history of the interaction. In particular, this decision can depend on estimates of the amount of errors that have occurred so far. Braverman and Rao [STOC'11] show that non-adaptively one can code for any constant error rate below 1/4 but not more. They asked whether this bound could be improved using adaptivity. We answer this open question in the affirmative (with a slightly different collection of resources): Our adaptive coding scheme tolerates any error rate below 2/7 and we show that tolerating a higher error rate is impossible. We also show that in the setting of Franklin et al. [CRYPTO'13], where parties share randomness not known to the adversary, adaptivity increases the tolerable error rate from 1/2 to 2/3. For list-decodable interactive communications, where each party outputs a constant size list of possible outcomes, the tight tolerable error rate is 1/2.
Our negative results hold even if the communication and computation are unbounded, whereas for our positive results communication and computations are polynomially bounded. Most prior work considered coding schemes with linear communication bounds, while allowing unbounded computations. We argue that studying tolerable error rates in this relaxed context helps to identify a setting's intrinsic optimal error rate. We set forward a strong working hypothesis which stipulates that for any setting the maximum tolerable error rate is independent of many computational and communication complexity measures. We believe this hypothesis to be a powerful guideline for the design of simple, natural, and efficient coding schemes and for understanding the (im)possibilities of coding for interactive communications.
Supplemental Material
- S. Agrawal, R. Gelles, and A. Sahai. Adaptive protocols for interactive communication. In arXiv:1312.4182, 2014.Google Scholar
- Z. Brakerski and Y. Kalai. Efficient interactive coding against adversarial noise. In FOCS, pages 160--166, 2012. Google ScholarDigital Library
- Z. Brakerski and M. Naor. Fast algorithms for interactive coding. In SODA, pages 443--456, 2013. Google ScholarDigital Library
- M. Braverman. Coding for interactive computation: progress and challenges. In Allerton, pages 1914--1921, 2012.Google ScholarCross Ref
- M. Braverman. Towards deterministic tree code constructions. In ITCS, pages 161--167, 2012. Google ScholarDigital Library
- M. Braverman and K. Efremenko. List and unique coding for interactive communication in the presence of adversarial noise. In ECCC:TR14--007, 2014.Google ScholarDigital Library
- M. Braverman and A. Rao. Towards coding for maximum errors in interactive communication. In STOC, pages 159--166, 2011. Google ScholarDigital Library
- K.-M. Chung, R. Pass, and S. Telang. Knowledge preserving interactive coding. FOCS, 2013. Google ScholarDigital Library
- M. Franklin, R. Gelles, R. Ostrovsky, and L. J. Schulman. Optimal coding for streaming authentication and interactive communication. In CRYPTO, pages 258--276, 2013.Google ScholarCross Ref
- M. Ghaffari and B. Haeupler. Optimal Error Rates for Interactive Coding II: Efficiency and List decoding. In arXiv:1312.1763, 2013.Google Scholar
- M. Ghaffari, B. Haeupler, and M. Sudan. Optimal Error Rates for Interactive Coding I: Adaptivity and other settings. In arXiv:1312.1764, 2013.Google Scholar
- L. J. Schulman. Coding for interactive communication. Trans. on Inf. Theory, 42(6):1745--1756, 1996. Google ScholarCross Ref
- M. Sudan. Decoding of reed solomon codes beyond the error-correction bound. Journal of Complexity, 13(1):180--193, 1997. Google ScholarDigital Library
Index Terms
- Optimal error rates for interactive coding I: adaptivity and other settings
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