We consider the problem of amplifying two-party coin-tossing protocols: given a protocol where it is possible to bias the common output by at most ρ, we aim to obtain a new protocol where the output can be biased by at most ρ* < ρ. We rule out the existence of a natural type of amplifiers called oblivious amplifiers for every ρ* < ρ. Such amplifiers ignore the way that the underlying ρ-bias protocol works and can only invoke an oracle that provides ρ-bias bits. We provide two proofs of this impossibility. The first is by a reduction to the impossibility of deterministic randomness extraction from Santha-Vazirani sources. The second is a direct proof that is more general and also rules outs certain types of asymmetric amplification. In addition, it gives yet another proof for the Santha-Vazirani impossibility.
@InProceedings{bitansky_et_al:LIPIcs.ITCS.2020.30, author = {Bitansky, Nir and Geier, Nathan}, title = {{On Oblivious Amplification of Coin-Tossing Protocols}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {30:1--30:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.30}, URN = {urn:nbn:de:0030-drops-117150}, doi = {10.4230/LIPIcs.ITCS.2020.30}, annote = {Keywords: Coin Tossing, Amplification, Lower Bound, Santha Vazirani} }
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