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IEEE International Symposium on Information Theory (ISIT)

Unfair coin tossing

Grégory Demay, Ueli M. Maurer

Published Jul 1, 2013• 6 citations
Publisher

Abstract

An ideal coin tossing resource for two parties outputs the same random bit to both parties. We introduce the notion of an unfair coin tossing resource by relaxing both the fairness and the non-influenceability guarantees that an ideal coin toss would provide. The presence of this non-ideal behavior is necessary in order to understand what coin tossing protocols really achieve in the setting of two distrustful parties, since it is known that such an ideal coin tossing resource cannot be constructed whenever a majority of players is dishonest.

DOI
10.1109/isit.2013.6620488

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An ideal coin tossing resource for two parties outputs the same random bit to both parties. We introduce the notion of an unfair coin tossing resource by relaxing both the fairness and the non-influenceability guarantees that an ideal coin toss would provide. The presence of this non-ideal behavior is necessary in order to understand what coin tossing protocols really achieve in the setting of two distrustful parties, since it is known that such an ideal coin tossing resource cannot be constructed whenever a majority of players is dishonest.
Coin-flipping protocols allow mutually distrustful parties to generate a random bit with uniform distribution. Minimizing bias and keeping fairness are two key requirements in designing such protocols. However, the best previously known fair coin-flipping protocol only guarantees with bias O (1/ r ). In the paper, we introduce a new approach for fair coin-flipping protocol construction by using Garay et al ’s “resource fairness” technique and propose a resource-fair two-party coin-flipping protocol according to the standard real/ideal simulation paradigm. Compared with the other coin-flipping protocols, our protocol enjoys an important advantage that its bias is 0.
We review the quantum version of a well known problem of cryptography called coin tossing (``flipping a coin via telephone''). It can be regarded as a game where two remote players (who distrust each other) tries to generate a uniformly distributed random bit which is common to both parties. The only resource they can use to perform this task is a classical or quantum communication channel. In this paper we provide a general overview over such coin tossing protocols, concerning in particular their security.
We review the quantum version of a well known problem of cryptography called coin tossing ("flipping a coin via telephone"). It can be regarded as a game where two remote players (who distrust each other) try to generate a uniformly distributed random bit which is common to both parties. The only resource they can use to perform this task is a classical or quantum communication channel. In this paper we provide a general overview over such coin tossing protocols, concerning in particular their security.
We address one of the foundational problems in cryptography: the bias of coin-\nipping pro-tocols. Coin-\nipping protocols allow mutually distrustful parties to generate a common unbiased random bit, guaranteeing that even if one of the parties is malicious, it cannot signicantly bias the output of the honest party. A classical result by Cleve [STOC '86] showed that for any two-party r-round coin-\nipping protocol there exists an efficient adversary that can bias the output of the honest party by Ω(1=r). However, the best previously known protocol only guarantees O(1= p r) bias, and the question of whether Cleve's bound is tight has remained open for more than twenty years. In this paper we establish the optimal trade-off between the round complexity and the bias of two-party coin-\nipping protocols. Under standard assumptions (the existence of oblivious transfer), we show that Cleve's lower bound is tight: we construct an r-round protocol with bias O(1=r).
In a multiparty fair coin-flipping protocol, the parties output a common (close to) unbiased bit, even when some corrupted parties try to bias the output. Cleve [STOC 1986] has shown that in the case of dishonest majority (i.e., at least half of the parties can be corrupted), in any m-round coin-flipping protocol, the corrupted parties can bias the honest parties' common output bit by Ω(1/m). For more than two decades, the best known coin-flipping protocols against dishonest majority had bias [EQUATION], where ℓ is the number of corrupted parties. This was changed by a recent breakthrough result of Moran et al. [TCC 2009], who constructed an m-round, two-party coin-flipping protocol with optimal bias Θ(1/m). In a subsequent work, Beimel et al. [Crypto 2010] extended this result to the multiparty case in which less than 2/3 of the parties can be corrupted. Still for the case of 2/3 (or more) corrupted parties, the best known protocol had bias [EQUATION]. In particular, this was the state of affairs for the natural three-party case.