Serial composition of quantum coin-flipping, and bounds on cheat detection for bit-commitment

Quantum protocols for coin-flipping can be composed in series in such a way that a cheating party gains no extra advantage from using entanglement between different rounds. This composition principle applies to coin-flipping protocols with cheat sensitivity as well, and is used to derive two results: There are no quantum strong coin-flipping protocols with cheat sensitivity that is linear in the bias (or bit-commitment protocols with linear cheat detection) because these can be composed to produce strong coin-flipping with arbitrarily small bias. On the other hand, it appears that quadratic cheat detection cannot be composed in series to obtain even weak coin-flipping with arbitrarily small bias.Comment: 7 pages, REVTeX 4 (minor corrections in v2

Fair Loss-Tolerant Quantum Coin Flipping

Coin flipping is a cryptographic primitive in which two spatially separated players, who in principle do not trust each other, wish to establish a common random bit. If we limit ourselves to classical communication, this task requires either assumptions on the computational power of the players or it requires them to send messages to each other with sufficient simultaneity to force their complete independence. Without such assumptions, all classical protocols are so that one dishonest player has complete control over the outcome. If we use quantum communication, on the other hand, protocols have been introduced that limit the maximal bias that dishonest players can produce. However, those protocols would be very difficult to implement in practice because they are susceptible to realistic losses on the quantum channel between the players or in their quantum memory and measurement apparatus. In this paper, we introduce a novel quantum protocol and we prove that it is completely impervious to loss. The protocol is fair in the sense that either player has the same probability of success in cheating attempts at biasing the outcome of the coin flip. We also give explicit and optimal cheating strategies for both players.Comment: 12 pages, 1 figure; various minor typos corrected in version

Multiparty Quantum Coin Flipping

We investigate coin-flipping protocols for multiple parties in a quantum broadcast setting: (1) We propose and motivate a definition for quantum broadcast. Our model of quantum broadcast channel is new. (2) We discovered that quantum broadcast is essentially a combination of pairwise quantum channels and a classical broadcast channel. This is a somewhat surprising conclusion, but helps us in both our lower and upper bounds. (3) We provide tight upper and lower bounds on the optimal bias epsilon of a coin which can be flipped by k parties of which exactly g parties are honest: for any 1 <= g <= k, epsilon = 1/2 - Theta(g/k). Thus, as long as a constant fraction of the players are honest, they can prevent the coin from being fixed with at least a constant probability. This result stands in sharp contrast with the classical setting, where no non-trivial coin-flipping is possible when g <= k/2.Comment: v2: bounds now tight via new protocol; to appear at IEEE Conference on Computational Complexity 200

A large family of quantum weak coin-flipping protocols

Each classical public-coin protocol for coin flipping is naturally associated with a quantum protocol for weak coin flipping. The quantum protocol is obtained by replacing classical randomness with quantum entanglement and by adding a cheat detection test in the last round that verifies the integrity of this entanglement. The set of such protocols defines a family which contains the protocol with bias 0.192 previously found by the author, as well as protocols with bias as low as 1/6 described herein. The family is analyzed by identifying a set of optimal protocols for every number of messages. In the end, tight lower bounds for the bias are obtained which prove that 1/6 is optimal for all protocols within the family.Comment: 17 pages, REVTeX 4 (minor corrections in v2

Flipping quantum coins

Coin flipping is a cryptographic primitive in which two distrustful parties wish to generate a random bit in order to choose between two alternatives. This task is impossible to realize when it relies solely on the asynchronous exchange of classical bits: one dishonest player has complete control over the final outcome. It is only when coin flipping is supplemented with quantum communication that this problem can be alleviated, although partial bias remains. Unfortunately, practical systems are subject to loss of quantum data, which restores complete or nearly complete bias in previous protocols. We report herein on the first implementation of a quantum coin-flipping protocol that is impervious to loss. Moreover, in the presence of unavoidable experimental noise, we propose to use this protocol sequentially to implement many coin flips, which guarantees that a cheater unwillingly reveals asymptotically, through an increased error rate, how many outcomes have been fixed. Hence, we demonstrate for the first time the possibility of flipping coins in a realistic setting. Flipping quantum coins thereby joins quantum key distribution as one of the few currently practical applications of quantum communication. We anticipate our findings to be useful for various cryptographic protocols and other applications, such as an online casino, in which a possibly unlimited number of coin flips has to be performed and where each player is free to decide at any time whether to continue playing or not.Comment: 17 pages, 3 figure