A new protocol for global coin flipping in the model of Ben-Or and Linial is presented. In this model, global coin flipping is considered to be an asynchronous perfect information game among n players, some of whom are dishonest. Each player possesses a fair coin and they wish to agree on the value of a single bit, which the honest players want to be random. A protocol is $\varepsilon $-robust for t dishonest players if no set of t dishonest players can bias the bit by more than $\varepsilon $. The protocol given here is $\varepsilon $-robust against $\theta (n/\log n)$ dishonest players for any fixed $\varepsilon $, improving on the $\theta (n^{.63 \cdots } )$ robust protocol given by Ben-Or and Linial.
A new protocol for global coin flipping in the model of Ben-Or and Linial is presented. In this model, global coin flipping is considered to be an asynchronous perfect information game among n players, some of whom are dishonest. Each player possesses a fair coin and they wish to agree on the value of a single bit, which the honest players want to be random. A protocol is ε-robust for t dishonest players if no set of t dishonest players can bias the bit by more than ε. The protocol given here is ε-robust against θ(n/logn) dishonest players for any fixed ε, improving on the θ(n.63⋯) robust protocol given by Ben-Or and Linial.
The full-information model was introduced by Ben-Or and Linial in 1985 to study collective coin-flipping: the problem of generating a common bounded-bias bit in a network of n players with t=t(n) faults. They showed that the majority protocol can tolerate t=O(n) adaptive corruptions, and conjectured that this is optimal in the adaptive setting. Lichtenstein, Linial, and Saks proved that the conjecture holds for protocols in which each player sends a single bit. Their result has been the main progress on the conjecture in the last 30 years. In this work we revisit this question and ask: what about protocols involving longer messages? Can increased communication allow for a larger fraction of faulty players? We introduce a model of strong adaptive corruptions, where in each round, the adversary sees all messages sent by honest parties and, based on the message content, decides whether to corrupt a party (and intercept his message) or not. We prove that any one-round coin-flipping protocol, regardless of message length, is secure against at most O~(n) strong adaptive corruptions. Thus, increased message length does not help in this setting. We then shed light on the connection between adaptive and strongly adaptive adversaries, by proving that for any symmetric one-round coin-flipping protocol secure against t adaptive corruptions, there is a symmetric one-round coin-flipping protocol secure against t strongly adaptive corruptions. Returning to the standard adaptive model, we can now prove that any symmetric one-round protocol with arbitrarily long messages can tolerate at most O~(n) adaptive corruptions. At the heart of our results lies a novel use of the Minimax Theorem and a new technique for converting any one-round secure protocol into a protocol with messages of polylog(n) bits. This technique may be of independent interest.
In a distributed coin-flipping protocol, Blum [ACM Transactions on Computer Systems ’83], the parties try to output a common (close to) uniform bit, even when some adversarially chosen parties try to bias the common output. In an adaptively secure full-information coin flip, Ben-Or and Linial [FOCS ’85], the parties communicate over a broadcast channel, and a computationally unbounded adversary can choose which parties to corrupt along the protocol execution. Ben-Or and Linial proved that the n -party majority protocol is resilient to O(n) corruptions, and conjectured this is a tight upper bound for any n -party protocol (of any round complexity). Their conjecture was proved to be correct, up to polylogarithmic factors, for single-turn (each party sends a single message) single-bit (a message is one bit) protocols Lichtenstein et al. [Combinatorica ’89], symmetric protocols Goldwasser et al. [ICALP ’15], and recently for (arbitrary message length) single-turn protocols Tauman Kalai et al. [DISC ’18]. Yet, the question of many-turn protocols was left entirely open. In this work, we close the above gap, proving that no n -party protocol (of any round complexity) is resilient to Ω(n⋅log3n) adaptive corruptions. Namely, majority is the optimal coin-flipping protocol against adaptive adversaries (up to polylogarithmic factors).
Coin flipping is a cryptographic primitive in which two spatially separated players, who in principle do not trust each other, wish to agree on a random bit. Classical and quantum coin flipping protocols have been studied extensively for more than twenty‐five years. However, until recently, quantum coin flipping protocols were designed without taking into consideration the losses of quantum information that would be unavoidable in any realistic implementation. We introduced in 2008 a novel protocol and proved its security even when losses are taken into account: no cheating player could obtain a desired outcome with a probability greater than (6+2)/8≈93%. Here, we refine our earlier protocol by making it fair in the sense that the optimal cheating strategies allow either player to bias the outcome by the same amount. Specifically, either player can cheat to obtain a desired outcome with probability exactly 90%, but no more. An implementation is underway.
Coin flipping is a cryptographic primitive in which two spatially separated players, who 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 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.
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 participants 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 can completely bias the outcome to his choosing. If we allow for 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 cannot tolerate realistic losses on the quantum channel between the participants or in their quantum storage and measurement apparatus. In this paper, we introduce a novel quantum protocol and we prove its unconditional security even when such losses are taken into account.
In a multi-party fair coin-flipping protocol, the parties output a common (close to) unbiased bit, even when some corrupted parties try to bias the output. In this work we focus on the case of dishonest majority, ie at least half of the parties can be corrupted. [19] [STOC 1986] has shown that in any m-round coin-flipping protocol the corrupted parties can bias the honest parties’ common output bit by Θ(1/m). For more than two decades the best known coin-flipping protocols against majority was the protocol of [9] [Manuscript 1985], who presented a t-party, m-round protocol with bias This was changed by the breakthrough result of [42] [TCC 2009], who constructed an m-round, two-party coin-flipping protocol with optimal bias Θ(1/m). Recently, [32] [STOC 14] constructed an m-round, three-party coin-flipping protocol with bias O(log3 m/m). Still for the case of more than three parties, against arbitrary number of corruptions, the best known protocol remained the protocol of [9]. We make a step towards eliminating the above gap, presenting a t-party, m-round coin-flipping protocol, with bias This improves upon the protocol of [9] for any t ≤ 1/2 · log log m, and in particular for t ∊ O(1), this yields an protocol. For the three-party case, this yields an protocol, improving over the the O(log3 m/m)-bias protocol of [32]. Our protocol generalizes that of [32], by presenting an appropriate “defense protocols” for the remaining parties to interact in, in the case that some parties abort or caught cheating ([32] only presented a two-party defense protocol, which limits their final protocol to handle three parties). We analyze our new protocols by presenting a new paradigm for analyzing fairness of coin-flipping protocols. We map the set of adversarial strategies that try to bias the honest parties outcome in the protocol to the set of the feasible solutions of a linear program. The gain each strategy achieves is the value of the corresponding solution. We then bound the the optimal value of the linear program by constructing a feasible solution to its dual.