Abstract In a multi-party fair coin-flipping protocol, the parties output a common (close to) unbiased bit, even when some adversarial parties try to bias the output. In this work, we focus on the case of an arbitrary number of corrupted parties. Cleve [20] [STOC 1986] has shown that in any such m -round coin-flipping protocol, the corrupted parties can bias the honest parties’ common output bit by $$\Theta (1/m)$$ Θ ( 1 / m ) . For more than two decades, however, the best-known coin-flipping protocol was the one of Awerbuch, Blum, Chor, Goldwasser, and Micali [10] [Manuscript 1985], who presented a t -party, m -round protocol with bias $$\Theta (t/\sqrt{m})$$ Θ ( t / m ) . This was changed by the breakthrough result of Moran, Naor, and Segev [51] [Journal of Cryptology 2016], who constructed an m -round, two -party coin-flipping protocol with optimal bias $$\Theta (1/m)$$ Θ ( 1 / m ) . More recently, Haitner and Tsfadia [37] [SIAM Journal on Computing 2017] constructed an m -round, three -party coin-flipping protocol with bias $$O(\log ^3m / m)$$ O ( log 3 m / m ) . Still for the case of more than three parties, the best-known protocol remained the $$\Theta (t/\sqrt{m})$$ Θ ( t / m ) -bias protocol of [10]. We make a step toward eliminating the above gap, presenting a t -party, m -round coin-flipping protocol, with bias $$O\left( \frac{t^4 \cdot 2^t \cdot \sqrt{\log m}}{m^{1/2+1/(2^{t-1}-2)}}\right) $$ O t 4 · 2 t · log m m 1 / 2 + 1 / ( 2 t - 1 - 2 ) for any $$t\le \tfrac{1}{2} \cdot \operatorname {loglog}m$$ t ≤ 1 2 · loglog m . This improves upon the

The semantic notion of dependent safety is a common generalization of the notion of absoluteness used in set theory and the notion of domain independence used in database theory for characterizing safe queries. This notion has been used in previous works to provide a unified theory of constructions and operations as they are used in different branches of mathematics and computer science, including set theory, computability theory, and database theory. In this paper we provide a complete syntactic characterization of general first-order dependent safety. We also show that this syntactic safety relation can be used for characterizing the set of strictly decidable relations on the natural numbers, as well as for characterizing rudimentary set theory and absoluteness of formulas within it.

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.