# Computational Hardness of Collective Coin-Tossing Protocols

## 3. Optimal Coin-Tossing Protocols: A Geometric Approach

This section introduces the original combinative proficiency of Khorasgani, Maji, and Mukherjee [ 14 ] for characterizing the “ most procure ” coin-tossing protocol .

### 3.3. Prior Approaches to the General Martingale Problem

Azuma–Hoeffding inequality [ 39, 40 ] states that, if |Xi−Xi−1|=o ( 1/n ), for all i∈ { 1, …, normality }, then, basically, |Xn−X0|=o ( 1 ) with probability 1. That is, the final information Xn remains close to the a priori information X0. however, in our trouble affirmation, we have Xn∈ { 0,1 }. In particular, this constraint implies that the final examination information Xn is significantly different from the a priori information X0. thus, the initial constraint “ for all i∈ { 1, …, normality } we have |Xi−Xi−1|=o ( 1/n ) ” must be violated. What is the probability of this irreverence ? For X0=1/2, Cleve and Impagliazzo [ 10 ] proved that there exists a cycle iodine such that |Xi−Xi−1|≥132n with probability 1/5. We emphasize that the turn one is a random variable and not a constant. however, the definition of the “ big jump ” and the “ probability to encounter large jumps ” are both exponentially small functions of X0. so, the approach of Cleve and Impagliazzo is only applicable to constant X0∈ ( 0,1 ). recently, in an independent work, Beimel et alabama. [ 41 ] demonstrate an identical bind for weak martingales ( that have some extra properties ), which is used to model multi-party coin-tossing protocols. For the upper-bound, on the other hand, Doob ’ second martingale, corresponding to the majority protocol, is the only know martingale for X0=1/2 with a small maximum susceptibility. In general, to achieve arbitrary X0∈ [ 0,1 ], one considers coin-tossing protocols, where the end product is 1 if the total number of heads in normality uniformly random coins surpasses an appropriate threshold .

### 3.4. Inductive Approach

This section presents a high-level overview of the inductive scheme to characterizing optimum coin-tossing protocols. In the sequel, we shall assume that we are working with discrete-time martingales ( X0, X1, …, Xn ) such that Xn∈ { 0,1 }. Given a dolphin striker ( X0, …, Xn ), its susceptibility is represented by the following measure supstoppingtimeτE [ |Xτ−Xτ−1| ] intuitively, if a dolphin striker has high susceptibility, then it has a discontinue time, such that the col in the martingale while encountering the stop time is boastfully. Our aim is to characterize the least susceptibility that a dolphin striker ( X0, …, Xn ) can achieve. More formally, given n and X0, characterize Cn ( X0 ) : =inf ( X0, …, Xn ) supstoppingtimeτE [ |Xτ−Xτ−1| ]. The overture proceeds by induction on newton to precisely characterize the swerve Cn ( X ), and our argument naturally constructs the best dolphin striker that achieves Cn ( X0 ) .

1. Base character. note that the base case is C1 ( X ) =2X ( 1−X ) ( see for this controversy ) .Open in a separate window
2. inductive tone. Given the bend Cn−1 ( X ), one identifies a geometric transformationT ( see Figure fig : transform-def ) that defines the curl Cn ( X ) from the curl Cn−1 ( X ). furthermore, for any n≥1, there exist martingales such that its susceptibility is precisely Cn ( X0 ) .

We shall prove the follow technical result in this section .Theorem 1. Fix any X0∈ ( 0,1 ) and n∈ℕ. Let X= ( X0, X1, …, Xn ) be a martingale, such that Xn∈ { 0,1 }. There exists a stop time τ in such that