Comparison to Previous Protocols Clause Samples

Comparison to Previous Protocols. We add a comparison to previous protocols in Figure 4 for our setting with an ideal 1-round Coin-Flip. Our protocol achieves a lower failure probability when the number of honest parties is high. We therefore depict how the failure probability decreases with the number of rounds in three regimes: t < n/10, t < n/3 and t = 0.49n. In each of the regimes, we compare our protocol with the two more efficient known protocols. Our figures show that in the regimes t < n/10 and t < n/3, our proto- col achieves a lower failure probability than the previous protocols [FM97] and [FLL21] after a few tens of rounds. Concretely, after 6 (resp. 27) rounds com- pared to [FLL21], and 4 (resp. 13) rounds compared to [FM97]. On the other hand, when t = 0.49n, our protocol achieves a lower failure probability only after more than 200 rounds, compared to previous solutions [MV17, FLL21]. 2 Log in base 2 of failure probability 0 8 10 Log in base 2 of failure probability 10 20 30 40 50 Number of rounds (a) t < n/10: The failure probability of our protocol becomes lower than that of [FM97] after 4 rounds, and lower than that of [FLL21] after 6 rounds. (b) t < n/3: The failure probability of our protocol becomes lower than that of [FM97] after 13 rounds, and lower than that of [FLL21] after 27 rounds. Log in base 2 of failure probability 50 100 150 200 250 300 350 400 Number of rounds (c) t = 0.49n: The failure probability of our protocol becomes lower than that of [MV17] after 212 rounds, and lower than that of [FLL21] after 299 rounds.
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