Lower Bounds for Leader Election and Collective Coin Flipping, Revisited

2025-04-02

AI Summary

This paper investigates the inherent complexity of collective coin-flipping and leader election in the full-information model under the stringent constraint that each participant sends only one bit per round. Method: It employs combinatorial game-theoretic analysis, a novel joint bias lemma for families of functions, and precise log*-complexity characterization. Results: The work establishes, for the first time, that any k-round protocol tolerates at most O(ℓ / log^{(k)} ℓ) malicious players, where ℓ is the total number of participants. It raises the lower bound on round complexity for linearly fault-tolerant protocols to log* ℓ − O(1), matching the optimal asymptotic order. It further proves that classical protocols—such as RZ and F—are nearly optimal in both round count and per-round communication. Finally, it constructs the first two-round, 1-bit-per-player protocol whose resilience strictly surpasses that of all one-round protocols.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We study the tasks of collective coin flipping and leader election in the full-information model. We prove new lower bounds for coin flipping protocols, implying lower bounds for leader election protocols. We show that any $k$-round coin flipping protocol, where each of $ell$ players sends 1 bit per round, can be biased by $O(ell/log^{(k)}(ell))$ bad players. For all $k>1$ this strengthens previous lower bounds [RSZ, SICOMP 2002], which ruled out protocols resilient to adversaries controlling $O(ell/log^{(2k-1)}(ell))$ players. Consequently, we establish that any protocol tolerating a linear fraction of corrupt players, with only 1 bit per round, must run for at least $log^*ell-O(1)$ rounds, improving on the prior best lower bound of $frac12 log^*ell-log^*log^*ell$. This lower bound matches the number of rounds, $log^*ell$, taken by the current best coin flipping protocols from [RZ, JCSS 2001], [F, FOCS 1999] that can handle a linear sized coalition of bad players, but with players sending unlimited bits per round. We also derive lower bounds for protocols allowing multi-bit messages per round. Our results show that the protocols from [RZ, JCSS 2001], [F, FOCS 1999] that handle a linear number of corrupt players are almost optimal in terms of round complexity and communication per player in a round. A key technical ingredient in proving our lower bounds is a new result regarding biasing most functions from a family of functions using a common set of bad players and a small specialized set of bad players specific to each function that is biased. We give improved constant-round coin flipping protocols in the setting that each player can send 1 bit per round. For two rounds, our protocol can handle $O(ell/(logell)(loglogell)^2)$ sized coalition of bad players; better than the best one-round protocol by [AL, Combinatorica 1993] in this setting.

Problem

• Proving lower bounds for collective coin flipping protocols
• Establishing round complexity for corrupt player resilience
• Improving constant-round coin flipping with limited communication

Innovation

• Proves new lower bounds for coin flipping protocols
• Shows protocols must run for log* rounds
• Introduces improved constant-round coin flipping protocols