For asynchronous binary agreement (ABA) with optimal resilience, prior private-setup free protocols (Cachin et al., CCS 2002; Kokoris-Kogias et al., CCS 2020) incur $O({lambda}n^4)$ bits and $O(n^3)$ messages; for asynchronous multi-valued agreement with external validity (VBA), Abraham et al. [2] very recently gave the first elegant construction with $O(n^3)$ messages, relying on public key infrastructure (PKI), but still costs $O({lambda} n^3 log n)$ bits. We for the first time close the remaining efficiency gap, i.e., reducing their communication to $O({lambda} n^3)$ bits on average. At the core of our design, we give a systematic treatment of reasonably fair common randomness: - We construct a reasonably fair common coin (Canetti and Rabin, STOC 1993) in the asynchronous setting with PKI instead of private setup, using only $O({lambda} n^3)$ bit and constant asynchronous rounds. The common coin protocol ensures that with at least 1/3 probability, all honest parties can output a common bit that is as if uniformly sampled, rendering a more efficient private-setup free ABA with expected $O({lambda} n^3)$ bit communication and constant running time. - More interestingly, we lift our reasonably fair common coin protocol to attain perfect agreement without incurring any extra factor in the asymptotic complexities, resulting in an efficient reasonably fair leader election primitive pluggable in all existing VBA protocols, thus reducing the communication of private-setup free VBA to expected $O({lambda} n^3)$ bits while preserving expected constant running time. - Along the way, we improve an important building block, asynchronous verifiable secret sharing by presenting a private-setup free implementation costing only $O({lambda} n^2)$ bits in the PKI setting. By contrast, prior art having the same complexity (Backes et al., CT-RSA 2013) has to rely on a private setup.
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