Byzantine broadcast (BB) and Byzantine agreement (BA) are two most fundamental problems and essential building blocks in distributed computing, and improving their efficiency is of interest to both theoreticians and practitioners. In this paper, we study extension protocols of BB and BA, i.e., protocols that solve BB/BA with long inputs of l bits using lower costs than l single-bit instances. We present new protocols with improved communication complexity in almost all settings: authenticated BA/BB with t n/2, authenticated BB with t (1-?μ)n, unauthenticated BA/BB with t n/3, and asynchronous reliable broadcast and BA with t n/3. The new protocols are advantageous and significant in several aspects. First, they achieve the best-possible communication complexity of ?~(nl) for wider ranges of input sizes compared to prior results. Second, the authenticated extension protocols achieve optimal communication complexity given the current best available BB/BA protocols for short messages. Third, to the best of our knowledge, our asynchronous and authenticated protocols in the setting are the first extension protocols in that setting.
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