Information-Theoretic Broadcast With Dishonest Majority For Long Messages

Byzantine broadcast is a fundamental primitive for secure computation. In a setting with n parties in the presence of an adversary controlling at most t parties, while a lot of progress in optimizing communication complexity has been made for t < n/2, little progress has been made for the general case t < n, especially for information-theoretic security. In particular, all information-theoretic secure broadcast protocols for l-bit messages and t < n and optimal round complexity O(n) have, so far, required a communication complexity of O(ln(2)). A broadcast extension protocol allows a long message to be broadcast more efficiently using a small number of single-bit broadcasts. Through broadcast extension, so far, the best achievable round complexity for t < n setting with the optimal communication complexity of O(ln) is O(n(4)) rounds.In this work, we construct a new broadcast extension protocol for t < n with information-theoretic security. Our protocol improves the round complexity to O(n(3)) while maintaining the optimal communication complexity for long messages. Our result shortens the gap between the information-theoretic setting and the computational setting, and between the optimal communication protocol and the optimal round protocol in the information-theoretic setting for t < n.