# Bounds On Information Exchange For Byzantine Agreement

Specifically, we prove that any protocol that tolerates up to faulty processes requires at least no-t-1 messages in its flawless execution – and therefore, in the worst case, at least [n`t)/2] messages in the worst case and min (P 0.P 1). (No.1) Values measured in the average case, where P v is the probability that the value of the bit the sender wants to send is v. We also give protocols that solve the problem using only the minimum number of measures for these three complexity measures. These protocols can be implemented using 1-bit messages. Since a lower limit for the number of messages is also a lower limit for the number of messbits, this means that the narrow limits mentioned above on the number of messages are also narrow limits for the number of messbits. Dolev D, Strong HR: Authenticated algorithms for Byzantine chords. SIAM J Comput 12 (4):656-666 (1983) Cook SA, Dwork C, Reischuk R: upper and lower limits for parallel RAM without simultaneous writing. SIAM J COMPUT 15 (1): 87-97 (1986). Feldman P, Micali S: Byzantine agreement in time to erer expected constantly (and do not trust anyone). Proceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science pp 267-276, 1985 Amdur ES: On the message complexity of byzantine agreement.

M.S. Thesis, Department of Computer Science, University of Toronto 1988 Pease M, Shostak R, Lamport L: Reaching agreement in the presence of errors. J ACM 27 (2): 228-234 (1980) Feldman P, Micali S: A optimal algorithm for synchronous Byzantine agreement. MIT/LCSTM-425 technical report. Labor for Computer Science, Massachusetts Institute of Technology 1990 Coan B, Welch J: A Byzantine agreement protocol with optimal message complexity. Minutes from the 27th Allerton Conference on Communication. Control and Computing Pp 1062-1071, 1989 Dwork C, Moses Y: Knowledge and common knowledge in a Byzantine environment I: crash failures. In Proceedings of the Conference on Theoretical Aspects of Reasoning About Knowledge se 149-169, 1986 Dolev D, Reischuk R: Bounds on information exchange for Byzantine agreement. Research Report RJ 3587, IBM San Jose Research Laboratory 1982 This document presents a new Byzantine agreement protocol that tolerates t processor errors with 3t-1 processors, t-o (t) towers, total message bits and O (t) maximum message size for each > 0. The protocol is optimal or almost optimal in all cost measurement sizes: the number of processors is optimal, the complexity of the bit is optimal, the number of rounds exceeds the lower limit by o (t), and the maximum size of messages exceeds the lower limit per O (t). The circular complexity is uniformly better than 2 (t-1) and is therefore also for small t.