Bounds on information exchange for Byzantine Agreement

Abstract

Byzantine Agreement has become increasingly important in establishing distributed properties when there may exist errors in the systems. Recent polynomial algorithms for reaching Byzantine Agreement provide us with feasible solutions for obtaining coordination and synchronization in distributed systems. In this paper we study the amount of information exchange necessary to ensure Byzantine Agreement. This is measured by the number of messages and the number of signatures appended to messages (in case of authenticated algorithms) the participating processors need to send, in the worse case, in order to reach Byzantine Agreement. The lower bound for the number of signatures in the authenticated case is ~(nt), where n is the number of participating processors and t is the upper bound on the number of faults. If n is large compared to t, it matches the upper bounds from previously known algorithms. The lower bound for the number of messages is ~(n+t2). We present an algorithm that achieves this bound and for which the number of phases does not exceed the minimum t+l by more than a constant factor.

Early-deciding consensus is expensive

“…To this end, we leverage the argument from [6] yielding the wellknown lower bound of Ω(t 2 ) on the message complexity of consensus algorithms. Essentially, we argue that not only Ω(t 2 ) messages need to be exchanged to achieve consensus, but (i) this has to happen in round r + 1 if we fail no node in rounds r and r + 1 (as the algorithm needs to verify the decision it must take by the end of round r + 1), and (ii) this can be exploited to do the "pivoting" such that indeed also in the modified execution with more faults many messages are sent in round r + 1.…”

“…The basic time lower bound in this setting states that min(f + 2, t + 1) rounds are required for termination, even if we consider crash faults only [7,8]; showing that f + 1 rounds are required for decision (in the worst case) follows from standard bivalency arguments in virtually any reasonable fault model [11]. A message complexity lower bound of Ω(nt) has been known for long [6]. The question whether stopping early is more costly in terms of communication was not considered up to the present date.…”

Transmission Line Theory

“…Our protocols are polylogarithmic in time and, succeed with high probability. 1 We overcome the lower bound of [11] by allowing for a small probability of error. This is necessary since this lower bound also implies that any randomized algorithm which always uses no more than o(n 2 ) messages must necessarily err with positive probability, since the adversary can guess the random coinflips and achieve the lower bound if the guess is correct.…”

Scalable and Secure Polling in Dynamic Distributed Networks

“…In addition, most of them assume the existence of a broadcast channel, which is rarely available in large scale networks. Simulating such a channel deterministically is possible but has a communication cost of Ω(n 2 ) [6], [7]. The main motivation of this work is to design a distributed polling protocol whose communication complexity is close to be linear in the number of nodes of the system.…”