Lecture Notes in Computer Science
A d-broadcast primitive is a communication primitive that allows a sender to send a value from a domain of size d to a set of parties. A broadcast protocol emulates the d-broadcast primitive using only point-to-point channels, even if some of the parties cheat, in the sense that all correct recipients agree on the same value v (consistency), and if the sender is correct, then v is the value sent by the sender (validity). A celebrated result by Pease, Shostak and Lamport states that such a
... ast protocol exists if and only if t < n/3, where n denotes the total number of parties and t denotes the upper bound on the number of cheaters. This paper is concerned with broadcast protocols for any number of cheaters (t < n), which can be possible only if, in addition to point-topoint channels, another primitive is available. Broadcast amplification is the problem of achieving d-broadcast when d -broadcast can be used once, for d < d. Let φn(d) denote the minimal such d for domain size d. We show that for n = 3 parties, broadcast for any domain size is possible if only a single 3-broadcast is available, and broadcast of a single bit (d = 2) is not sufficient, i.e., φ3(d) = 3 for any d ≥ 3. In contrast, for n > 3 no broadcast amplification is possible, i.e., φn(d) = d for any d. However, if other parties than the sender can also broadcast some short messages, then broadcast amplification is possible for any n. Let φ * n (d) denote the minimal d such that d-broadcast can be constructed from primitives d 1 -broadcast,. . . , d k -broadcast, where d = i d i (i.e., log d = i log d i ). Note that φ * n (d) ≤ φn(d). We show that broadcasting 8n log n bits in total suffices, independently of d, and that at least n−2 parties, including the sender, must broadcast at least one bit. Hence min(log d, n − 2) ≤ log φ * n (d) ≤ 8n log n.