Broadcast Amplification.

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 upsilon (consistency), and if the sender is correct, then upsilon is the value sent by the sender (validity). A celebrated result by Pease, Shostak and Lamport states that such a broadcast 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-to-point 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 phi(n)(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., phi 3(d) - 3 for any d >= 3. In contrast, for n > 3 no broadcast amplification is possible, i.e., phi(n)(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 phi*(n)(d) denote the minimal d' such that d-broadcast can be constructed from primitives d'(1)-broadcast,..., d'(k)-broadcast, where d' = Pi(i) d'(i) (i.e., log d' = Sigma(i) log d'(i)). Note that phi*(n)(d) <= phi(n)(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 phi*(n)(d) <= 8n log n.