Verifiable broadcasting and gossiping in communication networks

In network communication where messages may be corrupted in transmission, one way to verify the correctness of a given message is to arrange for nodes in the network to receive the message multiple times. For example, in broadcasting (one-to-all communication) from a given source node u, if a message sent by u is received by all other nodes at least k+1 times, then each node can perform k checks against the original message to verify that it has not been corrupted in transmission. Similar behavior would be useful for gossiping (all-to-all communication) where information held in each node is to be communicated to all other nodes. For an n-node network, we consider the problem of determining the minimum number of network links required to support this k-fold verifiability. We show that the minimum size β(n,k) of an n-vertex k-verifiable broadcast scheme is given by β(n,k)=⌈(k+2)(n−1)/2⌉. We also show that the minimum size γ(n,k) of an n-vertex k-verifiable gossip scheme satisfies ⌈(k+4)(n−1)/2⌉−⌊log2 n⌋⩽γ(n,k)⩽⌈(k+4)n/2⌉−4. The value for β(n,k) and lower bound for γ(n,k) yield lower bounds for the size of a k-fault tolerant broadcast and gossip scheme which meet and improve, respectively, the previously known lower bounds for these schemes.