  We consider the corner-stone broadcast task with an adaptive adversary that
controls a fixed number of $t$ edges in the input communication graph. In this
model, the adversary sees the entire communication in the network and the
random coins of the nodes, while maliciously manipulating the messages sent
through a set of $t$ edges (unknown to the nodes). Since the influential work
of [Pease, Shostak and Lamport, JACM'80], broadcast algorithms against
plentiful adversarial models have been studied in both theory and practice for
over more than four decades. Despite this extensive research, there is no round
efficient broadcast algorithm for general graphs in the CONGEST model of
distributed computing. We provide the first round-efficient broadcast
algorithms against adaptive edge adversaries. Our two key results for $n$-node
graphs of diameter $D$ are as follows:
  1. For $t=1$, there is a deterministic algorithm that solves the problem
within $\widetilde{O}(D^2)$ rounds, provided that the graph is 3
edge-connected. This round complexity beats the natural barrier of $O(D^3)$
rounds, the existential lower bound on the maximal length of $3$ edge-disjoint
paths between a given pair of nodes in $G$. This algorithm can be extended to a
$\widetilde{O}(D^{O(t)})$-round algorithm against $t$ adversarial edges in
$(2t+1)$ edge-connected graphs.
  2. For expander graphs with minimum degree of $\Omega(t^2\log n)$, there is
an improved broadcast algorithm with $O(t \log ^2 n)$ rounds against $t$
adversarial edges. This algorithm exploits the connectivity and conductance
properties of G-subgraphs obtained by employing the Karger's edge sampling
technique.
  Our algorithms mark a new connection between the areas of fault-tolerant
network design and reliable distributed communication.
