Secure communications guaranteeing reliability and privacy (without unproven assumptions) in networks with active adversaries has been an important research issue. It has been studied for point to point networks by Dolev-Dwork-Waarts-Yung (J. ACM 1993), Desmedt–Wang (Eurocrypt 2002), and Srinathan–Narayanan–Rangan (Crypto 2004). Dolev–Dwork–Waarts–Yung gave necessary and sufficient conditions for secure communication in networks with the condition that 1) all the channels are two-way; or 2) all the channels are one-way from the sender to the receiver. In this paper, we study the general case with a network modeled by a directed graph. In this general case, there are communication channels from the sender to the receiver and there are feedback channels from the receiver to the sender. We give necessary and sufficient bounds on the number of channels that are required from sender to receiver given a number of “feedback” channels from receiver to sender. We give these bounds for the case reliability is perfect, as well as for the case it is not perfect.
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