Unconditionally reliable and secure message transmission in undirected synchronous networks: possibility, feasibility and optimality

We study the interplay of network connectivity and the issues related to the possibility, feasibility and optimality for unconditionally reliable message transmission (URMT) and unconditionally secure message transmission (USMT) in an undirected synchronous network, under the influence of an adaptive mixed adversary A(tb,to,tf ,tp), who has unbounded computing power and can corrupt up to tb, to, tf and tp nodes in the network in Byzantine, omission, fail-stop and passive fashion respectively. In URMT problem, a sender S and a receiver R are part of a distributed network, where S and R are connected by intermediate nodes, of which at most tb, to, tf and tp nodes can be under the control of A(tb,to,tf ,tp). S wants to send a message m which is a sequence of ` (` ≥ 1) field elements from a finite field F to R. The challenge is to design a protocol, such that after interacting in phases as per the protocol, R should be able to obtain m with probability at least 1− δ, where 0 < δ < 1 2 , irrespective of any adversarial strategy of A(tb,to,tf ,tp). The USMT problem has an additional requirement that A(tb,to,tf ,tp) should not know anything about m in information theoretic sense. In this paper, we answer the following in context of URMT and USMT: (a) Possibility: when is a protocol possible in a given network? (b) Feasibility: Once the existence of a protocol is ensured then does there exist a polynomial time protocol on the given network? (c) Optimality: Given a message of specific length, what is the minimum communication complexity (lower bound) needed by any protocol to transmit the message and how to design a protocol whose total communication complexity matches the lower bound on the communication complexity? One of the important conclusions we arrive at from the answers of the above questions is that allowing a negligible error probability significantly helps in the possibility, feasibility and optimality of both reliable and secure message transmission protocols. To design our protocols, we propose several new techniques which are of independent interest.