Protocols for Reliable and Secure Message Transmission.

TLDR: Several new/improved/efficient/optimal solutions are reported, affirmative/negative answers to several significant open problems are given, and first solutions to several newly formulated problems are provided.

Abstract: Consider the following problem: a sender S and a receiver R are part of an unreliable, connected, distributed network. The distrust in the network is modelled by an entity called adversary, who has unbounded computing power and who can corrupt some of the nodes of the network (excluding S and R) in a variety of ways. S wishes to send to R a message m that consists of l elements, where l ≥ 1, selected uniformly from a finite field F. The challenge is to design a protocol, such that after interacting with S as per the protocol, R should output m without any error (perfect reliability). Moreover, this hold irrespective of the disruptive actions done by the adversary. This problem is called reliable message transmission or RMT in short. The problem of secure message transmission or SMT in short requires an additional constraint that the adversary should not get any information about the message what so ever in information theoretic sense (perfect secrecy). Security against an adversary with infinite computing power is also known as non-cryptographic or information theoretic or Shannon security and this is the strongest notion of security. Notice that since the adversary has unbounded computing power, we cannot solve RMT and SMT problem by using classical cryptographic primitives such as public key cryptography, digital signatures, authentication schemes, etc as the security of all these primitives holds good only against an adversary having polynomially bounded computing power. RMT and SMT problem can be studied in various network models and adversarial settings. We may use the following parameters to describe different settings/models for studying RMT/SMT: 1. Type of Underlying Network — Undirected Graph, Directed Graph, Hypergraph. 2. Type of Communication — Synchronous, Asynchronous. 3. Adversary capacity — Threshold Static, Threshold Mobile, Non-threshold Static, Non-threshold Mobile. 4. Type of Faults — Fail-stop, Passive, Byzantine, Mixed. Irrespective of the settings in which RMT/SMT is studied, the following issues are common: 1. Possibility: What are the necessary and sufficient structural conditions to be satisfied by the underlying network for the existence of any RMT/SMT protocol, tolerating a given type of adversary? 2. Feasibility: Once the existence of a RMT/SMT protocol in a network is ascertained, the next natural question is, does there exist an efficient protocol on the given network? 3. Optimality: Given a message of specific length, what is the minimum communication complexity (lower bound) needed by any RMT/SMT protocol to transmit the message and how to design a polynomial time RMT/SMT protocol whose total communication complexity matches the lower bound on the communication complexity (optimal protocol)? In this dissertation, we look into the above issues in several network models and adversarial settings. This thesis reports several new/improved/efficient/optimal solutions, gives affirmative/negative answers to several significant open problems and last but not the least, provides first solutions to several newly formulated problems.