This dissertation deals with three most important as well as fundamental problems in secure distributed computing, namely Verifiable Secret Sharing (VSS), Byzantine Agreement (BA) and Multiparty Computation (MPC).
VSS is a two phase protocol (Sharing and Reconstruction) carried out among $n$ parties in the presence of a centralized adversary who can
corrupt up to $t$ parties. Informally, the goal of the VSS protocol is
to share a secret $s$, among the $n$ parties during the sharing phase in a way that would later allow for a unique reconstruction of this secret in the reconstruction phase, while preserving the secrecy of $s$ until the reconstruction phase. VSS is used as a key tool in MPC, BA and many other secure distributed computing problems. It can take many different forms, depending on the underlying network (synchronous or asynchronous), the nature (passive or active) and computing power (bounded or unbounded) of the adversary, type of security (cryptographic or information theoretic) etc. We study
VSS in information theoretic setting over both synchronous as well as
asynchronous network, considering an active unbounded powerful adversary.
Our main contributions for VSS are:
\item In synchronous network, we carry out in-depth investigation on the round complexity of VSS by allowing a probability of error in computation and show that existing lower bounds for the round complexity of error-free VSS can be circumvented by introducing a negligible probability of error.
\item We study the communication and round efficiency of VSS in synchronous network and present a robust VSS protocol that is simultaneously communication efficient and round efficient. In addition, our protocol is the best known communication and round efficient protocol in the literature.
\item In asynchronous network, we study the communication complexity of VSS and propose a number of VSS protocols. Our protocols are highly communication efficient and show significant improvement over the existing protocols in terms of communication complexity.
The next problem that we deal with is Byzantine Agreement (BA). BA is considered as one of the most fundamental primitives for fault tolerant distributed computing and cryptographic protocols. BA among a set of $n$ parties, each having a private input value, allows them to reach agreement on a common value even if some of the malicious parties (at most $t$) try to prevent agreement among the parties.
Similar to the case of VSS, several models for BA have been proposed during the last three decades, considering various aspects like the underlying network, the nature and computing power of adversary, type of security. One of these models is BA over asynchronous network which is considered to be more realistic network than synchronous in many occasions. Though important, research in BA in asynchronous network has received much less attention in comparison to the BA protocols in synchronous network. Even the existing protocols for asynchronous BA involve high communication complexity and in general are very inefficient in comparison to their synchronous counterparts. We focus on BA in information theoretic setting over asynchronous network tolerating an active adversary having unbounded computing power and mainly work towards the communication efficiency of the problem. Our contributions for BA are as follows:
\item We propose communication efficient asynchronous BA protocols that show huge improvement over the existing protocols in the same setting. Our protocols for asynchronous BA use our VSS protocols
in asynchronous network as their vital building blocks.
\item We also construct a communication optimal asynchronous BA protocol for sufficiently long message size. Precisely, our asynchronous BA communicates O(\ell n) bits for \ell bit message, for sufficiently large \ell.
The studies on VSS and BA naturally lead one towards MPC problems. The MPC can model almost any known cryptographic application and uses VSS as well as BA as building blocks. MPC enables a set of $n$ mutually distrusting parties to compute some function of their private inputs, such that the privacy of the inputs of the honest parties is guaranteed (except for what can be derived from the function output) even in the presence of an adversary corrupting up
to $t$ of the parties and making them misbehave arbitrarily. Much like
VSS and BA, MPC can also be studied in various models. Here, we attempt to solve MPC in information theoretic setting over synchronous as well as asynchronous network, tolerating an active unbounded powerful adversary. As for MPC, our main contributions are:
\item Using one of our synchronous VSS protocol, we design a synchronous MPC that minimizes the communication and round complexity simultaneously, where existing MPC protocols try to minimize one complexity measure at a time (i.e the existing protocols minimize either communication complexity or round complexity).
\item We study the communication complexity of asynchronous MPC protocols and design a number of protocols for the same that show significant gain in communication complexity in comparison to the existing asynchronous MPC protocols.
\item We also study a specific instance of MPC problem called Multiparty Set Intersection (MPSI) and provide protocols for the same.
In brief, our work in this thesis has made significant advancement in the state-of-the-art research on VSS, BA and MPC by presenting several inherent lower bounds and efficient/optimal solutions for the problems in terms of their key parameters such as communication complexity and time/round complexity. Thus our work has made a significant contribution to the field of secure distributed computing by carrying out a foundation research on the three most important
problems of this field.