Studies on Verifiable Secret Sharing, Byzantine Agreement and Multiparty Computation

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: • 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. • 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. • 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