In recent decades, many researchers have investigated peer-to-peer security and cryptography such as symmetric encryption, public-key encryption, hashing, and digital signature. These techniques can protect private communications between two parties securely in a public channel. Since the Internet and networking have become more and more popular, many people have started to investigate security between groups of people such as secret sharing scheme, secure multicast, broadcasting security, group signatures and so on. However, cryptographic techniques used for peer-to-peer security are not effective for group security. This is one motivation for the research of secret sharing schemes....[ Read more ]

In recent decades, many researchers have investigated peer-to-peer security and cryptography such as symmetric encryption, public-key encryption, hashing, and digital signature. These techniques can protect private communications between two parties securely in a public channel. Since the Internet and networking have become more and more popular, many people have started to investigate security between groups of people such as secret sharing scheme, secure multicast, broadcasting security, group signatures and so on. However, cryptographic techniques used for peer-to-peer security are not effective for group security. This is one motivation for the research of secret sharing schemes.

A secret sharing scheme is a system designed to share a piece of information or a secret among a group of people such that only authorized people can reconstruct the secret from their shares. Since Blakley and Shamir independently proposed threshold secret sharing schemes in 1979, many secret sharing schemes have been constructed.

In this thesis, we will propose several variants and generalizations of Shamir’s secret sharing scheme. We give conditions to ensure that these variants and generalizations are threshold schemes. We study the information rates of these variants and generalizations and show that they are as efficient as Shamir’s original scheme.