Share Computing Protocols over Fields and Rings

Abstract

In this thesis, we explain linear secret sharing schemes, in particular multiplicative threshold linear secret sharing schemes, over fields and rings in a compact and concise way. We explain two characterisations of linear secret sharing schemes, and in particular, we characterise threshold linear secret sharing schemes. We develop an algorithm to generate all multiplicative $(t+1)$-out-of-$n$ threshold linear secret sharing schemes over a field $mathbb{Z}sb{p}$. For the ring $mathbb{Z}sb{2sp{32}}$, we explain the generation of secret sharing schemes for threshold access structures and prove the non-existence of $(t+1)$-out-of-$n$ threshold linear secret sharing schemes with $n > t+1$.