Secret-Sharing

Any $S S$ protocal for a set of $n$ parties $P = {P_{1}, \dots, P_{n}}$ consists two phases, a sharing phase and a reconstruction phase.
We denote ${view}_{i}$ the view of $P_{i}$ at the end of the protocal, which consists of the local inputs of $P_{i}$ , along with the messages sent and received by $P_{i}$ .
Each party $P_{i}$ will output a share $s_{i}$ , which is a publicly-known function of $v i e w_{i}$
Each party then applies a publicly-known reconstruction function on the revealed views, as determined by the protocol Rec and reconstructs some output.
Properties needed from any $S S$ protocal,
- Correctness
  - Any qualified subset of parties should be able to reconstruct back the shared secret by executing the protocal Rec.(also called access-sets, the collection of them is called access-structure)
  - Usually use minimal access-sets.
- Privacy
  - Informally demands that the view of any "forbidden subset" of parties should not reveal any information about the shared secret.
  - Usually use maximal forbidden-set.