Syntax and Properties
Syntax
A \((t,n)\) Verifiable Secret Sharing consists of two phases: Share and Reconstruct as follow:
Share: The dealer \(D\) has a secret value \(s\). Initially, there are \(n\) parties \(P_1, P_2, ..., P_n\). The sharing phase may consist of several rounds of interaction between parties. At the end of the phase, each party \(P_i\) holds a share \(s_i\) that will be required to reconstruct the secret of the dealer later.
Reconstruct: In this phase, each party \(P_i\) publishes its share \(s_i\) from the sharing phase. Then, from these shares, a reconstruction function will be applied to output the original secret.
Properties
As in [BKP11], a VSS need to have the following properties:
Correctness: If \(D\) is honest, then the reconstruction function outputs \(s\) at the end of the Reconstruct phase and all honest parties agree on the value \(s\).
Secrecy: If \(D\) is honest, then any adversary that control at most \(t\) parties does not learn any information about \(s\) during the Share phase.
Commitment: If \(D\) is dishonest, then at the end of the sharing phase there exists a value \(s^* \in \mathbb{F}_p\) such that at the end of the Reconstruct phase all honest parties agree on \(s^*\).