### 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 $$s^*$$.