### 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 \(s^*\).