A commitment scheme $\system$ is a tuple of algorithms $(\setup, \commit, \check)$ defined as follows:

• $\setup(1^n) \to pp$ generates public parameters $pp$ for security parameter $n$
• $\commit(pp,u) \to (c, r)$ takes message $u$ in message space $\mathcal{M}$, outputs a commitment $c$ in commitment space $\mathcal{C}$ and opening $r$ in space $\mathcal{R}$
• $\check(pp, c, u, r) \to \{0, 1\}$ Takes a commitment $c$, message $u$, randomness $r$, outputs $1$ if the commitment is valid, $0$ otherwise