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