A Survey of Multivariate Polynomial Commitment Schemes

A commitment scheme is a cryptographic tool that allows one to commit to a hidden value, with the option to open it later at requested places without revealing the secret itself. Commitment schemes have important applications in zero-knowledge proofs and secure multi-party computation, just to name a few. This survey introduces a few multivariate polynomial commitment schemes that are built from a variety of mathematical structures. We study how Orion is constructed using hash functions; Dory, Bulletproofs, and Vampire using the inner-product argument; Signatures of Correct Computation using polynomial factoring; DARK and Dew using groups of unknown order; and Orion+ using a CP-SNARK. For each protocol, we prove its completeness and state its security assumptions.