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.

翻译：承诺方案是一种密码学工具，允许承诺方对隐藏值进行承诺，并可在后续指定位置开启该值而无需泄露秘密本身。承诺方案在零知识证明和安全多方计算（仅举几例）中具有重要应用。本综述介绍了基于多种数学结构构建的若干多元多项式承诺方案。我们研究了如何利用哈希函数构造Orion方案；利用内积论证构造Dory、Bulletproofs和Vampire方案；利用多项式分解构造Signatures of Correct Computation方案；利用未知阶群构造DARK和Dew方案；以及利用CP-SNARK构造Orion+方案。针对每种协议，我们证明了其完备性并阐述了其安全假设。