We design and analyze new protocols to verify the correctness of various computations on matrices over the ring $\mathsf{F}[x]$ of univariate polynomials over a field $\mathsf{F}$. For the sake of efficiency, and because many of the properties we verify are specific to matrices over a principal ideal domain, we cannot simply rely on previously-developed linear algebra protocols for matrices over a field. Our protocols are interactive, often randomized, and feature a constant number of rounds of communication between the Prover and Verifier. We seek to minimize the communication cost so that the amount of data sent during the protocol is significantly smaller than the size of the result being verified, which can be useful when combining protocols or in some multi-party settings. The main tools we use are reductions to existing linear algebra verification protocols and a new protocol to verify that a given vector is in the $\mathsf{F}[x]$-row space of a given matrix.

我们设计和分析新协议，以验证环上矩阵上各种计算的正确性 $\mathsf{F}[X]$ 域上的单变量多项式 $\mathsf{F}$。为了提高效率，并且由于我们验证的许多属性特定于主要理想域上的矩阵，因此我们不能简单地将先前开发的线性代数协议用于某个领域的矩阵。我们的协议是交互式的，通常是随机的，并且在证明者和验证者之间具有恒定数量的通信循环。我们试图使通信成本最小化，以使协议期间发送的数据量大大小于要验证的结果的大小，这在组合协议或在某些多方设置中很有用。我们使用的主要工具是简化现有的线性代数验证协议，以及使用新协议来验证给定矢量在$\mathsf{F}[X]$给定矩阵的行空间。