Verification Protocols with Sub-Linear Communication for Polynomial Matrix Operations

TL;DR

This paper introduces efficient, interactive verification protocols with sub-linear communication complexity for checking the correctness of polynomial matrix computations over F[x], enhancing verification in algebraic settings.

Contribution

It develops novel protocols for verifying polynomial matrix operations that are more communication-efficient and tailored to matrices over principal ideal domains.

Findings

Protocols achieve sub-linear communication complexity

Verification of vector in row space over F[x] is introduced

Protocols are interactive and often randomized

Abstract

We design and analyze new protocols to verify the correctness of various computations on matrices over the ring F[x] of univariate polynomials over a field 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 F[x]-row space of a given matrix.