Title: Interactive Certificates for Polynomial Matrices with Sub-Linear Communication,
Abstract: We develop and analyze new protocols to verify the correctness of various
computations on matrices over F[x], where F is a field. The properties we
verify concern an F[x]-module and therefore cannot simply rely on
previously-developed linear algebra certificates which work only for vector
spaces. Our protocols are interactive certificates, often randomized, and
featuring 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 certificates and a new protocol to verify that a given vector is in the
F[x]-linear span of a given matrix.