Polynomial modular product verification and its implications

TL;DR

This paper explores efficient probabilistic algorithms for verifying polynomial products, especially under modular conditions, improving existing bounds and introducing the first quasi-optimal method for sparse cases.

Contribution

It introduces new bounds and algorithms for polynomial product verification, including the first quasi-optimal method for sparse polynomial cases.

Findings

Verification can be more efficient than modular multiplication under certain conditions.

New bounds on bit complexity for polynomial multiplication verification.

First quasi-optimal algorithm for verifying sparse polynomial products.

Abstract

Polynomial multiplication is known to have quasi-linear complexity in both the dense and the sparse cases. Yet no truly linear algorithm has been given in any case for the problem, and it is not clear whether it is even possible. This leaves room for a better algorithm for the simpler problem of verifying a polynomial product. While finding deterministic methods seems out of reach, there exist probabilistic algorithms for the problem that are optimal in number of algebraic operations. We study the generalization of the problem to the verification of a polynomial product modulo a sparse divisor. We investigate its bit complexity for both dense and sparse multiplicands. In particular, we are able to show the primacy of the verification over modular multiplication when the divisor has a constant sparsity and a second highest-degree monomial that is not too large. We use these results to obtain new bounds on the bit complexity of the standard polynomial multiplication verification. In particular, we provide optimal algorithms in the bit complexity model in the dense case by improving a result of Kaminski and develop the first quasi-optimal algorithm for verifying sparse polynomial product.