  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.
