  In this work we characterize all ambiguities of the linear (aperiodic)
one-dimensional convolution on two fixed finite-dimensional complex vector
spaces. It will be shown that the convolution ambiguities can be mapped
one-to-one to factorization ambiguities in the $z-$domain, which are generated
by swapping the zeros of the input signals. We use this polynomial description
to show a deterministic version of a recently introduced masked Fourier phase
retrieval design. In the noise-free case a (convex) semi-definite program can
be used to recover exactly the input signals if they share no common factors
(zeros). Then, we reformulate the problem as deterministic blind deconvolution
with prior knowledge of the autocorrelations. Numerically simulations show that
our approach is also robust against additive noise.
