  For the last almost three decades, since the famous Buchberger-M\"oller(BM)
algorithm emerged, there has been wide interest in vanishing ideals of points
and associated interpolation polynomials. Our paradigm is based on the theory
of bivariate polynomial interpolation on cartesian point sets that gives us
related degree reducing interpolation monomial and Newton bases directly. Since
the bases are involved in the computation process as well as contained in the
final output of BM algorithm, our paradigm obviously simplifies the computation
and accelerates the BM process. The experiments show that the paradigm is best
suited for the computation over finite prime fields that have many
applications.
