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We report on an approach to integration-by-parts reduction based on Gr\"obnerbases. We establish the underlying noncommutative rational double-shift algebrawherein the integration-by-parts relations form a left ideal. We describe indetail the one-loop massless box as an example where we achieved the fullreduction to master integrals by means of the Gr\"obner basis approach, andreport on the performance of the implementation. We also identify potentialbottlenecks in more complicated examples and elaborate on interesting furtherdirections.

IBP reduction via Gr\"obner bases in a rational double-shift algebra