A Signature-Based Gr\"obner Basis Algorithm with Tail-Reduced Reductors (M5GB)

Gr\"obner bases are an important tool in computational algebra and, especially in cryptography, often serve as a boilerplate for solving systems of polynomial equations. Research regarding (efficient) algorithms for computing Gr\"obner bases spans a large body of dedicated work that stretches over the last six decades. The pioneering work of Bruno Buchberger in 1965 can be considered as the blueprint for all subsequent Gr\"obner basis algorithms to date. Among the most efficient algorithms in this line of work are signature-based Gr\"obner basis algorithms, with the first of its kind published in the late 1990s by Jean-Charles Faug\`ere under the name F5. In addition to signature-based approaches, Rusydi Makarim and Marc Stevens investigated a different direction to efficiently compute Gr\"obner bases, which they published in 2017 with their algorithm M4GB. The ideas behind M4GB and signature-based approaches are conceptually orthogonal to each other because each approach addresses a different source of inefficiency in Buchberger's initial algorithm by different means. We amalgamate those orthogonal ideas and devise a new Gr\"obner basis algorithm, called M5GB, that combines the concepts of both worlds. In that capacity, M5GB merges strong signature-criteria to eliminate redundant S-pairs with concepts for fast polynomial reductions borrowed from M4GB. We provide proofs of termination and correctness and a proof-of-concept implementation in C++ by means of the Mathic library. The comparison with a state-of-the-art signature-based Gr\"obner basis algorithm (implemented via the same library) validates our expectations of an overall faster runtime for quadratic overdefined polynomial systems that have been used in comparisons before in the literature and are also part of cryptanalytic challenges.