A Signature-Based Gröbner Basis Algorithm with Tail-Reduced Reductors (M5GB)

Gröbner 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 Gröbner 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 Gröbner basis algorithms to date. Among the most efficient algorithms in this line of work are signature-based Gröbner basis algorithms, with the first of its kind published in the late 1990s by Jean-Charles Faugère under the name $\texttt{F5}$. In addition to signature-based approaches, Rusydi Makarim and Marc Stevens investigated a different direction to efficiently compute Gröbner bases, which they published in 2017 with their algorithm $\texttt{M4GB}$. The ideas behind $\texttt{M4GB}$ and signature-based approaches are conceptually orthogonal to each other because each approach addresses a different source of inefficiency in Buchberger\u27s initial algorithm by different means. We amalgamate those orthogonal ideas and devise a new Gröbner basis algorithm, called $\texttt{M5GB}$, that combines the concepts of both worlds. In that capacity, $\texttt{M5GB}$ merges strong signature-criteria to eliminate redundant S-pairs with concepts for fast polynomial reductions borrowed from $\texttt{M4GB}$. 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 Gröbner 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

A polyhedral approach to computing border bases

Border bases can be considered to be the natural extension of Gr\"obner bases that have several advantages. Unfortunately, to date the classical border basis algorithm relies on (degree-compatible) term orderings and implicitly on reduced Gr\"obner bases. We adapt the classical border basis algorithm to allow for calculating border bases for arbitrary degree-compatible order ideals, which is \emph{independent} from term orderings. Moreover, the algorithm also supports calculating degree-compatible order ideals with \emph{preference} on contained elements, even though finding a preferred order ideal is NP-hard. Effectively we retain degree-compatibility only to successively extend our computation degree-by-degree. The adaptation is based on our polyhedral characterization: order ideals that support a border basis correspond one-to-one to integral points of the order ideal polytope. This establishes a crucial connection between the ideal and the combinatorial structure of the associated factor spaces