Faugère's F4 and F5 algorithms

In computer algebra, the Faugère F4 algorithm, by Jean-Charles Faugère, computes the Gröbner basis of an ideal of a multivariate polynomial ring. The algorithm uses the same mathematical principles as the Buchberger algorithm, but computes many normal forms in one go by forming a generally sparse matrix and using fast linear algebra to do the reductions in parallel.

The Faugère F5 algorithm first calculates the Gröbner basis of a pair of generator polynomials of the ideal. Then it uses this basis to reduce the size of the initial matrices of generators for the next larger basis:

If G_prev is an already computed Gröbner basis (f₂, …, f_m) and we want to compute a Gröbner basis of (f₁) + G_prev then we will construct matrices whose rows are m f₁ such that m is a monomial not divisible by the leading term of an element of G_prev.

This strategy allows the algorithm to apply two new criteria based on what Faugère calls signatures of polynomials. Thanks to these criteria, the algorithm can compute Gröbner bases for a large class of interesting polynomial systems, called regular sequences, without ever simplifying a single polynomial to zero—the most time-consuming operation in algorithms that compute Gröbner bases. It is also very effective for a large number of non-regular sequences.

Implementations

The Faugère F4 algorithm is implemented

in FGb, Faugère's own implementation, which includes interfaces for using it from C/C++ or Maple,
in Maple computer algebra system, as the option method=fgb of function Groebner[gbasis]
in the Magma computer algebra system,
in the SageMath computer algebra system,

Study versions of the Faugère F5 algorithm is implemented in^{[citation needed]}

the SINGULAR computer algebra system;^[1]
the SageMath computer algebra system.
in SymPy Python package.^[2]

Applications

The previously intractable "cyclic 10" problem was solved by F5,^{[citation needed]} as were a number of systems related to cryptography; for example HFE and C^*.^{[citation needed]}