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.
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.
Study versions of the Faugère F5 algorithm is implemented in[citation needed] 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*.