On implementing the symbolic preprocessing function over Boolean polynomial rings in Gröbner basis algorithms using linear algebra

Linear algebra is introduced by Faugère in F4 to speed up the reduction procedure during Gröbner basis computations. Linear algebra has also been used in fast implementations of F5 and other signature-based Gröbner basis algorithms. To use linear algebra for reductions, an important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function (given in F4). This function can be very costly over boolean polynomial rings when the number of involved polynomials/monomials is huge, because multiplications of monomials and polynomials are contained in the Symbolic Preprocessing function, and moreover, the multiplication of a monomial and a boolean polynomial over a boolean polynomial ring is different from that over a polynomial ring. In this paper, a method of implementing the Symbolic Preprocessing function over boolean polynomial rings is reported. The experimental data show this method is very efficient.