GSoC with Flint – Week 6
This week I’ve been working on implementing Pernet and Stein’s algorithm to compute the HNF. The basic idea of this method (which is in turn due to Micciancio and Warinschi) is to use the given matrix to construct another axillary matrix with small determinant (so the HNF of the new matrix can be computed without coefficient explosion) from which the original HNF can be reconstructed. The algorithm splits in to a few separate stages, the computation of two determinants, the modular HNF of the smaller matrix, the addition of a column to the HNF and then the addition of two rows. Each of these steps can be optimised quite a bit using some clever tricks, which I’m currently working on doing. The adding of rows and columns to the HNF now seems to work well, and exploiting the similarities of the submatrices whose determinants are needed to compute them both faster should be done tomorrow. At that point it should be interesting to take stock and see how well the Pernet-Stein approach is working compared to previous methods I’ve worked on. I’ll likely spend a couple more days after that improving this approach even more.
This is definitely the most complex algorithm I’ve worked on so far, but when completed it should really blow the others away for large enough random matrices.