Linear Algebra for Gröbner Bases over GF(2): LELA

The Efficient Linear Algebra for Gröbner Basis Computations workshop in Kaiserslautern two weeks ago was a welcome opportunity to finally test out LELA, a library specifically written for linear algebra for Gröbner basis computations including for GF(2). The library implements the “Faugère-Lachartre” algorithm (a similar trick, though less developed, appeared before in PolyBoRi) and uses M4RI for dense parts over GF(2).

So, I ran my benchmark matrices through LELA, discovered a bug in the process, then Bradford returned the favour and discovered a bug in M4RI … Finally, below are the timings. The column PLE is the PLE algorithm as implemented in M4RI, M4RI is the M4RI algorithm as implemented in M4RI, GB is a very naive variant of the algorithm LELA uses and LELA is, well, LELA.

problem m n density PLE M4RI GB LELA
HFE 25 12307 13508 0.076 1.0 0.5 0.8 0.56
HFE 30 19907 29323 0.067 4.7 2.7 4.7 3.42
HFE 35 29969 55800 0.059 19.3 9.2 19.5 13.92
Mutant 26075 26407 0.184 5.7 3.9 2.1 12.07
n=24, m=26 37587 38483 0.038 20.6 21.0 19.3 7.72
n=24, m=26 37576 32288 0.040 18.6 28.4 17.0 4.09
SR(2,2,2,4) c 5640 14297 0.003 0.4 0.2 0.1 0.40
SR(2,2,2,4) c 13665 17394 0.013 2.1 3.0 2.0 1.78
SR(2,2,2,4) c 11606 16282 0.035 1.9 4.4 1.5 0.81
SR(2,2,2,4) 13067 17511 0.008 1.9 2.0 1.3 1.45
SR(2,2,2,4) 12058 16662 0.015 1.5 1.9 1.6 1.01
SR(2,2,2,4) 115834 118589 0.003 528.2 578.5 522.9 48.39

What this table means is that one can expect more than an order of magnitude of speed-up when using LELA – which is dedicated to these computations – instead of M4RI – which does not have the specialised algorithm implemented yet. For very small matrices sometimes M4RI/PLE win, but then not by a large margin. The only row where LELA doesn’t do so good is Mutant, which – btw. – is not an F4 matrix but comes from the MXL2 algorithm.  It is possible that LELA’s sparse data structures are not that well equipped to deal with this rather dense matrix.

I am in the process of implementing the algorithm LELA uses in M4RI and will report updated timings here.

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