## libFES : Fast Exhaustive Search for Polynomial Systems over F2

Charles Bouillaguet set up a nice shiny website for libFES the library for exhaustive search on polynomial systems over $\mathbb{F}_2$. The library has a Sage interface, so it’s easy to get started. It’s also integrated in Charles’ upcoming one-stop boolean system solving patch.

He calls his benchmarketing “bragging rights” … and boy has he earned those rights! Check it out, libFES is fast!

## Faugère-Lachartre implementation for linear algebra for Gröbner bases

Fayssal’s code which implements the Faugère-Lachartre approach to linear algebra for Gröbner bases is available on Github now. Fayssal did a Master’s project on linear algebra for Gröbner bases in the team of Jean-Charles Faugère.

## Linear Algebra for Gröbner Bases over GF(2): M4RI

Two days ago I wrote about LELA’s implementation of Gaussian elimination for Gröbner basis computations over $\mathbb{F}_2$. Yesterday, I implemented LELA’s algorithm (which is from Faugere & Lachartre paper) in M4RI. Continue reading “Linear Algebra for Gröbner Bases over GF(2): M4RI”

## 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.

## Report: Workshop on Efficient Linear Algebra for Gröbner Basis Computations

As you may know, today is the last day of the wokshop on Efficient Linear Algebra for Gröbner Basis Computations that Christian Eder, Burcin Eröcal, Alexander Dreyer and I organised.

I have to say that I am quite pleased with how the workshop played out. We planned the whole thing to be hands on: people were strongly encouraged to work on projects, i.e., to write code preferably together, in addition to attending talks. Those who attended a Sage Days workshop in the past, will know what workshop format I am referring to. Continue reading “Report: Workshop on Efficient Linear Algebra for Gröbner Basis Computations”

## Challenge matrices

Now, that we have a decent PNG reader/writer in M4RI, it’s much easier to get some challenge matrices out of the library. Below, I list and link a few such matrices as they appear during Gröbner basis computations.

 file matrix dimensions density PLE M4RI GB HFE 25 matrix 5 (5.1M) 12307 x 13508 0.07600 1.03 0.59 0.81 HFE 30 matrix 5 (16M) 19907 x 29323 0.06731 4.79 2.70 4.76 HFE 35 matrix 5 (37M) 29969 x 55800 0.05949 19.33 9.28 19.51 Mutant matrix (39M) 26075 x 26407 0.18497 5.71 3.98 2.10 random n=24, m=26 matrix 3 (30M) 37587 x 38483 0.03832 20.69 21.08 19.36 random n=24_ m=26 matrix 4 (24M) 37576 x 32288 0.04073 18.65 28.44 17.05 SR(2,2,2,4) compressed, matrix 2 (328K) 5640 x 14297 0.00333 0.40 0.29 0.18 SR(2,2,2,4) compressed, matrix 4 (2.4M) 13665 x 17394 0.01376 2.18 3.04 2.04 SR(2,2,2,4) compressed, matrix 5 (2.8M) 11606 x 16282 0.03532 1.94 4.46 1.59 SR(2,2,2,4) matrix 6 (1.4M) 13067 x 17511 0.00892 1.90 2.09 1.38 SR(2,2,2,4) matrix 7 (1.7M) 12058 x 16662 0.01536 1.53 1.93 1.66 SR(2,2,2,4) matrix 9 (36M) 115834 x 118589 0.00376 528.21 578.54 522.98

The first three rows are from GB computations for the hidden field equations cryptosystem (those matrices were provided by Michael Brickenstein). The “mutant” row is a matrix as it appears during a run of the MXL2 algorithm on a random system (I believe). It was contributed by Wael Said. The rows “random n=25,m=26” are matrices as they appear during a GB computation with PolyBoRi for a random system of equations in 24 variables and 26 equations. The remaining rows are matrices from PolyBoRi computations on small scale AES instances. Those rows which have “compressed” in their description correspond to systems where “linear variables” were eliminate before running the Gröbner basis algorithm.

The last three columns give running times (quite rough ones!) for computing an echelon form (not reduced) using (a) the M4RI algorithm, (b) PLE decomposition and (c) a first implementation of the TRSM for trivial pivots trick. As you can see, currently it’s not straight-forward to pick which strategy to use to eliminate matrices appearing during Gröbner basis computations: the best algorithm to pick varies between different problems and the differences can be dramatic.