M4RI is a library for fast arithmetic with dense matrices over F2. The name M4RI comes from the first implemented algorithm: The “Method of the Four Russians” inversion algorithm published by Gregory Bard. This algorithm in turn is named after the “Method of the Four Russians” multiplication algorithm which is probably better referred to as Kronrod's method. M4RI is available under the General Public License Version 2 or later (GPLv2+).

Build Status

Main Features

  • basic arithmetic with dense matrices over F2 (addition, equality testing, stacking, augmenting, sub-matrices, randomisation);

  • asymptotically fast $O(n^{log_2 7})$ matrix multiplication via the "Method of the Four Russians" (M4RM) & Strassen-Winograd algorithm;

  • asymptotically fast $O(n^{log_2 7})$ PLE factorisation (Gaussian elimination, system solving, ...);

  • fast row echelon form computation and matrix inversion via the "Method of the Four Russians" (M4RI, $O(n^{3/log n})$);

  • asymptotically fast Triangular System solving with Matrices (upper left, lower left, upper right, lower right),

  • support for the x86/x86_64 SSE2 instruction set where available;

  • preliminary support for parallelisation on shared memory systems via OpenMP;

  • and support for Linux, Solaris, and OS X (GCC).

See Further Reading for implemented algorithms.


See Performance.

OpenMP Support

OpenMP support for parallel multiplication and elimination is enabled with the


configure switch.


If you downloaded M4RI as a compressed tarball from its website, installation instructions are contained in the file INSTALL. If you downloaded M4RI by cloning the mainline tree at

you need to first run the following command:

autoreconf --install

Then do the usual

make check

For details see the instructions in the file INSTALL. Note that if your system does not have autoreconf installed, you can download release tarballs from which can be built and installed without it.


To build the reference manual, ensure that you have Doxygen installed. The HTML version of the reference manual can be built as follows:

cd src/

The built documentation is contained under the doc subdirectory of m4ri/. Once the HTML version is built, you can build the PDF version as follows:

cd doc/latex/

The documentation is also available here.


At least the following people have contributed to the M4RI library.

  • Tim Abbott: Debian-isation & advice on correct libtool versioning;
  • Martin Albrecht: maintainer, release manager, peformance tuning (M4RM, M4RI, Strassen, PLE), initial M4RM implementation, parallelisation, PLE factorisation (MMPF algorithm);
  • Gregory Bard: initial author, M4RI algorithm and initial implementation;
  • Marco Bodrato: new Strassen-like sequence for matrix multiplication and squaring which improves performance for squaring;
  • Michael Brickenstein: PolyBoRi author, standard conformity contributions for ANSIC, test data, discussion/suggestion of performance improvements, fast vector-matrix products;
  • Alexander Dreyer: PolyBoRi author, standard conformity contributions for ANSIC;
  • Jean-Guillaume Dumas: linear system resolution;
  • William Hart: many performance improvements for matrix multiplication and in general;
  • David Harvey: parallel parity function used in classical multiplication;
  • Jerry James: bug fixes, dealing with compiler warnings, Fedora Linux packaging;
  • David Kirkby: portability issues (Solaris, HP Unix);
  • Clément Pernet: PLE factorisation, triangular system solving (TRSM);
  • Wael Said: test cases, feedback;
  • Carlo Wood: bit-level optimisation (transpose, column swaps), refactoring, benchmark(et)ing framework, test code, build system clean-up;

We are grateful to William Stein for providing our hosting and general infrastructure in the past.

Citing M4RI

If you use our libraries in a non-trivial part of your research please consider citing them as follows:

    key          = "M4RI",
    author       = "Martin Albrecht and Gregory Bard",
    organization = "The M4RI~Team",
    title        = "{The M4RI Library -- Version 20121224}",
    year         = 2012,
    url          = "\url{}",

and cite the appropriate publications mentioned in Further Reading.


Please contact our mailinglist if there are bugs, questions, comments.


  • 2015/0417 Our hosting for at University of Washington. is discontinued and we’re moving everything over to A copy of the old website (except for large files) is available at

  • 2014/09/14 A new version of M4RI and M4RIE is available for download. The biggest change is that A->offset was dropped. Also, various small (multicore) performance improvements were implemented. The update for M4RIE is to maintain compatibility with M4RI. A few improvements were implemented for the mzd_poly module as well.

  • 2013/04/16 A new version of M4RI is available for download. A detailed changlog is available here for M4RI.

  • 2012/12/21 A new version of M4RI is available for download. A detailed changlog is available here for M4RI. See also this blog post for details.

  • 2012/06/13 New versions of both M4RI and M4RIE are available for download. A detailed changlog are available here for M4RI.

  • 2012/04/13 New versions of both M4RI and M4RIE are available for download. Detailed changlogs are available here for M4RI and here for M4RIE.

  • 2011/12/04 New versions of both M4RI and M4RIE are available for download. The highlight of this version for M4RI is support for reading and writing 1-bit PNG images. The highlight of this release of M4RIE is much improved performance for $4 < e \leq 8$. Detailed changlogs are available here for M4RI and here for M4RIE.

  • 2011/11/30 A technical report by Martin R. Albrecht is available describing the M4RIE library. In particular, Newton-John tables are introduced and our implementation of Karatsuba based matrix-matrix multiplication is described:

The M4RIE library for dense linear algebra over small fields with even characteristic

Abstract: In this work, we present the M4RIE library which implements efficient algorithms for linear algebra with dense matrices over GF(2^e) for 2 ≤ e ≤ 10. As the name of the library indicates, it makes heavy use of the M4RI library both directly (i.e., by calling it) and indirectly (i.e., by using its concepts). We provide an open-source GPLv2+ C library for efficient linear algebra over GF(2^e) for e small. In this library we implemented an idea due to Bradshaw and Boothby which reduces matrix multiplication over GF(p^k) to a series of matrix multiplications over GF(p). Furthermore, we propose a caching technique - Newton-John tables - to avoid finite field multiplications which is inspired by Kronrod's method ("M4RM") for matrix multiplication over GF(2). Using these two techniques we provide asymptotically fast triangular solving with matrices (TRSM) and PLE-based Gaussian elimination. As a result, we are able to significantly improve upon the state of the art in dense linear algebra over $F(2^e) with 2 ≤ e ≤ 10.

  • 2011/11/29 A technical report by Martin R. Albrecht, Gregory Bard and Clément Pernet is available describing the Gaussian elimination machinery (PLE decomposition) in the M4RI library:

Efficient Dense Gaussian Elimination over the Finite Field with Two Elements.

Abstract: In this work we describe an efficient implementation of a hierarchy of algorithms for Gaussian elimination upon dense matrices over the field with two elements. We discuss both well-known and new algorithms as well as our implementations in the M4RI library, which has been adopted into Sage. The focus of our discussion is a block iterative algorithm for PLE decomposition which is inspired by the M4RI algorithm. The implementation presented in this work provides considerable performance gains in practice when compared to the previously fastest implementation. We provide performance figures on x86_64 CPUs to demonstrate the alacrity of our approach.

  • 2011/10/10 A new release of M4RI is available for download. See the release notes for the list of changes. Also, a new release of M4RIE is also available for download. See the release notes for the list of changes.

  • 2011/07/14 A new release of M4RI is available for download. See the release notes for the list of changes. Also, a new release of M4RIE is also available for download. M4RIE now relies on M4RI for cache size and other hardware feature detection.

  • 2011/06/10 A new release of M4RI is available for download. This version fixes various issues when M4RI is built with OpenMP enabled.

  • 2011/06/01 A new release of M4RI is available for download. See the release notes for the list of changes. Also, a new release of M4RIE is also available for download. The only changes to M4RIE are to ensure compatibility with M4RI version 20110601 and up.

  • 2011/04/13 We now have a mailinglist.

  • 2010/08/14 A new release of M4RI is available for download. The main changes are improved automatic cache size detection and some clean ups necessary for M4RIE. A first official release of M4RIE is also available for download.

  • 2010/07/13 A new release is available for download. See the release notes for details.

  • 2009/11/04 A new release is available for download. See the release notes for details.

  • 2009/04/09 A new release is available for download. It heavily breaks backward compatibility but supports much bigger matrices than before. See the release notes for details.

  • 2009/01/05 A new release is available for download. It contains new features, performance enhancements and bug fixes. Release notes are available in the wiki.

  • 2008/11/12 A paper describing our matrix multiplication implementation is available as pre-print on the ArXiv. Also, M4RI is being packaged for Fedora Core. Finally, we updated the peformance data for GAP and Magma on the Core 2 Duo with improved timings.

  • 2008/10/28 A new release is available for download. It contains mainly bugfixes but starting with this release triangular solving with matrices (TRSM) is fully supported. Also LUP factorisation (i.e. on full rank matrices) seems to be working now but it is not optimised at all.

  • 2008/10/22 The slides for the Sage Days 10 talk about matrix multiplication in the M4RI library are available online.

  • 2008/09/22 A new release is available. It is identical to the version of M4RI shipped with Sage 3.1.2 and contains many build fixes for a wide range of platforms. Sage (and thus M4RI) supportes x86 Linux, x86_64 Linux, ia64 Linux, x86 OSX and ppc OSX. M4RI also supports Windows and Solaris 10.

  • 2008/08/26 This release is a pure bugfix release. Before this bugfix, if the input matrices were very non-square either wrong results or SIGSEGVs could be observed.

  • 2008/08/21 A new release is available. This release contains Clément Pernet's latest LQUP and TRSM development code. LQUP still lacks a basecase but TRSM should be fairly complete. No attempts were made so far to optimise things. Furthermore, this release contains an improved strategy for choosing $k$ in M4RM which improves performance on the Core2Duo.

  • 2008/08/17 A new release is available. This release adds a simple memory manager for systems with slow malloc/free syscalls. Also, the initialisation (m4ri_init) and finalisation (m4ri_fini) routines are now called automatically when the library is loaded/unloaded. This is tested with GCC and SunCC but not with MSVC. Matrix elimination got slightly faster across plattforms, multi-core support was extended to elimination and improved for multiplication. The README contains instruction how to enable multi-core support. This release does not contain Clément Pernet's latest LQUP patch.

  • 2008/06/24 A new release is available. This release uses the libtool -release mechanism to ensure binary (in)compatibility between releases since - again - the API changed: since the project is quite young do not expect the API to be stable anytime soon. Also the new version attempts to detect the L1 and L2 cache sizes and uses a Strassen-Winograd cutoff by default such that both source matrices fit in L2 (this is not optimal but a good compromise). This new version has some scratch/experimental code which is the beginning of asymptotically fast LQUP factorisation. Finally, elimination got slightly faster.

  • 2008/06/20 Thanks to Tim Abbott libM4RI is now in Debian/unstable.

  • 2008/06/13 It turns out our comparison with Magma on the Core2Duo was strongly biased, since we compared with a version of Magma that was optimised for AMD64 rather than Intel64. The correct times are given now and we apologise for this mix-up.

  • 2008/06/03 This release is a small bugfix release. Matrices are now printed correctly and a bug in mzd_gauss_delayed was reported and fixed by Wael Said and Mohammed Saied.

  • 2008/06/01 This release greatly improves the performance of M4RI: the reduction of a given matrix to (reduced) row echelon form. The speed-up over the last release can be as much as ten, we will provide performance data for this in the near future. However, the new implementation still isn't asymptotically fast. Also mzd_transpose is much faster now due to improved data locality.

  • 2008/05/21 Today's release fixes a severe bug found by Bill Hart, disables SSE2 on all CPUs except those manufactured by Intel (for performance reasons), improves performance on the Core2Duo and introduces a configure switch to enable OpenMP support.

  • 2008/05/20 A new release is available with massive speed improvements for matrix multiplication. These improvements were discussed and tested in this thread on the sage-devel mailing list. This release has also experimental and preliminary support for OpenMP. To activate it compile with GCC 4.2 and CFLAGS="-fopenmp -DHAVE_OPENMP“ Note however, that this release is still a developer preview since some automatic tuning is still not implemented, the performance on the Opteron isn't acceptable yet, and the parallel implementation is naive.

  • 2008/05/16 Release early, release often. This release fixes the unconditional use of _mm_free even when it is not available.

  • 2008/05/15 A new minor release is available which improves performance on Opterons. Also, the website has moved to