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

In this work, we present the M4RIE library which implements efficient algorithms for linear algebra with dense matrices over GF(2^e) for 2 <= 2 <= 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 Kronrods 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 GF(2^e) with 2 <= e <= 10.

F4/5

We describe an algorithm to compute Grobner bases which combines F4-style reduction with the F5 criteria. Both F4 and F5 originate in the work of Jean-Charles Faug`ere, who has successfully computed many Grobner bases that were previously considered intractable. Another description of a similar algorithm already exists in Gwenole Ars dissertation; unfortunately, this is only available in French, and although an implementation exists, it is not made available for study. We not only describe the algorithm, we also direct the reader to a study implementation for the free and open source Sage computer algebra system. We conclude with a short discussion of how the approach described here compares and contrasts with that of Ars dissertation.

Efficient Decomposition of Dense Matrices over GF(2)

In this work we describe an efficient implementation of a hierarchy of algorithms for the decomposition of dense matrices over the field with two elements (GF(2)). Matrix decomposition is an essential building block for solving dense systems of linear and non-linear equations and thus much research has been devoted to improve the asymptotic complexity of such algorithms. In this work we discuss an implementation of both well-known and improved algorithms in the M4RI library. The focus of our discussion is on a new variant of the M4RI algorithm - denoted MMPF in this work -- which allows for considerable performance gains in practice when compared to the previously fastest implementation. We provide performance figures on x86_64 CPUs to demonstrate the viability of our approach.

Efficient Multiplication of Dense Matrices over GF(2)

We describe an efficient implementation of a hierarchy of algorithms for multiplication of dense matrices over the field with two elements (GF(2)). In particular we present our implementation -- in the M4RI library -- of Strassen-Winograd matrix multiplication and the Method of the Four Russians multiplication (M4RM) and compare it against other available implementations. Good performance is demonstrated on on AMDs Opteron and particulary good performance on Intels Core 2 Duo. The open-source M4RI library is available stand-alone as well as part of the Sage mathematics software. In machine terms, addition in GF(2) is logical-XOR, and multiplication is logical-AND, thus a machine word of 64-bits allows one to operate on 64 elements of GF(2) in parallel: at most one CPU cycle for 64 parallel additions or multiplications. As such, element-wise operations over GF(2) are relatively cheap. In fact, in this paper, we conclude that the actual bottlenecks are memory reads and writes and issues of data locality. We present our empirical findings in relation to minimizing these and give an analysis thereof.