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.

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.