Efficient Decomposition of Dense Matrices over GF(2)

TL;DR

This paper presents an efficient implementation of matrix decomposition algorithms over GF(2), introducing a new variant called MMPF that significantly improves practical performance on x86_64 CPUs.

Contribution

The paper introduces the MMPF variant of the M4RI algorithm, enhancing practical efficiency for dense matrix decomposition over GF(2).

Findings

MMPF outperforms previous implementations in practice.

Performance benchmarks on x86_64 CPUs demonstrate viability.

The implementation improves the efficiency of solving linear systems over GF(2).

Abstract

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.