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In this work, we present the M4RIE library which implements efficientalgorithms 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 M4RIlibrary both directly (i.e., by calling it) and indirectly (i.e., by using itsconcepts). We provide an open-source GPLv2+ C library for efficient linearalgebra over GF(2^e) for e small. In this library we implemented an idea due toBradshaw and Boothby which reduces matrix multiplication over GF(p^k) to aseries of matrix multiplications over GF(p). Furthermore, we propose a cachingtechnique - Newton-John tables - to avoid finite field multiplications which isinspired by Kronrod's method ("M4RM") for matrix multiplication over GF(2).Using these two techniques we provide asymptotically fast triangular solvingwith matrices (TRSM) and PLE-based Gaussian elimination. As a result, we areable to significantly improve upon the state of the art in dense linear algebraover GF(2^e) with 2 <= e <= 10.

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